<p>I give below just an outline to orient oneself in the context of Pontryagin’s Principle, answering briefly your questions.</p>
<p>First of all, it must be noticed that what is called the <em><strong>Pontryagin's Maximum Principle</strong></em> is actually <strong>not a single theorem, but a collection of theorems,</strong> that differ according to different assumptions.</p>
<p>In particular, various cases of the Maximum Principle exist according to <strong>the interval of time under consideration,</strong>  which can be <em>finite</em> or <em>infinite</em> (the second extreme of integration becoming <span class="math-container">$\infty$</span>), and the <strong>characteristics of the boundary conditions</strong> we have.</p>
<p>In your case you have a <em><strong>finite time problem</strong></em>, and as for boundary conditions a distinction must be made between <strong><em>fixed end</em> problems</strong> and <strong><em>moving end</em> problems</strong>.
<span class="math-container">$$\;$$</span></p>
<blockquote>
<p>In the Hamiltonian equations for the state variables <strong>3.1</strong>, where
did the second equation come from?</p>
</blockquote>
<p>Are you sure there aren’t typos in your formulas? In <span class="math-container">$(3.1)$</span>, the second equation should be indeed <span class="math-container">$\dot{x}_i = \frac{\partial \mathcal{H}}{\partial \lambda_i}\; i=1,...,n$</span>.</p>
<p>In <span class="math-container">$(3.1)$</span> the first equation relates to the case there is only one constraint <span class="math-container">$g$</span>, that is <span class="math-container">$n=1$</span>, the second equation to the case we have <span class="math-container">$n&gt;1$</span> constraints <span class="math-container">$g_i$</span>.
<span class="math-container">$$\;$$</span></p>
<blockquote>
<p>In equation <strong>5.1</strong>, what is the <em>transversality condition</em>, and
when/how is it applied? Why is it not necessary when boundary
conditions are present, as shown in <strong>5.2</strong>?</p>
</blockquote>
<p><strong>Transversality conditions arise only in moving ends problems (and also in infinite time problems) and are absent in fixed end problems:</strong> they are present in the so-called <em>moving-end problems</em>, that is <strong>when the value of your <span class="math-container">$x(t)$</span> at the final time <span class="math-container">$t_f$</span> is not fixed</strong>, and they are not present in the <em>fixed-end problems</em>, when <strong>you know the final value of <span class="math-container">$x(t)$</span></strong>, as in <span class="math-container">$(5.2)$</span>.</p>
<p>Transversality conditions are <strong>mathematical conditions for a maximum,</strong> necessary conditions in the case of moving ends.</p>
<p>Their explanation is not trivial, as all proofs of the maximum principle. They have a geometrical meaning, they arise when the arrival of the trajectory is a so called <em>manifold</em>, think of it as a surface in <span class="math-container">$\mathbb{R}^n$</span>, and not a point.</p>
<p>In a sense, yes, from a practical point of view,  they are used as one uses  boundary conditions, but they are actual optimum conditions.</p>
<p>Proofs of the Maximum Principle are complex, consider that the proof of the main theorem in Pontryagin's original book<span class="math-container">$^1$</span> takes forty pages, the whole second chapter.
<span class="math-container">$$\;$$</span></p>
<blockquote>
<p>In the Hamiltonian equations for the costate variables <strong>3.2</strong>, where
does <span class="math-container">$\lambda_f$</span> come from? When is it used?</p>
</blockquote>
<p>The equation <span class="math-container">$\lambda_f = \frac{\partial \mathcal{H}}{\partial x_f} S(x_f)$</span>
in <span class="math-container">$(3.2)$</span> arises in the so-called <em><strong>Bolza Problem</strong></em>, where the functional to be minimized is the sum of an integral and a function depending on the boundary, in your case  (eq.<span class="math-container">$1.1$</span>) a function <span class="math-container">$S(x_f)$</span> of the final value <span class="math-container">$x_f$</span> (remember that in this problem the final value <span class="math-container">$x_f$</span> is a variable, it is not fixed).</p>
<p>It is a problem of Calculus of Variation, that has been dealt with the Pontryagin’s Maximum Principle, see<br />
<a href="https://encyclopediaofmath.org/wiki/Bolza_problem" rel="nofollow noreferrer">https://encyclopediaofmath.org/wiki/Bolza_problem</a></p>
<p><span class="math-container">$$\;$$</span></p>
<blockquote>
<p>In equations <strong>4.1</strong> and <strong>4.2</strong>, what does <span class="math-container">$m$</span> stand for? I
understand from my professors lectures that it means <em>multiplicity</em></p>
</blockquote>
<p>As for <span class="math-container">$m$</span> in <span class="math-container">$(4.1)$</span> and <span class="math-container">$(4.2$</span>) are you sure that it is not <span class="math-container">$n$</span> as in <span class="math-container">$(3.1)$</span> ? I see a gradient for <span class="math-container">$m&gt;1$</span> and a partial derivative for <span class="math-container">$m=1$</span>.</p>
<hr />
<p><span class="math-container">$^1$</span> <sup> Pontryagin L.S. and others, <em>The Mathematical Theory of Optimal Processes</em>, Pergamon Press, 1964.</sup></p>

