text:
\section{Introduction}

In this work, we consider fractional integrals and fractional derivatives with respect to another function $\psi$ (see \cite{Samko}).
To fix notation, in the following $\alpha>0$ is a real, $\psi\in C^1[a,b]$ is an increasing function, such that $\psi'(t)\not=0$, for all $t\in [a,b]$, and
$x:[a,b]\to\mathbb R$ is a function with some assumptions, so that the fractional operators that we deal with are well defined.
The left fractional integral of $x$, with respect to $\psi$ of order $\alpha$, is defined as
$${I_{a+}^{\a,\p}} x(t):=\frac{1}{\Gamma(\alpha)}\int_a^t \psi'(\tau)(\psi(t)-\psi(\tau))^{\alpha-1}x(\tau)\,d\tau,$$
and the right fractional integral of $x$ is
$${I_{b-}^{\a,\p}} x(t):=\frac{1}{\Gamma(\alpha)}\int_t^b \psi'(\tau)(\psi(\tau)-\psi(t))^{\alpha-1}x(\tau)\,d\tau.$$
Considering special cases for the kernel, that is, for the function $\psi$, we recover e.g. the Riemann--Liouville, the Hadamard and the
Erd\'{e}lyi--Kober integrals.
For fractional derivatives of Riemann--Liouville type, the left and right fractional derivatives of $x$ are defined as
$${D_{a+}^{\a,\p}} x(t):= \left(\frac{1}{\psi'(t)}\frac{d}{dt}\right)^n {I_{a+}^{n-\alpha,\psi}}x(t)$$
and
$${D_{b-}^{\a,\p}} x(t):=  \left(-\frac{1}{\psi'(t)}\frac{d}{dt}\right)^n {I_{b-}^{n-\alpha,\psi}}x(t)$$
respectively, where $n=[\alpha]+1$.
In this paper, we deal mainly with a Caputo-type fractional derivative. The concept is similar to the Riemann--Liouville derivative, but the
order of the dual integral/derivative is switched (see \cite{AlmeidaN}).

\begin{definition} Let $x\in C^n[a,b]$ be a function. The left Caputo fractional derivative of $x$ of order $\alpha$ with respect to $\psi$ is given by
$${^CD_{a+}^{\a,\p}} x(t):=   {I_{a+}^{n-\alpha,\psi}}\left(\frac{1}{\psi'(t)}\frac{d}{dt}\right)^nx(t),$$
while the right Caputo fractional derivative of $x$  is given by
$${^CD_{b-}^{\a,\p}} x(t):=   {I_{b-}^{n-\alpha,\psi}}\left(-\frac{1}{\psi'(t)}\frac{d}{dt}\right)^nx(t),$$
where
$$n=[\alpha]+1 \, \mbox{ for } \, \alpha\notin\mathbb N, \quad n=\alpha\, \mbox{ for } \, \alpha\in\mathbb N.$$
\end{definition}
It results that, if $\alpha=m$ is an integer, then
$${^CD_{a+}^{\a,\p}} x(t)=\left(\frac{1}{\psi'(t)}\frac{d}{dt}\right)^mx(t) \quad \mbox{and} \quad {^CD_{b-}^{\a,\p}} x(t)=\left(-\frac{1}{\psi'(t)}\frac{d}{dt}\right)^mx(t).$$
On the other hand, if $\alpha\in\mathbb R^+\setminus\mathbb N$, then
$${^CD_{a+}^{\a,\p}} x(t)=\frac{1}{\Gamma(n-\alpha)}\int_a^t \psi'(\tau)(\psi(t)-\psi(\tau))^{n-\alpha-1}\left(\frac{1}{\psi'(\tau)}\frac{d}{d\tau}\right)^nx(\tau)\,d\tau$$
and
$${^CD_{b-}^{\a,\p}} x(t)=\frac{1}{\Gamma(n-\alpha)}\int_t^b \psi'(\tau)(\psi(\tau)-\psi(t))^{n-\alpha-1} \left(-\frac{1}{\psi'(\tau)}\frac{d}{d\tau}\right)^nx(\tau)\,d\tau.$$
Since we are interested in a generalization of the ordinary derivatives, we will consider this second case only.
For example, we have the following formulas. Given $n<\beta\in\mathbb R$,
$${^CD_{a+}^{\a,\p}} (\psi(t)-\psi(a))^{\beta-1}=\frac{\Gamma(\beta)}{\Gamma(\beta-\alpha)}(\psi(t)-\psi(a))^{\beta-\alpha-1}$$
and
$${^CD_{b-}^{\a,\p}} (\psi(b)-\psi(t))^{\beta-1}=\frac{\Gamma(\beta)}{\Gamma(\beta-\alpha)}(\psi(b)-\psi(t))^{\beta-\alpha-1}.$$
Also, given $\lambda\in\mathbb R$, we have
$${^CD_{a+}^{\a,\p}} E_\alpha(\lambda(\psi(t)-\psi(a))^\alpha)=\lambda E_\alpha(\lambda(\psi(t)-\psi(a))^\alpha)$$
 and
 $${^CD_{b-}^{\a,\p}} E_\alpha(\lambda (\psi(b)-\psi(t))^\alpha)=\lambda E_\alpha(\lambda (\psi(b)-\psi(t))^\alpha).$$
There exists a relation between the fractional integral and the fractional derivative operators. In a certain sense, they are the inverse
operation of each other. In fact, we have that
$${I_{a+}^{\a,\p}} {^CD_{a+}^{\a,\p}} x(t)=x(t)-\sum_{k=0}^{n-1}\frac{ \left(\frac{1}{\psi'(t)}\frac{d}{dt}\right)^kx(a)}{k!}(\psi(t)-\psi(a))^k$$
and
$${I_{b-}^{\a,\p}} {^CD_{b-}^{\a,\p}} x(t)=x(t)-\sum_{k=0}^{n-1}\frac{ \left(-\frac{1}{\psi'(t)}\frac{d}{dt}\right)^kx(b)}{k!}(\psi(b)-\psi(t))^k.$$
For the converse, the relation is the following:
$${^CD_{a+}^{\a,\p}} {I_{a+}^{\a,\p}} x(t)=x(t)\quad \mbox{and} \quad {^CD_{b-}^{\a,\p}} {I_{b-}^{\a,\p}} x(t)=x(t).$$
One crucial formula, when dealing with variational problems, is a form of integration by parts, with respect to the fractional operators.
For these Caputo-type fractional derivatives, they are as follows.

\begin{theorem}\label{integrationparts} Given $x \in C[a,b]$ and $y\in C^n[a,b]$, the following holds:
\begin{align*}\int_a^b x(t)  {^CD_{a+}^{\a,\p}} y(t)\,dt &= \int_a^b {D_{b-}^{\a,\p}}  \left(\frac{x(t)}{\psi'(t)}\right) y(t)\psi'(t)\,dt\\
&\quad+\left[\sum_{k=0}^{n-1}\left(-\frac{1}{\psi'(t)}\frac{d}{dt}\right)^k {I_{b-}^{n-\alpha,\psi}}\left(\frac{x(t)}{\psi'(t)}\right)\cdot
\left(\frac{1}{\psi'(t)}\frac{d}{dt}\right)^{n-k-1}y(t)\right]_{t=a}^{t=b}\end{align*}
and
\begin{align*}\int_a^b x(t)  {^CD_{b-}^{\a,\p}} y(t)\,dt &= \int_a^b {D_{a+}^{\a,\p}} \left(\frac{x(t)}{\psi'(t)}\right) y(t)\psi'(t)\,dt\\
&\quad +\left[\sum_{k=0}^{n-1}(-1)^{n-k}\left(\frac{1}{\psi'(t)}\frac{d}{dt}\right)^k {I_{a+}^{n-\alpha,\psi}} \left(\frac{x(t)}{\psi'(t)}\right) \cdot
\left(\frac{1}{\psi'(t)}\frac{d}{dt}\right)^{n-k-1}y(t)\right]_{t=a}^{t=b}.\end{align*}
\end{theorem}

The paper is organized in the following way. In Section \ref{sec:main}, we present the main problem, and in Theorem \ref{teo:main} we prove an Euler--Lagrange type equation. In Section \ref{extension} we extend this result, by considering functionals where the lower bound of integration is greater than the lower bound of the fractional derivative. Next, in Section \ref{sec:iso}, we consider the variational problem subject to an integral constraint, in what is known as an isoperimetric problem, and in Section \ref{sec:legendre}, we deduce a second-order necessary condition that allows us to verify if the extremals are minimizers or not. In Sections \ref{sec:infinite} and \ref{sec:delay}, we consider the infinite horizon problem and the case where the Lagrange function has a delay, respectively. In Section \ref{sec:high}, we consider high-order derivatives in the functional, and derive the respective high-order Euler--Lagrange equation, and in Section \ref{sec:optimal}, we find a necessary condition that allows us to find the best fractional order to provide a minimum to the functional. Finally, in Section \ref{sec:suff}, we prove a sufficient condition  that guarantees the solutions of the Euler--Lagrange equations are almost  minimizers.

\section{Main results}

In this section, we study several variational problems, where the dynamic of the trajectories is described by a Caputo type fractional derivative.
We consider the initial point to be fixed, $x(a)=x_a$ ($x_a\in\mathbb R$), and the terminal point $T>a$ to be free, and thus it is also a variable of the problem.
We are interested in finding the optimal pair $(x,T)$ for the objective functionals.

\subsection{Fundamental problem}\label{sec:main}

The most important result in the calculus of variations is the so called Euler--Lagrange equation, which is a first order necessary
condition every extremizer of the functional must satisfy. For functionals depending on fractional operators, we find in the literature
numerous works already done for different kinds of fractional derivatives and initial/terminal conditions. Some examples are for the Riemann--Liouville
derivative \cite{AGRA1,Atanackovic,Baleanu2}, for the Caputo derivative \cite{AGRA2,Agnieszka1,withTatiana:Basia}, for the Riesz derivative
\cite{AGRA3,Almeida1,Baleanu1}.
We mention the recent books \cite{book2,book1}, where analytical and numerical methods are explained, respectively.

Our fractional variation problem with free terminal time is described in the following way. Let $L:[a,b]\times\mathbb R^2\to\mathbb R$
be a continuous function, such that there exist and are continuous the functions $\partial_2 L$ and $\partial_3 L$.
Define the functional
\begin{equation}\label{funct1}
J(x,T):=\int_a^T L(t,x(t),{^CD_{a+}^{\a,\p}} x(t))\,dt, \quad (x,T)\in\Omega\times[a,b],
\end{equation}
where $\Omega$ is the set
$$\Omega:=\{x\in C^1[a,b]: x(a)\, \mbox{ is a fixed real}\},$$
which we endow with the norm
$$\|x\|_\Omega:=\max_{t\in[a,b]}|x(t)|+\max_{t\in[a,b]}\left|{^CD_{a+}^{\a,\p}} x(t)\right|.$$
We say that $J$ assumes its minimum value at $(x^*,T^*)$ in $\Omega\times[a,b]$, relative to the norm
$$\|(x,t)\|:=\|x\|_\Omega+|t|,$$
provided that
$$\exists \epsilon>0 \, \forall (x,t)\in \Omega\times[a,b] \,: \, \|(x^*,T^*)-(x,t)\|<\epsilon \Rightarrow J(x^*,T^*) \leq J(x,t).$$
In this case, we say that  $(x^*,T^*)$ is a local minimum for $J$. An admissible variation for  $(x^*,T^*)$ is a pair
$(x^*+\epsilon v,T^*+\epsilon \triangle T)$, where $v\in C^1[a,b]$ and $v(a)=0$, $\epsilon,\triangle T \in\mathbb R$ and $|\epsilon| \ll 1$.
The next result provides a necessary condition that every local minimum for $J$ must  satisfy. In order to simplify the notation, we define $[x]$ as
$$[x](t):=(t,x(t),{^CD_{a+}^{\a,\p}} x(t)), \quad (x,t)\in \Omega\times[a,b].$$

\begin{theorem}\label{teo:main} Suppose that $(x^*,T^*)$ is a local minimum for $J$ as in \eqref{funct1} on the space $\Omega\times[a,b]$.
If there exists and is continuous the function $t\mapsto {D_{T^*-}^{\alpha,\psi}}\left(\partial_3 L[x^*](t)/\psi'(t)\right)$ on $[a,T^*]$, then
\begin{equation}\label{ELeq}\partial_2 L[x^*](t)+{D_{T^*-}^{\alpha,\psi}}\left(\frac{\partial_3 L[x^*](t)}{\psi'(t)}\right)\psi'(t)=0\end{equation}
for each $t\in[a,T^*]$, and at $t=T^*$, the following transversality conditions are satisfied:
\begin{equation}\label{TransCon}\left\{
\begin{array}{l}
{I_{T^*-}^{1-\alpha,\psi}}\left(\frac{\partial_3 L[x^*](t)}{\psi'(t)}\right)=0\\
L[x^*](t)=0.
\end{array}
\right.\end{equation}
\end{theorem}
\begin{proof} Consider an admissible variation of the optimal solution of the form  $(x^*+\epsilon v,T^*+\epsilon \triangle T)$.
If we define the function $j$ in a neighborhood of zero by the expression
$$j(\epsilon):=J(x^*+\epsilon v,T^*+\epsilon \triangle T),$$
we have that $j'(0)=0$. Differentiating $j$ at $\epsilon=0$, and using the integration by parts formula as in Theorem \ref{integrationparts}, we obtain
\begin{align*}
j'(0)&= \int_a^{T^*}\left[\partial_2 L[x^*](t) \cdot v(t)+\partial_3 L[x^*](t)  \cdot {^CD_{a+}^{\a,\p}} v(t)\right]dt+\triangle T\cdot L[x^*](T^*)\\
&= \int_a^{T^*}\left[\partial_2 L[x^*](t)+{D_{T^*-}^{\alpha,\psi}}\left(\frac{\partial_3 L[x^*](t)}{\psi'(t)}\right)\psi'(t)\right]v(t)\,dt\\
& \quad + \left[{I_{T^*-}^{1-\alpha,\psi}}\left(\frac{\partial_3 L[x^*](t)}{\psi'(t)}\right)\cdot v(t)\right]_{t=a}^{t=T^*} +\triangle T\cdot L[x^*](T^*).\\
\end{align*}
Since $v(a)=0$, if we consider $v(T^*)=0=\triangle T$, by the fundamental lemma of the calculus of variations (cf. \cite[Lemma 2.2.2]{Brunt}),
we conclude that for all $t\in[a,T^*]$, $x^*$ satisfies the condition
$$\partial_2 L[x^*](t)+{D_{T^*-}^{\alpha,\psi}}\left(\frac{\partial_3 L[x^*](t)}{\psi'(t)}\right)\psi'(t)=0.$$
Therefore, we have
$$\left[{I_{T^*-}^{1-\alpha,\psi}}\left(\frac{\partial_3 L[x^*](t)}{\psi'(t)}\right)\cdot v(t)\right]_{t=a}^{t=T^*} +\triangle T \cdot L[x^*](T^*).$$
Since $v(T^*)$ and $\triangle T$ are free, we obtain the two transversality conditions.
\end{proof}

Equations like \eqref{ELeq} are called Euler--Lagrange equations, and they provide a first-order necessary condition that all
minimizers of the problem must satisfy. Notice that, although the functional  \eqref{funct1} depends on a Caputo type fractional derivative,
the Euler--Lagrange equation \eqref{ELeq} involves a Riemann--Liouville fractional derivative. We can rewrite it in such a way that the fractional equation depends
on the Caputo derivative as well. Observe that, given a differentiable function $f$ and $\alpha\in(0,1)$, we have
\begin{align*}
{^CD_{T^*-}^{\alpha,\psi}}f(t)& = {D_{T^*-}^{\alpha,\psi}}(f(t)-f(T^*))\\
& = {D_{T^*-}^{\alpha,\psi}}f(t)-f(T^*)\cdot{D_{T^*-}^{\alpha,\psi}}1\\
& = {D_{T^*-}^{\alpha,\psi}}f(t)-\frac{f(T^*)}{\Gamma(1-\alpha)}(\psi(T^*)-\psi(t))^{-\alpha}.
\end{align*}
Using this new relation, the Euler--Lagrange equation is written in the form
$$\partial_2 L[x^*](t)+{^CD_{T^*-}^{\alpha,\psi}}\left(\frac{\partial_3 L[x^*](t)}{\psi'(t)}\right)\psi'(t)+
\frac{\partial_3 L[x^*](T^*)}{\psi'(T^*)\Gamma(1-\alpha)}(\psi(T^*)-\psi(t))^{-\alpha}\psi'(t)=0.$$
The variational problem involving several dependent variables is similar, and we omit the proof here.

\begin{theorem} Consider the functional
$$J(x_1,\ldots,x_m,T):=\int_a^T L[x](t)\,dt,$$
where $m\in\mathbb N$, the functions $x_i$ verify the two assumptions $x_i\in C^1[a,b]$ and $x_i(a)$ is a fixed real,  for all $i=1,\ldots,m$,
 the real $T$ belongs to the interval $[a,b]$, and
$$[x](t):=(t,x_1(t),\ldots,x_m(t),{^CD_{a+}^{\alpha_1,\psi}} x_1(t),\ldots,{^CD_{a+}^{\alpha_m,\psi}} x_m(t)), \quad \alpha_1,\ldots,\alpha_m\in(0,1).$$
 Suppose that $(x_1^*,\ldots,x_m^*,T^*)$ is a local minimum for $J$, and that there exist and
are continuous the functions $t\mapsto {D_{T^*-}^{\alpha_i,\psi}}\left(\partial_{i+1+m} L[x^*](t)/\psi'(t)\right)$ on $[a,T^*]$, for all $i=1,\ldots,m$. Then,
$$\partial_{i+1} L[x^*](t)+{D_{T^*-}^{\alpha_i,\psi}}\left(\frac{\partial_{i+1+m} L[x^*](t)}{\psi'(t)}\right)\psi'(t)=0$$
for all $i=1,\ldots,m$ and for all $t\in[a,T^*]$, and at $t=T^*$, the following holds:
$$\left\{
\begin{array}{l}
{I_{T^*-}^{1-\alpha_i,\psi}}\left(\frac{\partial_{i+1+m} L[x^*](t)}{\psi'(t)}\right)=0, \quad i=1,\ldots,m\\
L[x^*](t)=0.
\end{array}
\right.$$
\end{theorem}

\subsection{An extension}\label{extension}

In the previous problem, the lower limits of the cost functional and of the fractional derivative were the same, at $t=a$. In this section we generalize it,
by considering a cost functional starting at a point $t=A>a$.

\begin{theorem} Consider the functional
$$J(x,T):=\int_A^T L[x](t)\,dt,$$
where $x\in C^1[a,b]$ ($x(a)$ and $x(A)$ may be fixed or not) and $A,T\in[a,b]$ with $T>A$. Assume that $(x^*,T^*)$ is a local minimum for $J$, and that there
 exist and are continuous the functions $t\mapsto {D_{T^*-}^{\alpha,\psi}}\left(\partial_3 L[x^*](t)/\psi'(t)\right)$ on $[a,T^*]$ and
 $t\mapsto {D_{A-}^{\alpha,\psi}}\left(\partial_3 L[x^*](t)/\psi'(t)\right)$ on $[a,A]$. Then,
 $${D_{T^*-}^{\alpha,\psi}}\left(\frac{\partial_3 L[x^*](t)}{\psi'(t)}\right)-{D_{A-}^{\alpha,\psi}}\left(\frac{\partial_3 L[x^*](t)}{\psi'(t)}\right)=0,$$
for each $t\in[a,A]$, and
$$\partial_2 L[x^*](t)+{D_{T^*-}^{\alpha,\psi}}\left(\frac{\partial_3 L[x^*](t)}{\psi'(t)}\right)\psi'(t)=0$$
for each $t\in[A,T^*]$. At $t=T^*$, the following holds:
$$\left\{
\begin{array}{l}
{I_{T^*-}^{1-\alpha,\psi}}\left(\frac{\partial_3 L[x^*](t)}{\psi'(t)}\right)=0\\
L[x^*](t)=0.
\end{array}
\right.$$
Moreover, if $x(a)$ is free, then at $t=a$:
$${I_{T^*-}^{1-\alpha,\psi}}\left(\frac{\partial_3 L[x^*](t)}{\psi'(t)}\right)-{I_{A-}^{1-\alpha,\psi}}\left(\frac{\partial_3 L[x^*](t)}{\psi'(t)}\right)=0,$$
and if $x(A)$ is free, then at $t=A$:
$${I_{A-}^{1-\alpha,\psi}}\left(\frac{\partial_3 L[x^*](t)}{\psi'(t)}\right)=0.$$
\end{theorem}
\begin{proof} The first variation of the functional at an extremum must vanish, and so we conclude that
\begin{align*}
0&= \int_A^{T^*}\left[\partial_2 L[x^*](t) \cdot v(t)+\partial_3 L[x^*](t)  \cdot {^CD_{a+}^{\a,\p}} v(t)\right]dt+\triangle T\cdot L[x^*](T^*)\\
&= \int_a^{T^*}\left[\partial_2 L[x^*](t) \cdot v(t)+\partial_3 L[x^*](t)  \cdot {^CD_{a+}^{\a,\p}} v(t)\right]dt\\
& \quad -\int_a^A\left[\partial_2 L[x^*](t) \cdot v(t)+\partial_3 L[x^*](t)  \cdot {^CD_{a+}^{\a,\p}} v(t)\right]dt+\triangle T\cdot L[x^*](T^*)\\
&= \int_a^{T^*}\left[\partial_2 L[x^*](t)+{D_{T^*-}^{\alpha,\psi}}\left(\frac{\partial_3 L[x^*](t)}{\psi'(t)}\right)\psi'(t)\right]v(t)\,dt
 + \left[{I_{T^*-}^{1-\alpha,\psi}}\left(\frac{\partial_3 L[x^*](t)}{\psi'(t)}\right)\cdot v(t)\right]_{t=a}^{t=T^*}\\
&\quad -\int_a^A\left[\partial_2 L[x^*](t)+{D_{A-}^{\alpha,\psi}}\left(\frac{\partial_3 L[x^*](t)}{\psi'(t)}\right)\psi'(t)\right]v(t)\,dt
 - \left[{I_{A-}^{1-\alpha,\psi}}\left(\frac{\partial_3 L[x^*](t)}{\psi'(t)}\right)\cdot v(t)\right]_{t=a}^{t=A}\\
& \quad +\triangle T\cdot L[x^*](T^*)\\
&=\int_a^A\left[{D_{T^*-}^{\alpha,\psi}}\left(\frac{\partial_3 L[x^*](t)}{\psi'(t)}\right)-{D_{A-}^{\alpha,\psi}}\left(\frac{\partial_3 L[x^*](t)}{\psi'(t)}\right)\right]
\psi'(t)\cdot v(t)\,dt \\
& \quad +\int_A^{T^*}\left[\partial_2 L[x^*](t)+{D_{T^*-}^{\alpha,\psi}}\left(\frac{\partial_3 L[x^*](t)}{\psi'(t)}\right)\psi'(t)\right]\cdot v(t)\,dt\\
&\quad + \left[{I_{T^*-}^{1-\alpha,\psi}}\left(\frac{\partial_3 L[x^*](t)}{\psi'(t)}\right)\cdot v(t)\right]_{t=a}^{t=T^*}
- \left[{I_{A-}^{1-\alpha,\psi}}\left(\frac{\partial_3 L[x^*](t)}{\psi'(t)}\right)\cdot v(t)\right]_{t=a}^{t=A}\\
& \quad +\triangle T\cdot L[x^*](T^*).
\end{align*}
By the arbitrariness of $v$ and $\triangle T$, we obtain the necessary optimality conditions.
\end{proof}

\subsection{Isoperimetric problem}\label{sec:iso}

We formulate now the variational problem when in presence of an integral constraint. We refer to \cite{Almeida2,Odziehjewicz}, where similar
problems were solved involving fractional derivatives. This kind of problems are known in the literature as isoperimetric problems. The most
ancient problem of this type goes back to the Ancient Greece, with the question of finding out which of all closed planar curves of the same length would enclose the greatest area. Nowadays, an isoperimetric problem is a variational problem, restricted to a subclass of functions satisfying
a side condition of the form
$$\int_a^b M(t,x(t),x'(t))\,dt=\mbox{constant.}$$
Here, we replace the ordinary derivative by a fractional derivative, and since the terminal time is free, the integral value is not a constant,
but a function depending on the terminal time. Let $M:[a,b]\times\mathbb R^2\to\mathbb R$ be a continuous function, such that there exist and are
continuous the functions $\partial_2 M$ and $\partial_3 M$.

\begin{theorem}\label{teo:isop} Suppose that $(x^*,T^*)$ is a local minimum for $J$ as in \eqref{funct1} on the space $\Omega\times[a,b]$,
subject to the integral constraint
$$G(x,T):=\int_a^TM[x](t)\,dt=\Phi(T),$$
where $\Phi\in C^1[a,b]$. Suppose that $(x^*,T^*)$ is not a solution of the equation
\begin{equation}\label{aux1}\partial_2 M[x](t)+{D_{T^*-}^{\alpha,\psi}}\left(\frac{\partial_3 M[x](t)}{\psi'(t)}\right)\psi'(t)=0,
\quad \forall t \in[a,T^*],\end{equation}
and that there exist and are continuous the functions $t\mapsto {D_{T^*-}^{\alpha,\psi}}\left(\partial_{3} L[x^*](t)/\psi'(t)\right)$ and $t\mapsto {D_{T^*-}^{\alpha,\psi}}
\left(\partial_{3} M[x^*](t)/\psi'(t)\right)$ on $[a,T^*]$.
Then, there exists a real constant $\lambda$, such that if we define the augmented function $F:=L+\lambda M$, then $(x^*,T^*)$ satisfies the equation
$$\partial_2 F[x](t)+{D_{T^*-}^{\alpha,\psi}}\left(\frac{\partial_3 F[x](t)}{\psi'(t)}\right)\psi'(t)=0, \quad \forall t \in[a,T^*],$$
and the system
$$\left\{
\begin{array}{l}
{I_{T^*-}^{1-\alpha,\psi}}\left(\frac{\partial_3 F[x](t)}{\psi'(t)}\right)=0\\
F[x](t)=\lambda \Phi'(t)
\end{array}\right. \quad \mbox{at} \, \, t=T^*.$$
\end{theorem}
\begin{proof} Consider admissible variations of two parameters of kind  $(x^*+\epsilon_1 v_1+\epsilon_2v_2,T^*+\epsilon_2 \triangle T)$, where
$v_1,v_2\in C^1[a,b]$ with $v_1(a)=v_2(a)=0$, and $\epsilon_1,\epsilon_2,\triangle T\in\mathbb R$ with $|\epsilon_1|,|\epsilon_2|\ll1$. Define the two functions:
\begin{align*}j(\epsilon_1,\epsilon_2):=&J(x^*+\epsilon_1 v_1+\epsilon_2v_2,T^*+\epsilon_2 \triangle T)\\
g(\epsilon_1,\epsilon_2):=&G(x^*+\epsilon_1 v_1+\epsilon_2v_2,T^*+\epsilon_2 \triangle T)-\Phi(T^*+\epsilon_2 \triangle T).
\end{align*}
Since
\begin{align*}\frac{\partial g}{\partial \epsilon_1}(0,0)&=\int_a^{T^*}\left[\partial_2 M[x^*](t)+{D_{T^*-}^{\alpha,\psi}}
\left(\frac{\partial_3 M[x^*](t)}{\psi'(t)}\right)\psi'(t)\right]v_1(t)\,dt\\
& \quad+ \left[{I_{T^*-}^{1-\alpha,\psi}}\left(\frac{\partial_3 M[x^*](t)}{\psi'(t)}\right)\cdot v_1(t)\right]_{t=a}^{t=T^*} ,\end{align*}
and $(x^*,T^*)$ is not a solution for Eq. \eqref{aux1}, we deduce that there exists a function $v_1\in C^1[a,b]$ such that
 $\partial g/\partial \epsilon_1(0,0)\not=0$. We can appeal to the implicit function theorem, which asserts that there exists a function
 $\epsilon_1(\cdot)$, defined on a neighborhood of zero, such that $g(\epsilon_1(\epsilon_2),\epsilon_2)=0$. Thus, there exists a subfamily
 of admissible variations satisfying the integral constraint. Attending that $j$ is minimum at $(0,0)$ subject to the constraint $g(\cdot,\cdot)=0$,
 and since $\nabla g (0,0)\not=(0,0)$, by the Lagrange multiplier rule, there exists a real number $\lambda$ such that
$$\nabla(j+\lambda g)(0,0)=(0,0).$$
In particular, $\partial (j+\lambda g)/\partial \epsilon_2(0,0)=0$. Repeating the calculations as done before, we arrive at the desired formulas.
\end{proof}