CC BY-SA 4.0
2024-01-25T17:41:28.317Z
thank you! Your help has been crucial in terms of understanding this part of my course's curriculum.
astute-hoplite
CC BY-SA 4.0
2024-01-25T17:43:15.547Z
I'm happy having ben useful! Pontryagin maximum principle is not an easy subject.
BakerStreet
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2024-01-26T04:15:05.183Z
@BakerStreet :Would you mind mentioning ( should I start a new thread ? ) what texts you used that helped lead to your knowledge. Your answer above is MUCH, MUCH appreciated. I've never seen such an eloquent "seeing the forest through the trees" explanation of this material. Dorfman also wrote a nice, intro paper on the hamiltonian from an economic point of view. If I can find a free version, I'll send a link.
mark leeds
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2024-01-26T04:23:20.923Z
There was a free version of this somewhere at some point but now I can only find it on JSTOR. https://www.jstor.org/stable/1810679   Note that I mis-stated the title. It's "optimal control theory from an economic point of view" rather than the Hamiltonian. But there is material regarding the Hamiltonian in the paper.
mark leeds
CC BY-SA 4.0
2024-01-26T04:25:50.987Z
BakerStreet Note that I have the reference by L.S. Pontryagin but I found it not readable ( for me anyway ) so I was hoping that you'd mention something else. Thanks.
mark leeds
CC BY-SA 4.0
2024-01-26T10:06:17.100Z
@mark leeds,   I didn't mention Pontryagin as refererence, but only to say, indeed,  that  the proofs are complicated. A good illustration of the Principle in economics is in Takayama, *Analitical Methods in Economics*. If you want a pure mathematical approach, there are many books of control theory, many are advanced. A not advanced book, readable, is Liberzon, *Calculus of Variations and Optimal Control Theory: A Concise Introduction*., Priceton Univ. Press, 2012
BakerStreet
CC BY-SA 4.0
2024-01-26T10:08:53.847Z
@mark leeds Thank you very much for your appreciation of my answer! And thank you for the reference of Dorfman, I will read it!
BakerStreet
CC BY-SA 4.0
2024-01-26T13:43:35.260Z
Thanks BakerStreet. I have Liberzon in storage so I will someday read that. I don't have Takayama so I will check it out. Yes, Pontryagin's text is DIFFICULT. Thank you again for your great answer and I hope you like Dorfman's paper. It's basic but I found it to be a nice intro and way of looking at things.
mark leeds
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2024-01-26T13:51:03.623Z
You are welcome, and thank you again for your interesting reference to Dorfman.
BakerStreet
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2024-01-27T10:19:33.977Z
you're welcome bakerstreet. all of your nice answers are appreciated.
mark leeds
CC BY-SA 4.0
2024-01-27T11:32:37.867Z
You are really very kind mark leeds.
BakerStreet
CC BY-SA 4.0
2024-01-25T15:19:51.123Z
2024-01-25T15:37:00.690Z
2024-01-25T15:37:00.690Z
BakerStreet
<p>I am really struggling to grasp how the Hamiltonian Function and Pontryagin's Maximum Principle work in the context of Optimal Control Theory (Maths for Economics) course. I am given the following conditions:</p>
<p><span class="math-container">$$
\quad\space\max_{u} \mathcal{F} = \int_{t_0}^{t_f} f(x(t), u(t), t) \, dt + S(x_f) \tag{1.1}
$$</span></p>
<p><span class="math-container">$$
\begin{cases}
\begin{array}{l}
  \dot x(t) = g(x(t),u(t),t) \\
  x(t_i)=x_i \\
  x(t_f)=x_f \\
\end{array}
\end{cases}
\qquad
\begin{array}{l}
\text(constraints) \\
\text(known) \\
\text(unknown) \\
\end{array} 
\tag{1.2}
$$</span></p>
<p><span class="math-container">$$
\begin{cases}
\begin{array}{l}
  \overline{x} = (x_1,x_2,\dots,x_n) \\
  \overline{u} = (u_1,u_2,\dots,u_n) \\
  \overline{g} = (g_1,g_2,\dots,g_n) \\
\end{array}
\end{cases}
\qquad\quad
\begin{array}{l}
\text(state \space variables) \\
\text(control \space variables) \\
\text(constraint \space functions) \\
\end{array} 
\tag{1.3}
$$</span></p>
<p>So far I understand that we construct the <em>Hamiltonian</em> <span class="math-container">$\mathcal{H}$</span> from the <em>Hamiltonian Functional</em> <span class="math-container">$\mathcal{L}$</span> and the <em>costate functions</em> <span class="math-container">$\lambda$</span> such that:</p>
<p><span class="math-container">$$
\mathcal{H}(x,u,t,\lambda) = f(x,u,t) + {\lambda}(g(x,u,t)) \tag{2.1}
$$</span></p>
<p>Then, we find the Hamiltonian equations for the <em>state</em> <span class="math-container">$x$</span> and <em>costate</em> <span class="math-container">$\lambda$</span> variables:</p>
<p><span class="math-container">$$
\begin{cases}
\dot{x} = \frac{\partial \mathcal{H}}{\partial \lambda} \\
\dot{x}_i = \frac{\partial \mathcal{H}}{\partial x_i} \quad \text{for} \quad n &gt; 1
\end{cases}
\tag{3.1}
$$</span></p>
<p><span class="math-container">$$
\begin{cases}
\dot{\lambda} = -\frac{\partial \mathcal{H}}{\partial x} \\
\dot{\lambda}_i = -\frac{\partial \mathcal{H}}{\partial x_i} \quad \text{for} \quad n &gt; 1 \\
\lambda_f = \frac{\partial \mathcal{H}}{\partial x_f} S(x_f)
\end{cases}
\tag{3.2}
$$</span></p>
<p>We are then instructed to solve using Pontryagin's Maximum Principle:</p>
<p><span class="math-container">$$
\text{If } m = 1 \begin{cases}
\frac{\partial \mathcal{H}}{\partial u} = 0\\
\frac{\partial^2 \mathcal{H}}{\partial u^2} &lt; 0 \\
\end{cases}
\tag{4.1}
$$</span></p>
<p><span class="math-container">$$
\text{If } m &gt; 1 \begin{cases}
\nabla \mathcal{H} = \overline{0} \\
\nabla^2 \mathcal{H} &lt; 0 \quad \text{for} \quad \text{odd } |M_H| \\
\nabla^2 \mathcal{H} &gt; 0 \quad \text{for} \quad \text{even } |M_H| \\
\end{cases}
\tag{4.2}
$$</span></p>
<p>Finally, we are told that if we only have the initial conditions <span class="math-container">$x(t) = x_i$</span>, then:</p>
<p><span class="math-container">$$
\begin{cases}
x(t_i) = x_i \\
\lambda(t_f) = \frac{\partial \mathcal{H}}{\partial x_f} S(x_f) \quad \text{(transversality condition)} \\
\end{cases}
\tag{5.1}
$$</span></p>
<p>While, if we have both initial and boundary conditions, then:</p>
<p><span class="math-container">$$
\begin{cases}
x(t_i) = x_i \\
x(t_f) = x_f \\
\end{cases}
\tag{5.2}
$$</span></p>
<p>I fail to understand the following points:</p>
<ul>
<li><p>In the Hamiltonian equations for the state variables <strong>3.1</strong>, where did the second equation come from and how did it go from being the derivative of <span class="math-container">$\mathcal{H}$</span> with respect to <span class="math-container">$\lambda$</span> to being the derivative of <span class="math-container">$\mathcal{H}$</span> with respect to <span class="math-container">$\mathcal{x_i}$</span>? Shoulnd't it derivate with respect to <span class="math-container">$\lambda$</span> as well?</p>
</li>
<li><p>In the Hamiltonian equations for the costate variables <strong>3.2</strong>, where does <span class="math-container">$\lambda_f$</span> come from? When is it used?</p>
</li>
<li><p>In equations <strong>4.1</strong> and <strong>4.2</strong>, what does <span class="math-container">$m$</span> stand for? I understand from my professors lectures that it means <em>multiplicity</em>, but in a different sense than when used, for example, to talk about eigenvalues in the context of dynamical systems. What, then, is it alluding to?</p>
</li>
<li><p>In equation <strong>5.1</strong>, what is the <em>transversality condition</em>, and when/how is it applied? Why is it not necessary when boundary conditions are present, as shown in <strong>5.2</strong>?</p>
</li>
</ul>
<p>Overall, I am currently unable to solve this kind of problem. Any help  understanding how to approach it would be greatly appreciated. As a quick sidenote, any appropiate criticism on the formatting of this post and/or my usage of LaTex would also be appreciated.</p>