\subsection{Legendre condition}\label{sec:legendre}

We  now formulate a second-order necessary condition, usually called Legendre condition, which provides us with a necessary condition for minimization.
In \cite{Lazo}, by the first time, a Legendre type condition was obtained for fractional variational calculus. Here, we derive a similar condition to
a more general form of fractional derivative. Assume now that the Lagrange function $L$ is such that its second order partial derivatives
$\partial^2_{ij}L$, with $i,j\in\{2,3\}$, exist and are continuous.

\begin{theorem} Suppose that $(x^*,T^*)$ is a local minimum for $J$ as in \eqref{funct1} on the space $\Omega\times[a,b]$. Then for all $ t \in[a,T^*]$,
$$\partial^2_{33}L[x^*](t)\geq0. $$
\end{theorem}
\begin{proof} Let us consider variations over $x^*$ only, that is, we restrict to the case $\triangle T=0$. So, if we consider
$j(\epsilon):=J(x^*+\epsilon v,T^*)$, we have $j''(0)\geq0$, and so we conclude that
\begin{equation}\label{aux2}
\int_a^{T^*}\left[\partial^2_{22}L[x^*](t)\cdot v^2(t)+2\,\partial^2_{23}L[x^*](t)\cdot v(t){^CD_{a+}^{\a,\p}} v(t)+\partial^2_{33}L[x^*](t)\cdot
\left({^CD_{a+}^{\a,\p}} v(t)\right)^2\right]\,dt\geq0.
\end{equation}
Suppose that the Legendre condition is violated at some some $t_0\in[a,T^*]$:
$$\partial^2_{33}L[x^*](t_0)<0.$$
Then, there exists a subinterval $[c,d]\subseteq [a,T^*]$ and three real constants $C_1,C_2,C_3$ with $C_3<0$ such that
$$\partial^2_{22}L[x^*](t)\leq C_1, \quad \partial^2_{23}L[x^*](t)\leq C_2, \quad \partial^2_{33}L[x^*](t)\leq C_3, $$
for all $t \in[c,d]$. Define the function $h:[c,d]\to\mathbb R$ by the formula
\begin{align*}h(t)&:= (\alpha+2)(\psi(t)-\psi(c))^{\alpha+1}-2\frac{\alpha+4}{\psi(d)-\psi(c)}(\psi(t)-\psi(c))^{\alpha+2}\\
    & \quad +\frac{\alpha+10}{(\psi(d)-\psi(c))^2}(\psi(t)-\psi(c))^{\alpha+3}-\frac{4}{(\psi(d)-\psi(c))^3}(\psi(t)-\psi(c))^{\alpha+4}.
\end{align*}
Then, $h(c)=0=h(d)$ and $h'(c)=0=h'(d)$.
Once
\begin{align*}{^CD_{c+}^{\alpha,\psi}} h(t)&= \Gamma(\alpha+3)(\psi(t)-\psi(c))-\frac{(\alpha+4)\Gamma(\alpha+3)}{\psi(d)-\psi(c)}(\psi(t)-\psi(c))^{2}\\
    & \quad +\frac{(\alpha+10)\Gamma(\alpha+4)}{6(\psi(d)-\psi(c))^2}(\psi(t)-\psi(c))^{3}-\frac{\Gamma(\alpha+5)}{6(\psi(d)-\psi(c))^3}(\psi(t)-\psi(c))^{4},
\end{align*}
we also have ${^CD_{c+}^{\alpha,\psi}} h(c)=0={^CD_{c+}^{\alpha,\psi}} h(d)$. Besides this, for every $t\in[c,d]$,
\begin{align*}h(t)&\leq  (\alpha+2)(\psi(t)-\psi(c))^{\alpha+1}+\frac{\alpha+10}{(\psi(d)-\psi(c))^2}(\psi(t)-\psi(c))^{\alpha+3}\\
 & \leq (2\alpha+12) (\psi(d)-\psi(c))^{\alpha+1}\leq 14  (\psi(d)-\psi(c))^{\alpha+1}
\end{align*}
and
\begin{align*}{^CD_{c+}^{\alpha,\psi}} h(t)&\leq \Gamma(\alpha+3)(\psi(t)-\psi(c))+\frac{(\alpha+10)\Gamma(\alpha+4)}{6(\psi(d)-\psi(c))^2}(\psi(t)-\psi(c))^{3}\\
  & \leq \frac{\Gamma(\alpha+3)(\alpha^2+13\alpha+36)}{6}(\psi(d)-\psi(c))\leq 50(\psi(d)-\psi(c)).
\end{align*}
Define the function $v:[a,b]\to\mathbb R$ by the rule
$$v(t):=\left\{\begin{array}{ll}
h(t), & \quad \mbox{if } t \in [c,d]\\
0, & \quad \mbox{if } t \notin [c,d].\\
\end{array}\right.$$
By the properties of function $h$, we have that $v\in C^1[a,b]$, $v(a)=0$ and
$$ {^CD_{a+}^{\a,\p}} v(t):=\left\{\begin{array}{ll}
{^CD_{c+}^{\alpha,\psi}}v(t), & \quad \mbox{if } t \in [c,d]\\
0, & \quad \mbox{if } t \notin[c,d].\\
\end{array}\right.$$
Note that, for $t>d$, ${^CD_{a+}^{\a,\p}} v(t)= {^CD_{c+}^{\alpha,\psi}}h(d)=0$. Replacing this variation into Eq. \eqref{aux2}, we get
\begin{align*}0&\leq\int_a^{T^*}\left[\partial^2_{22}L[x^*](t)\cdot v^2(t)+2\,\partial^2_{23}L[x^*](t)\cdot v(t){^CD_{a+}^{\a,\p}} v(t)+\partial^2_{33}L[x^*](t)\cdot({^CD_{a+}^{\a,\p}} v(t))^2\right]\,dt\\
&\leq \int_a^{T^*}\left[ 14^2 C_1(\psi(d)-\psi(c))^{2\alpha+2}+2\cdot 14\cdot 50 C_2(\psi(d)-\psi(c))^{\alpha+2}+50^2C_3 (\psi(d)-\psi(c))^2 \right]\,dt\\
&=(\psi(d)-\psi(c))^2 (T^*-a)\left[ 196 C_1 (\psi(d)-\psi(c))^{2\alpha}+1400C_2(\psi(d)-\psi(c))^\alpha+2500C_3 \right]<0\end{align*}
if we assume that $|d-c|\ll1$, and thus we obtain a contradiction.
\end{proof}

\subsection{Infinite horizon problem}\label{sec:infinite}

We study now a new problem, important when we want to consider the effects at a long term.  This issue is especially pertinent when the
 planning horizon is assumed to be of infinite length.
The objective functional is given by an  improper integral, the initial state $x(a)$ is fixed and the terminal state (at infinity) is free, that is,
no constraints are imposed on the behaviour of the admissible trajectories at large times.
 This kind of problems are known as infinite horizon problems, where the objective functional is given by
\begin{equation}\label{funct2}
J(x):=\int_a^\infty L(t,x(t),{^CD_{a+}^{\a,\p}} x(t))\,dt, \quad x\in\Omega_\infty,
\end{equation}
where $\Omega_\infty$ is the set
$$\Omega_\infty:=\{x\in C^1[a,\infty): x(a)\, \mbox{ is a fixed real}\},$$
endowed with the norm
$$\|x\|_{\Omega_\infty}:=\sup_{t\in[a,\infty)}|x(t)|+\sup_{t\in[a,\infty)}\left|{^CD_{a+}^{\a,\p}} x(t)\right|.$$
We have to be careful when defining a minimal curve for functional \eqref{funct2}, since any admissible function for which the improper integral
 diverges to $-\infty$ would be a minimal path, according to the usual definition of minimum.
Here, we follow the one presented in \cite{brock}.
A curve $x^*$ in $\Omega_\infty$ is a local weakly minimal for $J$ as in \eqref{funct2} if there exists some $ \epsilon>0$ such that, for all
 $x\in \Omega_\infty$, if $\|x^*-x\|_{\Omega_\infty}<\epsilon$, then the lower limit
$$\lim_{T\to\infty}\inf_{T^* \geq T}\int_{a}^{T^*}[L[x^*](t)-L[x](t)] \, dt \leq 0.$$
For the following result, we will need some extra functions. Fixed two functions $x^*,v \in C^1[a,\infty)$, and given $|\epsilon|\ll 1$ and
$T^*\geq a$, define
\begin{align*}
 A(\epsilon, T^*)& := \int_a^{T^*} \frac{L[x^*+\epsilon v](t)- L[x^*](t)}{\epsilon}dt;\\
V(\epsilon, T)& := \inf_{T^* \geq T}\int_a^{T^*} [L[x^*+\epsilon v](t)-L[x^*](t)]dt;\\
W(\epsilon)&:= \lim_{T\to\infty} V(\epsilon,T).
\end{align*}

\begin{theorem} Let $x^*$ be a local weakly minimal for $J$ as in \eqref{funct2}. Suppose that:
\begin{enumerate}
\item $\displaystyle \lim_{\epsilon \to 0} \frac{V(\epsilon, T) }{\epsilon}$ exists for all $T$;
\item $\displaystyle \lim_{T\to\infty}\frac{V(\epsilon, T) }{\epsilon}$ exists uniformly for all $\epsilon$;
\item For every $T^*> a$ and $\epsilon\not=0$, there exists a sequence $\left(A(\epsilon, T^*_n)\right)_{n \in \mathbb{N}}$
such that
$$\lim_{n \to \infty} A(\epsilon, T^*_n)= \inf_{T^*\geq T} A(\epsilon, T^*)$$
uniformly for $\epsilon$.
\end{enumerate}
If there exists and is continuous the function $t\mapsto {D_{T^*-}^{\alpha,\psi}}\left(\partial_3 L[x^*](t)/\psi'(t)\right)$ on $[a,T^*]$, for all $T^*\geq t\geq a$,
 then
$$\partial_2 L[x^*](t)+{D_{T^*-}^{\alpha,\psi}}\left(\frac{\partial_3 L[x^*](t)}{\psi'(t)}\right)\psi'(t)=0,$$
for all $T^*\geq t\geq a$. Also, we have
$$\lim_{T\to\infty}\inf_{T^* \geq T}{I_{T^*-}^{1-\alpha,\psi}}\left(\frac{\partial_3 L[x^*](t)}{\psi'(t)}\right)=0 \quad \mbox{at} \, \, t=T^*.$$
\end{theorem}

\begin{proof} By the definition of minimum curve for the infinite horizon problem, we have that $W(\epsilon)\geq0$ in a neighborhood of zero,
and $W(0)=0$. Thus, $W'(0)=0$ and so we have the following:
\begin{align*}
0 &=\lim_{\epsilon\to 0} \frac{W(\epsilon)}{\epsilon}= \lim_{\epsilon\to 0}\lim_{T\to \infty}\frac{V(\epsilon, T) }{\epsilon}\\
&= \lim_{T\to\infty}\lim_{\epsilon\to 0} \frac{V(\epsilon, T) }{\epsilon}=\lim_{T\to\infty}\lim_{\epsilon\to 0}\inf_{T^* \geq T} A(\epsilon, T^*)\\
&= \lim_{T\to\infty}\lim_{\epsilon\to 0} \lim_{n\to \infty} A(\epsilon, T^*_n)=\lim_{T\to\infty}\lim_{n\to \infty} \lim_{\epsilon\to 0} A(\epsilon, T^*_n) \\
&= \lim_{T\to\infty}\inf_{T^*\geq T} \lim_{\epsilon\to 0} A(\epsilon, T^*)=\lim_{T\to\infty}\inf_{T^* \geq T}\lim_{\epsilon\to 0}\int_{a}^{T^*} \frac{L[ x^*
 + \epsilon v](t)- L[x^*](t)}{\epsilon} \, dt\\
&= \lim_{T\to\infty}  \inf_{T^* \geq T}\int_{a}^{T^*}\left[\partial_2 L[x^*](t) \cdot v(t)+\partial_3 L[x^*](t)  \cdot {^CD_{a+}^{\a,\p}} v(t)\right]dt\\
&= \lim_{T\to\infty}  \inf_{T^* \geq T}\left[\int_a^{T^*}\left[\partial_2 L[x^*](t)+{D_{T^*-}^{\alpha,\psi}}
\left(\frac{\partial_3 L[x^*](t)}{\psi'(t)}\right)\psi'(t)\right]v(t)\,dt\right.\\
& \quad \left.+\left[{I_{T^*-}^{1-\alpha,\psi}}\left(\frac{\partial_3 L[x^*](t)}{\psi'(t)}\right)\cdot v(t)\right]_{t=a}^{t=T^*}\right].
\end{align*}
If we assume that $v(T^*)=0$, we deduce that
$$\lim_{T\to\infty}  \inf_{T^* \geq T}\int_a^{T^*}\left[\partial_2 L[x^*](t)+{D_{T^*-}^{\alpha,\psi}}\left(\frac{\partial_3 L[x^*](t)}{\psi'(t)}\right)
\psi'(t)\right]v(t)\,dt=0,$$
and so (see \cite{AlmeidaIn})
$$\partial_2 L[x^*](t)+{D_{T^*-}^{\alpha,\psi}}\left(\frac{\partial_3 L[x^*](t)}{\psi'(t)}\right)\psi'(t)=0,$$
for all $t\geq a$ and for all $T^* \geq t$. Also, using this last condition, we get
$$\lim_{T\to\infty}\inf_{T^* \geq T}{I_{T^*-}^{1-\alpha,\psi}}\left(\frac{\partial_3 L[x^*](t)}{\psi'(t)}\right)=0 \quad \mbox{at} \, \, t=T^*.$$
\end{proof}