CC BY-SA 4.0
2024-01-24T05:19:28.653Z
maybe you could send the question to math.stackexchange. someone over there might be happy to answer.
mark leeds
CC BY-SA 4.0
2024-01-24T12:36:33.200Z
Are you sure there aren’t typos in your formulas? In  $(3.1)$, the second equation should be indeed  
$\dot{x}_i = \frac{\partial \mathcal{H}}{\partial \lambda_i},\; i=1,...,n$. In $(3.1)$ the first equation relates to the case  there is only one constraint $g$, that is $n=1$, the second equation
 to the case  we have $n>1$ constraints $g_i$.
BakerStreet
CC BY-SA 4.0
2024-01-24T12:38:35.197Z
As for transversality conditions, they are present in the so-called *moving-end* problems, that is when the value of your $x(t)$ at the final time $t_f$  is not fixed, and they are not present in the *fixed-end* problems, when you know the final value of $x(t)$ as in  $(5.2)$.
BakerStreet
CC BY-SA 4.0
2024-01-25T06:19:32.260Z
@BakerStreet, I will leave the typos in order for the comments to still make sense, but thank you for your clarification and for your explanation of transversality conditions. From what I understand they are used when solving the same way one would use $x(t_f)$?
astute-hoplite
CC BY-SA 4.0
2024-01-25T06:22:59.063Z
@markleeds, I have sent the question there as well, haven't gotten any traction yet, though.
astute-hoplite
CC BY-SA 4.0
2024-01-25T07:51:03.717Z
@  astute-hoplite Transversality conditions arise in moving ends problems (and also in infinite time problems) and they are mathematical conditions for a maximum, necessary conditions in the case of moving ends. Their explanation is not trivial, as all proofs of the maximum principle. They have  a geometrical meaning, they arise when the arrival of the trajectory is a so called *manifold*, think of it as a surface in $\mathbb{R}^n$, and not a point.  In a sense, from a pratical point of view, yes, they are used as one uses a boundary conditions, but they are actual optimum conditions.
BakerStreet
CC BY-SA 4.0
2024-01-25T07:54:57.400Z
Proofs of the Maximum Principle are complex, consider that the proof of the main theorem in Pontryagin's original book takes forty page, the whole second chapter.
BakerStreet
CC BY-SA 4.0
2024-01-25T08:20:11.537Z
As for $m$
in $(4.1)$ and $(4.2)$ are you sure that it is not $n$ as in $(3.1)$ ? I see a gradient for $m>1$ and a partial derivative for $m=1$.
BakerStreet
CC BY-SA 4.0
2024-01-25T13:50:36.027Z
The equation $ \lambda_f = \frac{\partial \mathcal{H}}{\partial x_f} S(x_f)$ in $(3.2)$ arises in the so-called Bolza Problem, where the functional to be minimized is the sum of an integral and a function depending on the boundary, in your case the final value $x_f$. It is a problem of Calculus of Variation, that has been dealt with the Pontryagin’s Maximum Principle,  see https://encyclopediaofmath.org/wiki/Bolza_problem
BakerStreet
CC BY-SA 4.0
2024-01-25T15:21:52.293Z
I synthetise my comments in an answer, for clarity, and because this way your question has an answer.
BakerStreet
CC BY-SA 4.0
2024-01-25T17:38:29.883Z
@BakerStreet, yes, it seems multiplicity refers to the number of constraint functions. Thank you very much for your help, I am now aware that my note's and those provided by my lecturer contain many errors.
astute-hoplite
CC BY-SA 4.0
2024-01-25T17:41:05.983Z
You are welcome!
BakerStreet
<mathematical-economics><optimal-control><hamiltonian>
How can I apply the Hamiltonian function and Pontryagin's maximum principle in the context of Optimal Control Theory?
astute-hoplite
# How can I apply the Hamiltonian function and Pontryagin's maximum principle in the context of Optimal Control Theory?