\subsection{Variational principles with delay}\label{sec:delay}

In this section we consider time-delay variational problems. This is an important subject, since in many systems there is almost always
a time delay \cite{Richard,Salamon}. A natural generalization of such theory is to replace ordinary derivatives by fractional derivatives,
since fractional operators contain memory, and their present state is determined by all past states. There exist
already some works dealing with fractional operators, for example \cite{AlmeidaD,BaleanuD1,BaleanuD2}.
Let $L:[a,b]\times\mathbb R^3\to\mathbb R$ be a continuous function such that there exist and are continuous the functions $\partial_i L$, for $i=2,3,4$.
Given $\tau>0$ such that $\tau<b-a$, define the functional
\begin{equation}\label{functDelay}
J(x,T):=\int_a^T L(t,x(t),x(t-\tau),{^CD_{a+}^{\a,\p}} x(t))\,dt, \quad (x,T)\in\Omega_\tau\times[a,b],
\end{equation}
where
$$\Omega_\tau:=\{x\in C^1[a-\tau,b]: x(t)=\theta(t)\, \mbox{ for }\, t\in[a-\tau,a]\},$$
and $\theta$ is a given function. Let $[x]_\tau$ denote the vector
$$[x]_\tau(t):=(t,x(t),x(t-\tau),{^CD_{a+}^{\a,\p}} x(t)).$$

\begin{theorem}\label{teo:Delay} Let the pair $(x^*,T^*)$ be local minimum for $J$ as in \eqref{functDelay}.
If there exist and are continuous the functions $t\mapsto {D_{(T^*-\tau)-}^{\alpha,\psi}}\left(\partial_4 L[x^*](t)/\psi'(t)\right)$ and $t\mapsto {D_{T^*-}^{\alpha,\psi}}\left(\partial_4 L[x^*](t)/\psi'(t)\right)$
 on $[a,T^*]$, then for all $t\in[a,T^*-\tau]$,
$$\partial_2 L[x^*]_\tau(t)+\partial_3 L[x^*]_\tau(t+\tau)+{D_{(T^*-\tau)-}^{\alpha,\psi}}\left(\frac{\partial_4 L[x^*]_\tau(t)}{\psi'(t)}\right)\psi'(t)$$
$$-\frac{1}{\Gamma(1-\alpha)}\left(\frac{1}{\psi'(t)}\frac{d}{dt}\right)\int_{T^*-\tau}^{T^*}\psi'(s)(\psi(s)-\psi(t))^{-\alpha}\frac{\partial_4 L[x^*]_\tau(s)}{\psi'(s)}
\, ds \cdot \psi'(t)=0,$$
and for all $t\in[T^*-\tau,T^*]$,
$$\partial_2 L[x^*]_\tau(t)+{D_{T^*-}^{\alpha,\psi}}\left(\frac{\partial_4 L[x^*]_\tau(t)}{\psi'(t)}\right)\psi'(t)=0.$$
Also, at $t=T^*$, it is true that
$$\left\{
\begin{array}{l}
{I_{T^*-}^{1-\alpha,\psi}}\left(\frac{\partial_4 L[x^*]_\tau(t)}{\psi'(t)}\right)=0\\
L[x^*]_\tau(t)=0.
\end{array}
\right.$$
\end{theorem}
\begin{proof} Let $v\in C^1[a-\tau,b]$ be such that $v(t)=0$, for all $t\in[a-\tau,a],$ and consider variations of the form
$(x^*+\epsilon v,T^*+\epsilon \triangle T)$. Since the first variation of the functional must vanish at an extremum point, we have
$$\int_a^{T^*}\left[\partial_2 L[x^*]_\tau(t) \cdot v(t)+\partial_3 L[x^*]_\tau(t) \cdot v(t-\tau)+
\partial_4 L[x^*](t)  \cdot {^CD_{a+}^{\a,\p}} v(t)\right]dt+\triangle T\cdot L[x^*](T^*)=0.$$
Observe that
$$\int_a^{T^*} \partial_3 L[x^*]_\tau(t) \cdot v(t-\tau)\,dt=\int_a^{T^*-\tau} \partial_3 L[x^*]_\tau(t+\tau) \cdot v(t)\,dt$$
since $v(t)=0$, for all $t\in[a-\tau,a]$.
Also, since
\begin{align*}{D_{T^*-}^{\alpha,\psi}}\left(\frac{\partial_4 L[x^*]_\tau(t)}{\psi'(t)}\right)&
={D_{(T^*-\tau)-}^{\alpha,\psi}}\left(\frac{\partial_4 L[x^*]_\tau(t)}{\psi'(t)}\right)\\
& \quad-\frac{1}{\Gamma(1-\alpha)}\left(\frac{1}{\psi'(t)}\frac{d}{dt}\right)\int_{T^*-\tau}^{T^*}\psi'(s)(\psi(s)-\psi(t))^{-\alpha}\frac{\partial_4 L[x^*]_\tau(s)}{\psi'(s)}\,ds
\end{align*}
we obtain the following
$$\int_a^{T^*} \partial_4 L[x^*]_\tau(t) \cdot {^CD_{a+}^{\a,\p}} v(t)\,dt=\int_a^{T^*} {D_{T^*-}^{\alpha,\psi}}\left(\frac{\partial_4 L[x^*]_\tau(t)}{\psi'(t)}\right)\psi'(t)
\cdot v(t)\,dt+\left[ {I_{T^*-}^{1-\alpha,\psi}}\left(\frac{\partial_4 L[x^*]_\tau(t)}{\psi'(t)}\right)\cdot v(t)\right]_a^{T^*}$$
$$=\int_a^{T^*-\tau}\left[{D_{(T^*-\tau)-}^{\alpha,\psi}}\left(\frac{\partial_4 L[x^*]_\tau(t)}{\psi'(t)}\right)\psi'(t)
-\frac{1}{\Gamma(1-\alpha)}\left(\frac{1}{\psi'(t)}\frac{d}{dt}\right)\int_{T^*-\tau}^{T^*}\psi'(s)(\psi(s)-\psi(t))^{-\alpha} \frac{\partial_4 L[x^*]_\tau(s)}{\psi'(s)}\,ds\right.$$
$$\cdot \psi'(t)\Bigg]v(t)\,dt
+\int_{T^*-\tau}^{T^*} {D_{T^*-}^{\alpha,\psi}}\left(\frac{\partial_4 L[x^*]_\tau(t)}{\psi'(t)}\right)\psi'(t)
\cdot v(t)\,dt+\left[ {I_{T^*-}^{1-\alpha,\psi}}\left(\frac{\partial_4 L[x^*]_\tau(t)}{\psi'(t)}\right)\cdot v(t)\right]_a^{T^*}.$$
Finally, combining all the previous formulas, we obtain
$$\int_a^{T^*-\tau}\left[ \partial_2 L[x^*]_\tau(t)+\partial_3 L[x^*]_\tau(t+\tau)+{D_{(T^*-\tau)-}^{\alpha,\psi}}
\left(\frac{\partial_4 L[x^*]_\tau(t)}{\psi'(t)}\right)\psi'(t)\right.$$
$$\left. -\frac{1}{\Gamma(1-\alpha)}\left(\frac{1}{\psi'(t)}\frac{d}{dt}\right)\int_{T^*-\tau}^{T^*}\psi'(s)(\psi(s)-\psi(t))^{-\alpha}\frac{\partial_4 L[x^*]_\tau(s)}{\psi'(s)}
\, ds \cdot \psi'(t) \right]v(t)dt$$
$$+\int_{T^*-\tau}^{T^*}\left[ \partial_2 L[x^*]_\tau(t)+{D_{T^*-}^{\alpha,\psi}}\left(\frac{\partial_4 L[x^*]_\tau(t)}{\psi'(t)}\right)\psi'(t)\right]v(t)dt$$
$$+\left[ {I_{T^*-}^{1-\alpha,\psi}}\left(\frac{\partial_4 L[x^*]_\tau(t)}{\psi'(t)}\right)\cdot v(t)\right]_a^{T^*}+\triangle T\cdot L[x^*]_\tau(T^*)=0.$$
Since $v$ is arbitrary on the interval $[a,T^*]$, as well as $\triangle T$, we obtain the desired result.
\end{proof}

\subsection{High order derivatives}\label{sec:high}

So far we considered a fractional order as a real between 0 and 1. Using similar techniques as the ones presented in the proof of Theorem
\ref{teo:main}, we can generalize the previous results in order to include high order derivatives. We show how to do it for the basic problem
 of the calculus of variations, and we deduce the respective Euler--Lagrange equation.

\begin{theorem} Consider the functional
$$J(x,T):=\int_a^T L(t,x(t),{^CD_{a+}^{\alpha_1,\psi}} x(t),\ldots, {^CD_{a+}^{\alpha_m,\psi}} x(t))\,dt, \quad (x,T)\in\Omega_m\times[a,b],$$
where
\begin{enumerate}
\item $m\in\mathbb N$, and for all $n\in\{1,\ldots,m\}$, we have $\alpha_n\in(n-1,n)$;
\item $L:[a,b]\times\mathbb R^{m+1}\to\mathbb R$ is a continuous function;
\item there exist and are continuous the functions $\partial_2 L$, $\partial_3 L$, $\ldots$, and  $\partial_{m+2} L$;
\item $\Omega_m:=\{x\in C^m[a,b]: x(a), x^{(1)}(a),\ldots, x^{(m-1)}(a) \, \mbox{ are fixed reals}\}$.
\end{enumerate}
Suppose that $(x^*,T^*)$ is a local minimum for $J$. If, for all $n\in\{1,\ldots,m\}$, there exist and are continuous the functions
$t\mapsto {D_{T^*-}^{\alpha_n,\psi}}\left(\partial_{n+2} L[x^*](t)/\psi'(t)\right)$ on $[a,T^*]$, then
$$\partial_2 L[x^*]_m(t)+\sum_{n=1}^m{D_{T^*-}^{\alpha_n,\psi}}\left(\frac{\partial_{n+2} L[x^*]_m(t)}{\psi'(t)}\right)\psi'(t)=0,$$
for each $t\in[a,T^*]$, and at $t=T^*$, we have
$$\left\{
\begin{array}{l}
\sum_{n=k}^m \left(-\frac{1}{\psi'(t)}\frac{d}{dt}\right)^{n-k}{I_{T^*-}^{n-\alpha_n,\psi}}\left(\frac{\partial_{n+2} L[x^*]_m(t)}{\psi'(t)}\right)=0,
\quad \, k=1,\ldots,m,\\
L[x^*]_m(t)=0,
\end{array}
\right.$$
where
$$[x^*]_m(t):=(t,x^*(t),{^CD_{a+}^{\alpha_1,\psi}} x^*(t),\ldots, {^CD_{a+}^{\alpha_m,\psi}} x^*(t)).$$
\end{theorem}
\begin{proof} Consider admissible variations of the form  $(x^*+\epsilon v,T^*+\epsilon \triangle T)$, where $v\in C^m[a,b]$ and $v(a)=0=v^{(n)}(a)$,
 for all $n=1,\ldots,m-1$. Since the first variation of the functional must vanish at an extremum point, we deduce the following:
$$\int_a^{T^*}\left[\partial_2 L[x^*]_m(t) \cdot v(t)+\sum_{n=1}^m\partial_{n+2} L[x^*]_m(t)\cdot{^CD_{a+}^{\alpha_n,\psi}} v(t)\right]dt+\triangle T\cdot
 L[x^*]_m(T^*)=0.$$
Integrating by parts, we get
$$\int_a^{T^*}\left[ \partial_2 L[x^*]_m(t)+\sum_{n=1}^m{D_{T^*-}^{\alpha_n,\psi}}\left(\frac{\partial_{n+2} L[x^*]_m(t)}{\psi'(t)}\right)\psi'(t)\right]v(t)\,dt+
\triangle T\cdot L[x^*]_m(T^*)$$
$$+\left[\sum_{n=1}^m \left(-\frac{1}{\psi'(t)}\frac{d}{dt}\right)^{n-1}{I_{T^*-}^{n-\alpha_n,\psi}}\left(\frac{\partial_{n+2} L[x^*]_m(t)}{\psi'(t)}\right)
\cdot v(t)\right.$$
$$+\sum_{n=2}^m \left(-\frac{1}{\psi'(t)}\frac{d}{dt}\right)^{n-2}{I_{T^*-}^{n-\alpha_n,\psi}}\left(\frac{\partial_{n+2} L[x^*]_m(t)}{\psi'(t)}\right)\cdot
\left(\frac{1}{\psi'(t)}\frac{d}{dt}\right)v(t)$$
$$\left.+\ldots+{I_{T^*-}^{m-\alpha_m,\psi}}\left(\frac{\partial_{m+2} L[x^*]_m(t)}{\psi'(t)}\right)\cdot \left(\frac{1}{\psi'(t)}\frac{d}{dt}\right)^{m-1}v(t)
\right]_{t=a}^{t=T^*}=0.$$
Choosing appropriate variations, we deduce the result.
\end{proof}

\subsection{Optimal fractional order}\label{sec:optimal}

One advantage of considering fractional derivatives in modelling phenomena is that they may describe more efficiently the dynamics than ordinary derivatives.
So, a natural question to pose is what should be the order of the fractional operator, in order to minimize the functional. The next result provides necessary
conditions to determine the fractional order.

\begin{theorem}\label{FracOrder} Consider the functional
$$J(x,T,\alpha):=\int_a^T L[x](t)\,dt,$$
where $x\in C^1[a,b]$ with $x(a)$ fixed, $T\in[a,b]$ and $\alpha\in(0,1)$. Assume that $(x^*,T^*,\alpha^*)$ is a local minimum for $J$, and that there
 exists and is continuous the function $t\mapsto {D_{T^*-}^{\alpha,\psi}}\left(\partial_3 L[x^*](t)/\psi'(t)\right)$ on $[a,T^*]$. Given $t\in[a,T^*]$,
 define a function $ \Lambda_t:(0,1)\to\mathbb R$ by the rule $\Lambda_t(\alpha):={^CD_{a+}^{\a,\p}} [x^*](t)$. Then,
$$\partial_2 L[x^*](t)+{D_{T^*-}^{\alpha,\psi}}\left(\frac{\partial_3 L[x^*](t)}{\psi'(t)}\right)\psi'(t)=0$$
for each $t\in[a,T^*]$, and at $t=T^*$, the following holds:
$$\left\{
\begin{array}{l}
{I_{T^*-}^{1-\alpha,\psi}}\left(\frac{\partial_3 L[x^*](t)}{\psi'(t)}\right)=0\\
L[x^*](t)=0.
\end{array}
\right.$$
Also, we have the following equality:
$$\int_a^{T^*}\partial_3 L[x^*](t)\cdot \Lambda_t'(\alpha^*)\,dt=0.$$
\end{theorem}
\begin{proof} Define
$$j(\epsilon):=J(x^*+\epsilon v, T^*+\epsilon \triangle T, \alpha^*+\epsilon \triangle \alpha),$$
where $v\in C^1[a,b]$ with $v(a)=0$ and $\triangle T,\triangle \alpha$ are two fixed reals.
Since $j'(0)=0$, using integration by parts, we arrive at the formula
$$\int_a^{T^*}\left[\partial_2 L[x^*](t)+{D_{T^*-}^{\alpha,\psi}}\left(\frac{\partial_3 L[x^*](t)}{\psi'(t)}\right)\psi'(t)\right]\cdot v(t)\,dt
+ \left[{I_{T^*-}^{1-\alpha,\psi}}\left(\frac{\partial_3 L[x^*](t)}{\psi'(t)}\right)\cdot v(t)\right]_{t=a}^{t=T^*}$$
$$+\triangle T\cdot L[x^*](T^*)+\triangle \alpha\cdot \int_a^{T^*}\partial_3 L[x^*](t)\cdot \Lambda_t'(\alpha^*)\,dt=0.$$
By the arbitrariness of $v$, $\triangle T$ and $\triangle \alpha$, we prove the desired result.
\end{proof}

\subsection{Sufficient conditions}\label{sec:suff}

So far we deduced necessary conditions that every minimizer of the functional must verify. Now, we will deal with a sufficient condition, and for that some conditions of convexity over the Lagrangian are needed.

\begin{theorem} Assume that $L$ is convex in $[a,b]\times\mathbb R^2$, in the sense that
$$L(t,x+v,y+w)-L(t,x,y)\geq \partial_2 L(t,x,y)v+\partial_3 L(t,x,y)w,$$
for all $t\in[a,b]$ and $x,y,v,w\in\mathbb R$. If  $(x^*,T^*)$ satisfies the Euler--Lagrange equation \eqref{ELeq} and the transversality
conditions \eqref{TransCon}, then
$$\forall\epsilon>0\,\exists\delta>0 \, : \, \|(v,\triangle T)\|<\delta \Rightarrow J(x^*+v,T^*+\triangle T)-J(x^*,T^*)\geq-\epsilon$$
on the space $\Omega\times[a,b]$, where functional $J$ is given by \eqref{funct1}.
\end{theorem}

\begin{proof}
Observe that
\begin{align*}
J(x^*+v,T^*+\triangle T)&-J(x^*,T^*)=\int_a^{T^*+\triangle T}L[x^*+v](t)\,dt-\int_a^{T^*}L[x^*](t)\,dt\\
&=\int_a^{T^*}\left[L[x^*+v](t)-L[x^*](t)\right]\,dt+\int_{T^*}^{T^*+\triangle T}L[x^*+v](t)\,dt\\
&\geq\int_a^{T^*}\left[\partial_2 L[x^*](t) \cdot v(t)+\partial_3 L[x^*](t)  \cdot {^CD_{a+}^{\a,\p}} v(t)\right]dt+\triangle T\cdot L[x^*+v](c)=\star,
\end{align*}
for some $c$ between $T^*$ and $T^*+\triangle T$. Integrating by parts, we get
\begin{align*}
\star&=\int_a^{T^*}\left[\partial_2 L[x^*](t)+{D_{T^*-}^{\alpha,\psi}}\left(\frac{\partial_3 L[x^*](t)}{\psi'(t)}\right)\psi'(t)\right]v(t)\,dt\\
& \quad+ \left[{I_{T^*-}^{1-\alpha,\psi}}\left(\frac{\partial_3 L[x^*](t)}{\psi'(t)}\right)\cdot v(t)\right]_{t=a}^{t=T^*} +\triangle T\cdot L[x^*+v](c)\\
& =\triangle T\cdot L[x^*+v](c).
\end{align*}
Since $L[x^*](T^*)=0$ and the map $t\mapsto L[x^*+v](t)$ is continuous, given $\epsilon>0$, there exists some $\delta>0$ such that
$$\|(v,\triangle T)\|<\delta \Rightarrow \left| L[x^*+v](c)\right|<\frac{\epsilon}{|\triangle T|}$$
(the case $\triangle T=0$ is obvious). Thus,
$$J(x^*+v,T^*+\triangle T)-J(x^*,T^*)\geq-\epsilon.$$
\end{proof}

We remark that we can not conclude that
$$J(x^*+v,T^*+\triangle T)-J(x^*,T^*)\geq0,$$
under these assumptions.
For example, if we consider
$$J(x,T):=\int_0^T(1-t)\,dt,$$
with the initial condition $x(0)=x_0\in\mathbb R$, it is easy to verify that $L$ is convex and that $(x^*,T^*)=(x^*,1)$, where $x^*\in C^1[a,b]$ is an arbitrary function, verifies Eqs  \eqref{ELeq} and \eqref{TransCon}. However, we have the following:
$$J(x^*+v,1+\triangle T)-J(x^*,1)=-\frac12(\triangle T)^2,$$
which is a negative number when $\triangle T\not=0$.

\subsection{Examples}

\begin{example} Consider the functional
$$J(x,T):=\int_0^T \left[\left({^CD_{0+}^{\alpha,\psi}} x(t)-\frac{(\psi(t)-\psi(0))^{1-\alpha}}{\Gamma(2-\alpha)}\right)^2+t^2-1\right]\,dt$$
subject to the restriction $x(0)=0$. From Theorem \ref{teo:main}, the necessary condition that every minimizer of the functional must fulfill is the following
$${D_{T-}^{\alpha,\psi}}\left(2\frac{{^CD_{0+}^{\alpha,\psi}} x(t)-\frac{(\psi(t)-\psi(0))^{1-\alpha}}{\Gamma(2-\alpha)}}{\psi'(t)}\right)\psi'(t)=0,$$
for all $t\in[0,T]$, and also the two next transversality conditions
$$\left\{
\begin{array}{l}
{I_{T-}^{1-\alpha,\psi}}\left(2\frac{{^CD_{0+}^{\alpha,\psi}} x(t)-\frac{(\psi(t)-\psi(0))^{1-\alpha}}{\Gamma(2-\alpha)}}{\psi'(t)}\right)=0\\
L[x](t)=0.
\end{array}
\right.$$
must be meet at  $t^=T^*$. Once
$${^CD_{0+}^{\alpha,\psi}} \left(\psi(t)-\psi(0)\right)=\frac{(\psi(t)-\psi(0))^{1-\alpha}}{\Gamma(2-\alpha)},$$
we see that the pair
$$(x^*,T^*):=(\psi(t)-\psi(0),1)$$
satisfies all three conditions. Also, using the Legendre condition, a local minimizer for the functional must verify the condition
$$\partial^2_{33}L[x](t)\geq0, $$
which for our example is verified, since $\partial^2_{33}L[x](t)=2$. We remark that, for every curve $x$ and for every endpoint $T$, the inequality
$$J(x,T)\geq\int_0^T (t^2-1)\,dt\geq \int_0^1 \left[t^2-1\right]\,dt=-\frac23,$$
holds, and that
$$J(x^*,T^*)=-\frac23,$$
and so $(x^*,T^*)$ is actually the global minimizer of $J$.
\end{example}

\begin{example} For our second example, let

$$J(x,T):=\int_0^T \left[\left({^CD_{0+}^{\alpha,\psi}} x(t)\right)^2+\left(\frac{(\psi(t)-\psi(0))^{1-\alpha}}{\Gamma(2-\alpha)}\right)^2+t^2-1\right]\,dt$$
subject to the restrictions $x(0)=0$ and
$$\int_0^T \left[{^CD_{0+}^{\alpha,\psi}} x(t)\cdot\frac{(\psi(t)-\psi(0))^{1-\alpha}}{\Gamma(2-\alpha)}\right]\,dt=\Phi(T),$$
where the function $\Phi$ is defined as
$$\Phi(T):=\int_0^T \left[\frac{(\psi(t)-\psi(0))^{1-\alpha}}{\Gamma(2-\alpha)}\right]^2\,dt.$$
Considering $\lambda=-2$, the augmented function is given by
$$F(x)=\left({^CD_{0+}^{\alpha,\psi}} x(t)-\frac{(\psi(t)-\psi(0))^{1-\alpha}}{\Gamma(2-\alpha)}\right)^2+t^2-1,$$
and as we have seen in the previous example, the function
$$x^*=\psi(t)-\psi(0)$$
satisfies the two first necessary conditions, as stated in Theorem \ref{teo:isop}. Using the second transversality condition of Theorem \ref{teo:isop}, we determine the optimal time $T^*$ by solving the equation
$${T^*}^2-1=-2.\left[\frac{(\psi(T^*)-\psi(0))^{1-\alpha}}{\Gamma(2-\alpha)}\right]^2.$$
\end{example}

\begin{example}
For our next example, we determine the best fractional order to minimize a given functional, using the necessary conditions given by Theorem \ref{FracOrder}.
Let
$$J(x,T,\alpha):=\int_a^T \Bigg[\frac{(\psi(t)-\psi(a))^\alpha}{2\Gamma(\alpha+2)}\cdot({^CD_{a+}^{\a,\p}} x(t))^2-(\psi(t)-\psi(a))^{\alpha+1}\cdot {^CD_{a+}^{\a,\p}} x(t)+20\Gamma(\alpha+2)\Bigg]dt,$$
subject to the initial condition $x(a)=0$.
If we define $x^*(t):= (\psi(t)-\psi(a))^{\alpha+1}$, then ${^CD_{a+}^{\a,\p}} x^*(t)=\Gamma(\alpha+2)(\psi(t)-\psi(a))$ and so $\partial_3 L[x^*](t)=0$, for all $t$.
Thus, the three next conditions
$$\partial_2 L[x^*](t)+{D_{T^*-}^{\alpha,\psi}}\left(\frac{\partial_3 L[x^*](t)}{\psi'(t)}\right)\psi'(t)=0, \quad t\in[a,T^*],$$
$${I_{T^*-}^{1-\alpha,\psi}}\left(\frac{\partial_3 L[x^*](t)}{\psi'(t)}\right)=0, \quad t=T^*$$
and
$$\int_a^{T^*}\partial_3 L[x^*](t)\cdot \Lambda_t'(\alpha^*)\,dt=0,$$
are verified for all $T^*$ and $\alpha^*$. It remains to solve the equation $L[x^*](T^*)=0$. Since
\begin{equation}\label{auxFracOr}L[x^*](t)=0 \Leftrightarrow \psi(t)=\psi(a)+40^{1/(\alpha+2)},\end{equation}
we establish a relation between $T^*$ and $\alpha^*$ given by condition \eqref{auxFracOr}.
Observe that
$$J(x^*,T^*,\alpha)=\int_a^{T^*} \Gamma(\alpha+2)\left[20-\frac{(\psi(t)-\psi(a))^{\alpha+2}}{2}\right]\,dt,$$
and if we differentiate with respect to $\alpha$, and then  put the resulting function equal to zero, we get
\begin{equation}\label{auxFracOr1}\int_a^{T^*} \Gamma(\alpha+2)\left[\Psi(\alpha+2)\left(20-\frac{(\psi(t)-\psi(a))^{\alpha+2}}{2}\right)
-\frac{\ln(\psi(t)-\psi(a))}{2}\psi'(t)(\psi(t)-\psi(a))^{\alpha+2}\right]\,dt=0,\end{equation}
where $\Psi$ denotes the digamma function. For example, for $\psi_1(t)=t$ and $\psi_2(t)=\sqrt{t+1}$, with initial point $a=0$,
we  have $T_1^*=40^{1/(\alpha^*+2)}$ and $T_2^*=1+40^{1/(\alpha^*+2)}$, respectively. The graphs of the functions
$$\alpha\mapsto J(x^*,T^*,\alpha), \quad \alpha\in(0,1),$$
are given in Figure \ref{FracOrder5}.
\begin{figure}[h]
  \centering
  \subfigure[Graph with respect to $\psi_1$.]{\includegraphics[width=6cm]{FracOrder.eps}}
\subfigure[Graph with respect to $\psi_2$.]{\includegraphics[width=6cm]{FracOrder2.eps}}
  \caption{Best fractional order.}\label{FracOrder5}
\end{figure}
Solving numerically Eq. \eqref{auxFracOr1}, we obtain an approximation of the fractional order that minimizes the problem:
$$\alpha_1\approx 0.2677 \quad \mbox{and} \quad \alpha_2\approx 0.2827.$$
\end{example}

\section*{Acknowledgments}

Work supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications, and the Portuguese
Foundation for Science and Technology (FCT-Funda\c{c}\~ao para a Ci\^encia e a Tecnologia), within project UID/MAT/04106/2013.