(Asked by: username_0 on 2024-01-22)

I am really struggling to grasp how the Hamiltonian Function and Pontryagin's Maximum Principle work in the context of Optimal Control Theory (Maths for Economics) course. I am given the following conditions:

$$
\quad\space\max\_{u} \mathcal{F} = \int\_{t\_0}^{t\_f} f(x(t), u(t), t) \, dt + S(x\_f) \tag{1.1}
$$

$$
\begin{cases}
\begin{array}{l}
\dot x(t) = g(x(t),u(t),t) \\
x(t\_i)=x\_i \\
x(t\_f)=x\_f \\
\end{array}
\end{cases}
\qquad
\begin{array}{l}
\text(constraints) \\
\text(known) \\
\text(unknown) \\
\end{array}
\tag{1.2}
$$

$$
\begin{cases}
\begin{array}{l}
\overline{x} = (x\_1,x\_2,\dots,x\_n) \\
\overline{u} = (u\_1,u\_2,\dots,u\_n) \\
\overline{g} = (g\_1,g\_2,\dots,g\_n) \\
\end{array}
\end{cases}
\qquad\quad
\begin{array}{l}
\text(state \space variables) \\
\text(control \space variables) \\
\text(constraint \space functions) \\
\end{array}
\tag{1.3}
$$

So far I understand that we construct the *Hamiltonian* $\mathcal{H}$ from the *Hamiltonian Functional* $\mathcal{L}$ and the *costate functions* $\lambda$ such that:

$$
\mathcal{H}(x,u,t,\lambda) = f(x,u,t) + {\lambda}(g(x,u,t)) \tag{2.1}
$$

Then, we find the Hamiltonian equations for the *state* $x$ and *costate* $\lambda$ variables:

$$
\begin{cases}
\dot{x} = \frac{\partial \mathcal{H}}{\partial \lambda} \\
\dot{x}\_i = \frac{\partial \mathcal{H}}{\partial x\_i} \quad \text{for} \quad n > 1
\end{cases}
\tag{3.1}
$$

$$
\begin{cases}
\dot{\lambda} = -\frac{\partial \mathcal{H}}{\partial x} \\
\dot{\lambda}\_i = -\frac{\partial \mathcal{H}}{\partial x\_i} \quad \text{for} \quad n > 1 \\
\lambda\_f = \frac{\partial \mathcal{H}}{\partial x\_f} S(x\_f)
\end{cases}
\tag{3.2}
$$