redpajama_set_name: RedPajamaArXiv


Like in any other profession, the life of a designer is (or should be) about fixing problems. Many (the vast majority) are solving market problems. However, the number of people who are relying on design to find the tools and the skills needed to solve the problems of humanity is steadily growing. It's design being used in social impact projects.
Before we start to dwell on the application of design itself, we should remember what social impact projects are, in which the results are not measured by profit or by market potential, rather by behavior changes that can benefit individuals or entire communities.
Okay, but what actually changes when one uses design for social impact? Even though the philosophy, methodologies and tools are virtually the same, there are at least four points that make this approach to design, when used in social impact projects, different from the marketing approach that we are used to.
Focusing or prioritizing groups of specific users can be counterproductive.
In a commercial context, focusing your product on a specific niche can be the key to success. The trolley is a classic example of something that was designed to meet the needs of a small group of people: flight attendants. Of course, after a while, many people realized that they had the very same problems and the product became a success, but it only had a highly-segmented audience in the beginning, when it was conceived.
When we're designing with the intent to prompt some kind of social outcome, we don't have the luxury to restrict our projects to a niche. A public service, for instance, has to serve all citizens, so if that solution is only focused on a specific share of people, we may end up interfering in the project's original purpose, creating advantages for those who need our creation the least.
Developing empathy requires humility and sensitivity.
Historically, social impact projects have been known to pay little attention to what people really want, what motivates them and what challenges they face. That's where design steps in!! Methodologies such as Human-Centered Design, for instance, are crucial to understanding the people involved in the project.
If you're looking to create something disruptive, then include in the process those who will have to cope with that disruption on a daily basis. Keep this in mind, the goal of a designer is to help people help themselves, not creating things for them. Rely on design for social impact is an opportunity to work WITH people and not FOR people.
Designers must also know the difference and the limits between walking a mile in someone else's shoes and feel pity for that person. Even though it may seem a blatant difference, many times the temptation of wanting to be the solution to the problem is just something big.
Designers are not heroes. We will not save the world with our own abilities. Always make it clear for your team that they need to create solutions that bring independence and autonomy for the community.
Getting social results implies psychological motivations that are much deeper than to change someone's consumption habits.
To make someone consume a product or service, especially in today's consumer society, seldom involve deep motivations. A good example is to look at the advertisements around, most of them work as "entertainment", leading people to feel a superficial and momentary state of pleasure and satisfaction, enough to trigger the consumer interest.
Now imagine that you are working to solve the following challenge: "How can we, based on mobile technology, improve access to health?". Market-wise speaking, perhaps developing and encouraging people to download an app can be deemed a case of success. Nevertheless, you should keep in mind that we don't want to promote consumption, but a behavior change. Well before you make the app available, in this case, we need to identify the motivations, fears, desires and wishes to make sure that our product is just a tool to support this change.
Making mistakes or focusing on false assumptions can be lethal.
Companies like Apple can "make mistakes". An iPhone recall, for instance, can even annoy some customers, but Apple's brand and value are so strong that they can topple that situation almost unscathed.
Now, when you think about social impact projects, people are not that acquainted with it. Imagine a system failure of a major welfare program like Bolsa Família, or a bug in the app you use to fill your income tax return. Can you imagine the turmoil that would take place in social media? That happens because people have a lower tolerance when they are dealing with public and social projects. A tolerance that is much lower than when they are struggling with a phone or car problem.
That's why design is so important as an experimentation and validation process. We always need to take grab our chances and find a way to know if they are true. We don't have that many chances to make an error. Generating an inappropriate and/or ineffective solution may fully compromise the trust that people had in you during the project's earlier stages, which in many cases is difficult to get back.


Right now, you're super-motivated to make healthy and lasting changes in your life.
Now that you're pregnant, are you afraid to start or keep exercising? This myth-busting information will help you get the activity you need.
Ease pregnancy discomforts and stay in shape safely with this cool pool workout.
Tweak your body's alignment for a pain-free pregnancy.
Prepare your body and mind for labor and delivery with our prenatal yoga workout.
Healthy prenatal eating isn't just about avoiding. It's also about choosing wisely. Stick to these 10 basic do's and don'ts.
Move over, brown rice. You might not know about them yet, but these whole grains are good for you and your baby—and taste great, too.
Get a host of essential pregnancy nutrients with six surprising (and scrumptious!) seasonal vegetables.
Easy recipes, full of flavor and loaded with fiber, lean protein, calcium and folate.
Eating healthy is important for you and your baby. Search our database for nutritious recipes to take you through your pregnancy and beyond.


text:
Working Title
Andy Gets…
FRAK ATTACK!!
LOSTWATCH!!
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The Year End Review: New TV
Posted on December 21, 2009 by jimmybing | Leave a comment
Every year, TV networks try shoving an endless parade of new television shows down our throats. Usually, these new shows follow the same formula as the shows that were popular the year before. Sometimes it works (Parks and Recreation), and sometimes it doesn't (Day Break). More often it doesn't, so finding a new show that works on different levels and is able to suck you in is a rare and beautiful thing, akin to your wedding day or your first child being born. This year, two new TV shows grabbed our attention and win our awards, which really aren't awards and are empty of any sort of meaning to anyone who doesn't read this blog. In the category of drama, the award goes to Southland!
When we first heard about Southland, we said, "A show about cops! I'm sold!" But it was in a really ironic way so you know we didn't mean it. Still, we checked it out and were very impressed. With great performances by lead actors Ben Mackenzie, Michael Cudlitz, and Regina King, along with realistic storylines that aren't always wrapped up at the end of the hour, we see Southland as a successor to that other great cop drama, NYPD Blue. Thankfully TNT picked up the show after NBC completely dropped the ball and canceled it two weeks before the premiere of its second season. Hopefully it'll catch on and do much better than The Closer, because somebody needs to put Kyra Sedgwick in her place. So, on to comedy. This year, the Working Title award for best new comedy series goes to Modern Family!
Any of the buzz you've heard about this show, it's all true. While characters like Michael Scott and Liz Lemon can be funny because of how over-the-top zany they sometimes are, the characters on Modern Family are funny because how familiar they are to us. While some shows (Community comes to mind) struggle to find their identity, we think Modern Family has been pretty solid right out of the gate. Ratings have been pretty good and we think this one will be an easy greenlight for a second season.
Honorable mentions this year go to no one, because these two shows are far and away the best we've seen. New dramas such as FlashForward and V, and new comedies like The Cleveland Show and Cougar Town are still trying to find their footing. The dramas are trying too desperately to be the next Lost while completely forgetting what makes Lost such a great show, while the comedies feel like they're either stuck telling the same joke, or are trying to be funny by going way over the top. We're looking at you, Cleveland Brown. Sex jokes are a scalpel, not a club.
Stay tuned for the The Year End Review: Movies, as well as our mid-season write-ups on the shows we've been watching.
This entry was posted in The Year End Review and tagged ABC, comedy, drama, entertainment, Modern Family, NBC, Southland, television, TNT, TV. Bookmark the permalink.
Previously, on Twitter…
@keeneTV Holy shit, you weren't kidding. Felt like the whole show just turned everything up to 11. 13 hours ago
@itslizhannah In summation, this is one of my all-time favorites. 20 hours ago
@itslizhannah "It's. Your. Mother." 20 hours ago
@AlisonNBCBoston @MittRomney @NBC10Boston Interesting that a rich white man's American experience might differ from… twitter.com/i/web/status/1… 21 hours ago
I asked Alexa what the weather was like today and she said partly sunny with a 100% chance of great deals on Amazon Prime Day. 1 day ago
A change is gonna come.
Lights Out: How Many Boxing Metaphors Can I Cram Into This Review?
The Year End Review: Movies
The Year End Review: TV
"I can do nothing for you, son."
"Watch the way she moves."
It's Always Sunny, "Charlie Kelly: King of the Rats," and "The Gang Gets Stranded in the Woods"
The Walking Dead, "Wildfire": I'm okay!
The Walking Dead, "Vatos": I'm glad that's over.
King is a Fink
Never Seen Lost
Progressive Boink
Sklarbro Country
The Ack Attack!
The Good Ship Phaeton
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Television Critics
James Poniewozik
Myles A. McNutt
The Random!

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text:
Drive and Thrive
Goals and Proposals
Back to Governor's Page
Wyoming's Strategy to Survive, Drive and Thrive
Watch the ARP Virtual Town Hall from December 3, 2021 here
View the presentation from the Governor's Strike Team here
Wyoming has been through historic challenges the last two years, and during this time Governor Mark Gordon has been planning the state's recovery. The governor understood that there were actions the government needed to take immediately to stop additional negative impacts from the global pandemic. At the same time, he recognized that the state would have an opportunity to tackle problems long-term if it was strategic with its own funds and with funds made available through additional federal stimulus bills.
When Congress passed the $1.9 trillion American Rescue Plan Act (Rescue Plan or ARPA) into law, the governor was ready to deploy his planning efforts. The governor convened a group he called the Strike Team to create a strategy to best maximize the COVID-19-related federal funds in the near-term, mid-term, and long-term. He directed the Strike Team to determine what the state needed to do to survive, and what could be done to better drive to a future where all of Wyoming can thrive.
In addition, the governor identified criteria by which he believes the state should evaluate projects eligible for American Rescue Plan Act funding. The governor believes each project should:
Have a long-term impact or a return on investment;
Not replenish budget cuts unless the replenishment can be sustained;
Be sustainable and not add to the state's ongoing financial responsibilities;
Support stimulus over relief;
Where possible, leverage the dollars through matching or buy-in programs;
Create capacity for the future; and
Benefit a wide group of citizens.
The governor asked the full Strike Team to continue to assist him with the planning and study, as well as the evaluation of solutions, required in the Drive and Thrive phases. That led to the proposals that are outlined here on this website and available for public comment. These are the proposals that have been developed to date. The State Legislature can also consider other ideas when it meets in February.
From the outset of the planning process, the governor voiced his desire to work with the Wyoming Legislature to best deploy federal funds made available to the state. This mirrored the process for CARES Act funding. Thus, the timeframe of planning efforts largely aligns with the work of the Legislature. The Survive phase has been on-going and will continue through the 2022 Legislative Session. The Drive phase has been ongoing in 2021. Most of the planning and study efforts are culminating with recommendations the Governor will make to the Legislature in December 2021. Some areas identified for the Drive phase will require additional time and are expected to continue throughout 2022. The Thrive phase encompasses actions with long-term benefit.
The American Rescue Plan requires that all funds be obligated by the end of 2024 and spent by 2026.
Comments on these proposals can continue through the Legislative session, which starts February 14, 2022.
The American Rescue Plan Act of 2021 is a $1.9 trillion relief and stimulus package to support the nation's recovery from the many negative impacts caused by the Covid-19 pandemic. The Rescue Plan establishes the Coronavirus State Fiscal Recovery Fund and the Coronavirus Local Fiscal Recovery Fund which provide billions of dollars in aid to states and local governments. The State of Wyoming, through the Coronavirus State Fiscal Recovery Fund, expects to receive $1.68 billion. Wyoming counties and cities, through the Coronavirus Local Fiscal Recovery Fund, expect to receive $182 million.
This federal act does have many requirements and directs funding to be used for only allowable projects. However, there are other funds available because of this federal act and some of them could be used for the proposals outlined here with Legislative approval.
In addition to the Fiscal Recovery Funds, the State of Wyoming expects to receive up to $109 million from the Capital Projects Fund for critical projects tied to connectivity. The Rescue Plan also provides funds for education: $303 million to K-12 school districts and $44 million to higher education. Further, the Rescue Plan provides millions of dollars to Wyoming state agencies directly combating the pandemic (the Wyoming Department of Health) and the negative impacts of the pandemic - housing insecurity, child care fragility, food insecurity etc. (Department of Family Services) among others.
The Rescue Plan also provides millions of dollars to Wyoming citizens, businesses, and other entities outside of government through federal relief and stimulus programs, direct payments, tax credits, and benefits/ services.
To evaluate projects eligible for funding through the Coronavirus State Fiscal Recovery Fund, we must consider whether the projects could be funded with other available funds or whether there is access to other available funds. This will allow the state to maximize the benefits it receives from each different funding source.
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Our World Famous Tortuga "Coconut" Rum Cakes are baked fresh daily in the Island of Jamaica with only the finest natural ingredients. The only preservative used is our specially blended "5-year-old", oak barrel aged Tortuga Rum. Each cake is individually hand glazed then vacuum packed, ensuring a shelf life of 6 months, or indefinitely if refrigerated or frozen.


Sure, we all want our hair to look good, and ads tell us certain products will make our tresses shine and bounce. Besides having lovely locks, don't we want the women--and people--who work for the companies have family friendly workplaces with parental leave, flex-time, on-site daycare, etc etc.
If you want to shop with a gender equality lens, download Buy Up Index today. We'll show you which brands serve women. That will REALLY make you look awesome.


text:
Carlos Armando Gruezo Quiñónez (born 1975-09-18 in Quinindé) is a retired football forward from Ecuador, who earned one cap for the Ecuador national team during his career. His only appearance came on 14 October 1998 when Ecuador lost 5–1 in a friendly against Brazil, and Gruezo was a second-half substitute for Ariel Graziani. He is the father of Carlos Armando Gruezo Arboleda.

Honors

Club
 Deportivo Cuenca
 Serie A de Ecuador: 2004

References

External links
 Carlos Armando Gruezo Quiñónez at 11v11.com

1975 births
Living people
People from Quinindé Canton
Association football forwards
Ecuadorian footballers
Ecuador international footballers
Barcelona S.C. footballers
S.D. Aucas footballers
C.D. ESPOLI footballers
C.D. El Nacional footballers
C.D. Cuenca footballers
L.D.U. Portoviejo footballers
S.D. Quito footballers

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A popular and coveted street food the Vada Pav is perfect for vegetarians and those who want a break from the meat scene. Battered and fried potatoes are slathered in your favorite chutney and placed between sliced soft buns. Messy and delicious!
This cornflower cake can be stuffed with a number of meats and vegetables. Add chili sauce and the unique sandwich, known as Arepa, can deliver quite a kick! You can find them in most South American countries served in a variety of ways.
Vegemite is a yeasty paste that only an Australian can love. Spread on buttered toast with some cheese and you have the most unique sandwich you'll ever eat. Packed with vitamins and minerals it's one of the most healthful spreads out there, if you can get used to the taste.
Zapiekanka is an open-faced baguette popular throughout Poland. Typically found at street vendors the extra-long baguette is covered with sauteed mushroom, onions, and cheese then baked in the oven and drizzled with ketchup. Another messy delight.
The Gatsby is a simple concoction with massive size. A ridiculous amount of french fries (chips) and bologna are stuffed into an Italian roll and served with ketchup, piri piri, and lettuce. Have an empty stomach before ordering one of these!
This meat lover's dream is a favorite street food in Puerto Rico. The Tripleta sandwich gets its name from the trio of grilled chicken, ham, and beef served on a sweetened sandwich roll. Adding mayo, ketchup, mustard, or hot sauce gives it added dimension. I love mine plain with just the juices form the meats dripping down my arm!
Well there you have it…I could go on and on but those are some of my favorites. Have you ever tried one of these delicious sandwiches? Do you have a unique one of your own? Share your experiences with us!


Feeling overwhelmed? Just do the next thing.
Concerned over your future – of things not working out?
You don't need to know it all.
God knows. He will direct your path – if you ask Him.
So, if you've asked God to be with you today, to direct all your ways, then take heart.
Just do the next thing.
"Tucked in the back of the Book of Mark, following the story of the Crucifixion, we read this lovely little story.
Can't you imagine the disciples and Mary and Martha and the other bewildered women, sitting in absolute dejection and perplexity when their Lord and Master and King had just died?
They couldn't think of one single thing to do.
Here came this Godly man, who looked forward to the kingdom of God, who bravely went in and asked for the body of Jesus.
He could think of one thing to do. He did the next thing.
Elizabeth Elliot is the author of more than 20 books.
Keep that in mind when you're dealing with disappointment, worry or unease over the timing or waiting.
We all have dreams. Sometimes those dreams are seeds of what God has put into our hearts. However, HIS PLANS are always greater than what we, as flawed humans, can dream or imagine.
Ask the One who loves you more than any other to trust in HIS plan for you. How do you know it's from God? It'll be filled with hope.
His PLANS are greater than my dreams. Be patient. And while you're waiting, be the reason someone believes in the goodness of people!
Prayer provides strength. And isn't strength what we all really need as we face challenges and questions?
As good as venting can feel, it doesn't provide anything long term.
For those who have uncovered the joy of praying, any and every challenge (your own or others) has the potential for building long-term strength.
Because unlike a "vent", answered prayer unfolds into a long, long, stretch of experience that builds confidence. Prayer provides strength.
Your efforts in prayer don't just disappear. Those efforts remain as an experience of tapping into the power that originates and flows from the Creator, the One who loves you.
Is prayer easy? Sometimes. Not always. Prayer can be work. However, the effort to pray for something for a very long time, regardless, is sweeter and more satisfying than anything else.
Put your energy into prayer.
Want more encouragement to pray? Read this.
It's not as though we won't experience pain just because we're optimistic people… every person on this planet will suffer in some way, at some time. This hurt that could be physical, mental or spiritual will afflict all of us.
It's not going to be possible to be cheery all the time. How can an optimistic person deal with these afflictions that cause an ache?
You can treat pain with the love of God.
So, to paraphrase C.S. Lewis — just a little bit of the Love of God when dealing with pain is more effective than anything else.
Where do you go to feel the Love of God?
and when I honestly force myself to list all the blessings God has provided for me. Being thankful helps me feel God's Love.
How about you – where do you feel the Love of God?
Finally, brothers and sisters, rejoice!
Want more about love? Read this post or Read this post.
When positive people take action it flows from love.
Positive People know they're loved. And their actions flow from love.
There's an amazing power in love – both accepting love and giving it.
Positive people are able to love others and receive love. They know that they're created by God, who loves them and has put them here on this earth, at this time, for His purpose.
Their belief in this truth of God's love helps others' outlook by demonstrating, despite circumstances, mistakes and imperfection, love is possible.
This is the 8th of eight ways positive people improve others' outlook. It concludes the series! Here are links to the 1st, 2nd, 3rd, 4th, 5th, 6th, and 7th ways positive people improve others' outlook on life.
Positive People acknowledge the truth and manage the overflow.
Positive People look for things for which to be thankful.
Positive People bring out the best in others.
Positive People build a tribe of other positive people.
Positive People go out of their way.
Positive People are aware of boundaries and also put them in place.
In his post, he's talking about toxic people and then contrasts them with positive people.
We are to surround ourselves with other positive people.
I imagine there are at least 8 ways positive people help improve other people's outlook on life. They are probably more than eight… maybe you can add some in the comments.
We won't all be good at all eight!
There's every chance we won't be experienced or proficient at all eight – even if we are pursuing positivity. No worries. Optimists always know there's room for improvement.
We were also chosen to belong to him. God decided to choose us long ago in keeping with his plan. He works out everything to fit his plan and purpose.


Our Budget Camping Safaris in Tanzania provide the highest level of comfort.
4 Day Safari to Tarangire, Ngorongoro Crater and Lake Manyara National Park.
A wonderful way to relax on one of our safaris through Northern Tanzania's most famous national parks and conservation areas. We offer a great deal of safari itineraries to suit any schedule. All our safaris use well-maintained Toyota Land Cruisers or Land Rovers TDI, with roof hatches for unobstructed and up-close game-viewing. Our safari itineraries include all park fees, an English speaking driver/guide, meals and lodge accommodations. Price per person varies depending on group size.
Morning breakfast at East African hotel in Arusha/moshi, safari briefing and after wards we depart for Tarangire National Park. Your first experience of safari will be the thrilling game drive through the vastness of Tarangire National Park, arrive in good time for lunch. Afternoon game driving through the park followed by dinner and overnight at Lodge.
Today we visit the largest intact volcanic crater in the world, the magnificent Ngorongoro Crater. At more than 2,000 feet deep and 12 miles in diameter, the Ngorongoro Crater has one of the largest concentrations of wildlife in Africa. 30,000 animals make their home in this crater, making it an ideal place for sightings of the 'big 5' – lion, elephant, rhino, buffalo and leopard. We enjoy a picnic lunch in this truly world-class attraction, also known as the '8th Wonder of the World'. Our dinner and overnight stay is at the rustic Ngorongoro Crater Lodge.
With prior arrangement to the Airport.
Day1: Arusha/ moshi to Lake Manyara National Park.
We depart from your hotel in Arusha/moshi and proceed to Lake Manyara National Park. Although small in size, this park is one of the most diverse reserves in the country. Lake Manyara covers two thirds of the park. At the backdrop is the wall of the Great Rift Valley, before which lies the ground water forest, areas of open grassland near the lake foreshore, and the Soda Lake. We enjoy a picnic lunch in this area, which consists of open grassy areas, hot springs, dense woodlands and steep mountainsides. Lake Manyara National Park is a birding paradise that has more than 350 species of bird. The park is also famed for its unique and elusive tree-climbing lions. Our dinner and overnight stay is at the rustic Rhino Lodge, situated near the Ngorongoro Crater rim.
Day 2: Ngorongoro Crater via Olduvai Gorge to Serengeti National Park.
We proceed to Olduvai Gorge, known as the 'Cradle of Mankind'. It was here, in the Great Rift Valley, that the remains of early hominids dating back 2 million years were uncovered. Afterwards, we visit the most famous wildlife sanctuary in the world – Serengeti National Park. The word 'Serengeti' means 'endless plains', which describes the park's 5,600 square mile boundary. Yet for its sprawling area, the Serengeti does not disappoint when it comes to rich game-viewing. The Serengeti ecosystem is one of the oldest on earth. We make our way to Ikoma Bush Camp for dinner and overnight stay.
Day3: A full day game view in the Serengeti National Park.
We embark on an encore game drive in the expansive Serengeti National Park. The Serengeti boasts one of the most varied collections of wildlife on the planet. Annually, from May to December, millions of animals embark on a migration which brings them from central Serengeti to Kenya's Masai Mara and back again in an incredible spectacle. Even during the 'down-time', vast herds of wildebeest, zebra and gazelle roam here, along with groups of elephant, giraffe, and buffalo, as the lions, leopard, cheetah, hyenas and jackals stalk their meals. We return to Ikoma Bush Camp for dinner and overnight stay.
We depart from the Serengeti to visit the largest intact volcanic crater in the world, the magnificent Ngorongoro Crater. At more than 2,000 feet deep and 12 miles in diameter, the Ngorongoro Crater has one of the largest concentrations of wildlife in Africa. 30,000 animals make their home in this crater, making it an ideal place for sightings of the 'big 5' – lion, elephant, rhino, buffalo and leopard. This truly world-class attraction is also known as the '8th Wonder of the World'. Our dinner and overnight stay is at the rustic Rhino Lodge.
After breakfast, we depart the Ngorongoro Conservation Area and head back to your hotel in Arusha/ moshi. With prior arrangement, we can arrange a private transfer to Kilimanjaro International Airport (JRO) after the safari.
Early in the morning drives straight to the Serengeti arriving in time for an afternoon game drive. Catching the great migration in the Serengeti is a spectacular sight, and if you time your safari right, you might witness over a million wildebeest hustling together as the intensity of the migration grows, dinner and overnight at the Campsite.
Full day on game viewing and you will have both morning and afternoon game drives through the Serengeti exploring the wildlife in the great open plains, dinner and overnight at the Campsite.
After breakfast you leave Lake Manyara and drive to Tarangire. In the afternoon you'll be taken on a game drive through the park. Many of the animals congregate around the only source of permanent water in the park, the Tarangire River. This is the best place in Tanzania to see really big herds of elephant – up to 300 at a time. After lunch transfer to Arusha or Moshi and overnight, If you are departing this evening, we are happy to arrange a dayroom and airport transfer.
meet us at the airport and pick up with private transfer to Arusha/moshi.
we will gather in the evening for a pre-safari briefing and an equipment check. We will also confirm you have the appropriate mandatory medical coverage and travel insurance.
Day 3: NsiyaTented Lodge to Lake Manyara National Park.
Day 4: Lake Manyara to Serengeti National Park.
Arrive at Kilimanjaro airport – transfer to Tarangire Safari Lodge Located at Tarangire National Park, Dinner and stay for overnight.
Pick up after breakfast from Hotel which is inside the park, with your picnic lunch, start game drive in the park, afternoon drive to MtowaMbu; Twiga Campsite.
Night safari; Safari start after dinner around 9:00pm, can takes up to three hours or more.
Hot air Balloon ride Safari; Balloon Safari is probably the most beautiful balloon flight in the world and the ultimate Safari Experience which makes you enjoy the most beautiful migration of wildebeests through the endless plain of Serengeti.
About 5:00 am, Serengeti Balloon Safaris team will collect you from Ikoma Safari Lodge, and transfer Masai Kopjes. Here you will meet your pilot, receive a briefing and watch the inflation of your balloon.
Your pilot can precisely control the altitude of your balloon: sometimes flying at treetop height, sometimes lower, offering a unique perspective and great photographic opportunities of the wildlife below. After bush breakfast, you are taken to the rendezvous point to meet with your driver guide and continue with your day game drive.
Late in the afternoon, proceed driving to Ngorongoro. Arrive at Rhino lodgefor dinner and overnight.
After breakfast descends to the Ngorongoro Crater floor for gamedrive, The Ngorongoro Crater provides great game-viewing all year-round. The wildlife here makes up a permanent population because grazing and water supplies are plentiful throughout the seasons.
The only animal you won't see in here is a giraffe, whose long legs are unable to cope with the steeps sides of the crater.
8 Day Safari to Arusha National Park, Tarangire, Lake Manyara, Serengeti and Ngorongor Crater.
Arrive Kilimanjaro and transfer to Arusha / moshi. The rest of the day is free to relax and recover from your flight.
Day 2: Arusha National Park.
We depart from your hotel in Arusha/ moshi. We go through Arusha National Park, where we view the herds of buffalo in the spectacular Ngurdoto Crater, the flamingos of the Momela Lakes, and the imposing peak of rugged Mount Meru. Arusha National park houses a variety of wildlife, including giraffe, zebra, colobus monkeys, leopards and hyenas, in the park's small but bust ling 53 square miles of natural habitats. Enjoy a picnic lunch while gazing at the rolling hills and wooded savannahs of this national treasure. At the end of the day, we return to your hotel in Arusha/ moshi accomodation for dinner and overnight.
Day 3: Arusha/ moshi to Tarangire National Park.
Day 4: A full day game drive in the Lake Manyara National Park.
Day 5: Lake Manyara to Serengeti National Park.
Day 6: Another full day game drive in the Serengeti National Park.
A Serengeti balloon safari is a fantastic once in a lifetime experience, particularly suitable for a bird's eye-view game viewing experience withPlanet Guide SafariCo Ltd.
Hot air balloon rides leave from three sites in this prime national park where In Central Serengeti include theSeroneraRiver Valley where there is permanent water and the Great Migration passes around the said area through May and June.
At the Western Serengeti includes the Grummet River, where the Great migration crossingson June and July. In South Serengeti plains, the wildebeests assemble from December onwards, to calve in February.
The Balloon safari is particularly convenient, letting you get closer and distinctly accessible to the wild animals than you could be by a game-viewing safari vehicle.
Conditions are favorable, but you are advised to pre-book this popular activity for your safari holiday to Serengeti due to its limited seating availability. The balloons are immense; Baskets are compartmented for comfort and security capable to accommodating 16 people for each one-hour voyage. The price reflects the enormous cost of the balloons due to a safety replacement every few hundred hours, equivalent to swapping with a high-end automobile.
A skilled, experienced and licensed Pilot and staff of 20 are needed for each trip, including launching and recovery. Each flight is unique, some high enough to view the landscape for miles ahead. The Balloon safari will make your northern Tanzania vacation one among the most memorable of your life. It is possible to charter special private balloon flights in the Grummet Reserve for Honeymoon Couples on their romantic trip in the Great Plains of Africa.
Want to experience the Wild Africa in an authentic way? A camping safari is for you! A Budgeted camping safari packages are the most pocket friendly compared lodge safari packages or Luxury safari Packages. Some of our visitors choose to stay in tents in order to be close to nature and to the National Parks' beautiful landscapes.
Planet Guide safaris clients who choose this option will have their tents assembled for them each evening and collapsed again in the morning by our staff, with the tents being packed away and stored on the vehicle, for use again the following evening. Our Camping safaris are rewarding experiences because visitors feel like they're participating in a real African adventure, but still have all the services of safari drivers, guides, and a chef to take care of all their needs, while on camping safari you will have a chef who will be preparing delicious fresh foods for breakfast, Lunch and dinner.
Safety while camping inside the national parks with wild animals. Well be assured that it's very safe as we have National Park armed rangers patrolling and guarding the campsites at all times.
Leave Arusha to Ngorongoro for a half day Crater tour then proceed to Nainokanoka village where you set your camp. Walk to Olmoti Crater and Munge Waterfalls, in the afternoon you'll return to the camp for dinner and overnight.
Hike through the bushland and forest to the rim of the Empakaai crater where you set up your camp. Walk to the crater to see the volcanic lake, dense forest, different bird species and wild animals like buffalos, waterbucks, reedbucks. Return to the rim of the crater for dinner and overnight.
Day 5: Naiyobi village-Muonwongoni village.
After breakfast we head to Tarangire National Park, where we reach around noon. We pitch our tents at a campsite in a remote area of the park. First experience of an African landscape we get on an afternoon walk close to the park border. Dinner and overnight at Campsite.
Full day Walking Safaris guided by an armed ranger and game drives in the park. The park owns a rich bird life, depending on season you can see elephants, zebras, wildebeest and even lions. A visit to Tarangire, whatever the season, is full of experience. Dinner and overnight at Lake Manyara Campsite.
Today we visit the largest intact volcanic crater in the world, the magnificent Ngorongoro Crater. At more than 2,000 feet deep and 12 miles in diameter, the Ngorongoro Crater has one of the largest concentrations of wildlife in Africa. 30,000 animals make their home in this crater, making it an ideal place for sightings of the 'big 5' – lion, elephant, rhino, buffalo and leopard. We enjoy a picnic lunch in this truly world-class attraction, also known as the '8th Wonder of the World'. Our dinner and overnight stay is at the rustic on top of Ngorongoro Crater rim Campsite.
After breakfast depart to Olmoti Crater for a half day walk. Often we meet Maasai on the way to the water falls at the crater.
At afternoon we proceed to our base camp at Empakaai Crater (Special Campsite).
Experiencing the serenity and quiet beauty of this wild spot. Surrounding the lake you will reach springs, where you frequently see lots of wildlife. In the lake there are many flamingoes. Overnight at the special Campsite, Empakaai.
Day 6: Empakaai to Lake Natron.
Today we start walking from Empakaai to Lake Natron. We descend the outer hills of the Great Rift Valley; we walk through the open plains of the Embulbul depression, passing through Naiobi village. We have also the great opportunity to meet the local Maasai.
OlDoinyoLengai and Mount Meru are seen along the way. Donkeys carry our luggage.
Afterward we proceed to the Acacia Campsite for dinner and overnight stay.
Today we walk to OldonyoLengai, the single active volcano mountain in Tanzania this means (the holy mountain of the Maasai). The trail will lead us down into the Rift Valley escarpment, and most of the parts are quite steep. After 5-6 hours of walking we reach the foot of the inspiring mountain. Here you can decide if you want to camp at the ChiniYaMlima campsite, or want to proceed to the Engaresero campsite, which is much more comfortable (showers!) and less hot. We can also pick you up by our car at the mountain, and drive back to the Engaresero Campsite.
Day 8: Lake Natron to Arusha.
Afternoon lunch at Moivaro Lodge. Drive back to Kilimanjaro International airport (JRO) ready for the evening flight.
Chimpanzees share about 98% of their genes with humans and no scientific expertise is required to distinguish between the individual repertoires of pants, hoots and screams that define the celebrities, the powerbrokers, and the supporting characters. Perhaps you will see a flicker of understanding when you look into a chimp's eyes, assessing you in return – a look of apparent recognition across the narrowest of species barriers.
The bulk of Katavi supports a hypnotically featureless cover of tangled brachystegia woodland, home to substantial but elusive populations of the localized eland, sable and roan antelopes. But the main focus for game viewing within the park is the Katuma River and associated floodplains such as the seasonal Lakes Katavi and Chada. During the rainy season, these lush, marshy lakes are a haven for myriad water birds. Seasons define much of the park's ecohydrology: while Lake Chada and Lake Katavi are grasslands during the dry season, they transform into shallow lakes with the onset of heavy rains during the rainy season (October to April). The average rainfall is approximately 930 mm and follows a bimodal pattern with short rains in slight low in February. The vital lifeline of the Park is the Katumariver which feeds Lake Katavi in the north and Lake Chada in the center as well as the huge Katisunga floodplain (425 km²). In recent years, this river as well as the Kapapa and Ngolima rivers, which feed lakeChada, tend to dry out earlier due to illegal damming upstream outside the national park.
Lake Manyara National Park is another of Tanzania's small parks with an area of not more than 330square kilometers. The beautiful nature, the varied flora and the exiting fauna make Lake Manyara an obvious place to spend a day or so on the way to Serengeti and Ngorongoro. Lake Manyara is situated in the bottom of the Great Rift Valley, which stretches all the way from Jordan to Mozambique, and it has an intriguing ecology and a greatly varied bird life, perfect for bird watching. 380 species have been registered, and at most times of the year, thousands of flamingoes settle down, feeding on algae as a pink ribbon along the lake.
The vegetation is also very unique, with groundwater forest with high fig and mahogany trees, acacia forest, marsh area and grassland, as well as hot springs in the south. The park is especially famous for its lions, which have a habit of climbing the trees. Except from in the Ruwenzori Park in Uganda, the lions in Lake Manyara are the only ones who have this peculiar behavior, which is inherited from generation to generation. Zoologists believe that they have attributed this habit to avoid the biting flies and the heat at the ground. You are lucky if you spot these lions, though.
But there are good chances of spotting elephants. Lake Manyara used to have the largest concentration in Africa, but because of poaching, the number of animals has decreased. You are also sure find hippos, buffaloes, giraffes, zebras, antelopes, gazelles and a lot of baboons. Lake Manyara is about 120 km (1.5-2 hours) from Arusha on the way to Ngorongoro and Serengeti. One can also take a 1-hour flight from Arusha. This park can be visited year round.
Although comparatively small in area, Mikumi is rich in wildlife and contains buffalo, elephant and lions. There are also wildebeests, zebras, impalas, wart hogs, hippos, cheetahs and giraffes. Most of the animals can be seen at all seasons within a short drive of Mikumi Wildlife Lodge and in the vicinity of the Hippo Pool and Mikumi Wildlife Camp. Elephants can be seen throughout the area, and buffalo totaling some 3,000 are to be found on the treeless grassland. A family of hippo lives at Hippo Pool 5km from the park gate.
Covering approximately 2,600 square kilometers, Tarangire lies to the south of large open grass plains of the southern Maasai land. Tarangire National Park is named after the Tarangire River, where great flocks of animals gather together in the dry season from June to November. This is the best time to visit Tarangire. Most of the other waterholes are dried out, so the flood is the only place where the animals can extinguish their thirst. After the short rainy season in November, the animals wander in big flocks of wildebeests, zebras, gazelles and hartebeests, outside the borders of the park – some wander off to Lake Manyara, others all the way to Lake Natron close to the border to Kenya. First when the rainy season has passed, the animals turn back to Tarangire, where they arrive during the summer.
Several circuits are possible, Burungi to the west, Lamarkau to the south and Matete along the river. Acacia and baobab trees shelter animals seeking refuge near the river. The park is especially known for its elephants, which are more numerous than in the other northern parks. Apart from this, there are good chances of spotting buffaloes, giraffes, waterbucks, kudus, Oryx, lions, leopards, hyenas, and if you are lucky, you'll even spot the famous tree climbing pythons! The park has a varied landscape consisting of open grassland with acacia trees and baobab trees, bush steppe, forests and swamps. In the whole area there are tsetse flies, which make the area unsuitable for cattle holding. This explains why Tarangire today has a status as a National Park. Tarangire is not far away from the tarred main road between Arusha and Dodoma. The distance from Arusha to the gate of Tarangire is 115 km, or about 1 hour driving.
In the south of Tanzania, stretching inland from the coast, lays Africa's largest wildlife sanctuary, the Selous Game Reserve – bigger than Denmark. It lies south of Mikumi, covers 6% of Tanzania's total area and is four times the size of Serengeti. It contains the greatest concentration of big game left on earth. Selous game reserve is internationally famous for its animals. It has the world's largest number of big game: – elephant 28,000, buffaloes 160,000, hippos 2,000, and sable antelopes 14,000. There are about 350 species of birdlife which include: – Fish eagle, secretary bird, king fisher, sun bird, herms, bill stork and hammer kop.
The climate is tropical, with two rainy seasons; from January – March it is hot with occasional showers. The heaviest rains are between March – May. The best time to visit is from June – October when the weather is cool and dry, the vegetation lush and the animals easy to see.
Selous has the finest virgin bush, unspoiled by time. Although it remains one of the least scientifically researched areas in Africa, some 1700 botanical species have been identified. In the far south, the vastness of the reserve and its general inaccessibility has turned the Selous into a magnificent refuge for animals, birds, insects and reptiles.
Tanzania is one of the World's best places for wildlife lovers hoping to see the Big Five. It's also a fantastic destination for bird lovers with a myriad of colorful species. The Serengeti, Ngorongoro Crater, Lake Manyara and Tarangire national parks are all within easy reach of Kilimanjaro Airport and can be combined into a fantastic safari at the end of a Mountain Kilimanjaro climb.
We have shared some of the itineraries that previous customers have loved but we can arrange anything you can imagine. So go on, spoil yourself and do it your way!


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Locations > Instructors > Tanner McAteer
About Tanner McAteer
Tanner McAteer
Instructs: Bass Guitar, Drums, Guitar, Music Theory, Ukulele
Education: Tanner received his B.A. in General Music studies at the University of North Carolina in Wilmington in May of 2020, and his M.M. in Guitar Performance at Appalachian State University in August of 2022.
Performance Experience: Tanner began performing in blues and rock groups in 2008, and continues to do so today. He has also performed as a solo classical guitarist at UNCW and Appalachian State performance seminars, and at Cameron Art Museum in Wilmington and the Raleigh Museum of Art. During his time at UNCW, he sang in both the chamber and concert choirs.
Teaching Experience: Tanner has been a teacher of guitar, bass. ukulele, drums. voice and harmonica since 2012. He has taught classical and jazz guitar at the University of North Carolina in Wilmington from September of 2017 until May of 2020 and at Appalachian State University from August of 2020 until May of 2022.
Teaching Philosophy: "I believe in meeting students where they are and working with them to discover and achieve their unique goals as musicians of any skill level."
Get to know Tanner McAteer
Tanner McAteer is a multi-instrumentalist teacher and performer of several genres and styles of music including rock, pop, jazz, classical, blues and bluegrass. Based in Fayetteville North Carolina, he has been teaching, writing and performing music since 2012. He completed his M.M. in guitar performance at Appalachian State University in August of 2022.
Tanner McAteer teaches at:
Fayetteville NC
5075 Morganton Rd Unit 10B
With over 150 locations nationwide, Music & Arts is the nation's leading provider of private music lessons, band and orchestra instrument sales, rentals and repairs. Our remarkable story of growth has everything to do with the countless unshakable relationships we've formed with students, parents, music educators and musicians of every stripe in the local communities we serve across the country. Since opening our first store back in 1952, we've been owned and operated by the same family, and there's nothing we do as a company without our values in mind. For whatever you're doing in music, we're here to help. We hope you visit our Fayetteville location soon! Our friendly staff can't wait to meet you.


Encouraging employee development might seem like a waste of time. After all, you want them to be productive and doing their job. And whilst of course it is important to remain productive, employee development is equally as important. A recent study suggests that 40% of people who receive poor training leave within the first year of their employment. Encouraging employee development can actually increase your staff retention, as well as staff productivity, giving your staff the right knowledge and tools to work more effectively.
If you're looking for ways to upskill your workforce, here are some tips on how to set them up with a robust employee development plan.
Setting goals from the start puts our employees on a clear path of progression with clear goals to achieve. It allows for more focus in areas of development. Sit down with your employees and work out what areas they want to develop their skills in and how they can do that.
Knowledge is sharing. And sharing of knowledge at work can create a happier, close knit group sharing their newly developed skills. It also enables your employees to talk, get advice, tips and share ideas; building a strong community or productive workers.
Recognising the efforts and accomplishments of your team can go a long way in making your employees feel valued. It can also show that their recent training has tangible results in their day-to-day work. If an employee is doing well think about how this could be rewarded with growth within the company.
How are your employees coping with their development and their work? Are they still on course? What else can they do to achieve their goals (see first point)? Regular catch ups with your employees and reviews of their progress keeps them on the right path whilst also showing them you're invested in their futures.
Encouraging skill development doesn't have to be costly, and doesn't mean employees are quicker to leave once they've gained vital skills either. It can mean upskilled, loyal staff who are more productive. It can also mean your staff is invested in their roles and their future careers. See how you can encourage employee development today.


Jason began playing the piano at age 3, was in concert band on percussion from 3rd grade through high school, and studied voice as a music major for five years in college. In his free time he composes, records and produces his own music. He's been performing professionally since he was 21. His biggest claim to fame, before Ajammination, was his role as the multi-instrumentalist for the infamous "Yasgur's Farm" in which he has played for many years. Some influences include Radiohead, Tori Amos, and Sting, as well as Impressionist composers Ravel, Debussy, and Stravinsky. To this day, he has never had a day job and, God willing, never will!!!