We are then instructed to solve using Pontryagin's Maximum Principle:

$$
\text{If } m = 1 \begin{cases}
\frac{\partial \mathcal{H}}{\partial u} = 0\\
\frac{\partial^2 \mathcal{H}}{\partial u^2} < 0 \\
\end{cases}
\tag{4.1}
$$

$$
\text{If } m > 1 \begin{cases}
\nabla \mathcal{H} = \overline{0} \\
\nabla^2 \mathcal{H} < 0 \quad \text{for} \quad \text{odd } |M\_H| \\
\nabla^2 \mathcal{H} > 0 \quad \text{for} \quad \text{even } |M\_H| \\
\end{cases}
\tag{4.2}
$$

Finally, we are told that if we only have the initial conditions $x(t) = x\_i$, then:

$$
\begin{cases}
x(t\_i) = x\_i \\
\lambda(t\_f) = \frac{\partial \mathcal{H}}{\partial x\_f} S(x\_f) \quad \text{(transversality condition)} \\
\end{cases}
\tag{5.1}
$$

While, if we have both initial and boundary conditions, then:

$$
\begin{cases}
x(t\_i) = x\_i \\
x(t\_f) = x\_f \\
\end{cases}
\tag{5.2}
$$

I fail to understand the following points:

* In the Hamiltonian equations for the state variables **3.1**, where did the second equation come from and how did it go from being the derivative of $\mathcal{H}$ with respect to $\lambda$ to being the derivative of $\mathcal{H}$ with respect to $\mathcal{x\_i}$? Shoulnd't it derivate with respect to $\lambda$ as well?
* In the Hamiltonian equations for the costate variables **3.2**, where does $\lambda\_f$ come from? When is it used?
* In equations **4.1** and **4.2**, what does $m$ stand for? I understand from my professors lectures that it means *multiplicity*, but in a different sense than when used, for example, to talk about eigenvalues in the context of dynamical systems. What, then, is it alluding to?
* In equation **5.1**, what is the *transversality condition*, and when/how is it applied? Why is it not necessary when boundary conditions are present, as shown in **5.2**?

Overall, I am currently unable to solve this kind of problem. Any help understanding how to approach it would be greatly appreciated. As a quick sidenote, any appropiate criticism on the formatting of this post and/or my usage of LaTex would also be appreciated.

**Tags:** mathematical-economics, optimal-control, hamiltonian

> **Comments:**
> **username_1**: maybe you could send the question to math.stackexchange. someone over there might be happy to answer.  
> **username_2**: Are you sure there aren’t typos in your formulas? In  $(3.1)$, the second equation should be indeed   $\dot{x}_i = \frac{\partial \mathcal{H}}{\partial \lambda_i},\; i=1,...,n$. In $(3.1)$ the first equation relates to the case  there is only one constraint $g$, that is $n=1$, the second equation  to the case  we have $n>1$ constraints $g_i$.  
> **username_2**: As for transversality conditions, they are present in the so-called *moving-end* problems, that is when the value of your $x(t)$ at the final time $t_f$  is not fixed, and they are not present in the *fixed-end* problems, when you know the final value of $x(t)$ as in  $(5.2)$.  
> **username_0**: @username_2, I will leave the typos in order for the comments to still make sense, but thank you for your clarification and for your explanation of transversality conditions. From what I understand they are used when solving the same way one would use $x(t_f)$?  
> **username_0**: @username_1, I have sent the question there as well, haven't gotten any traction yet, though.  
> **username_2**: @  username_0 Transversality conditions arise in moving ends problems (and also in infinite time problems) and they are mathematical conditions for a maximum, necessary conditions in the case of moving ends. Their explanation is not trivial, as all proofs of the maximum principle. They have  a geometrical meaning, they arise when the arrival of the trajectory is a so called *manifold*, think of it as a surface in $\mathbb{R}^n$, and not a point.  In a sense, from a pratical point of view, yes, they are used as one uses a boundary conditions, but they are actual optimum conditions.  
> **username_2**: Proofs of the Maximum Principle are complex, consider that the proof of the main theorem in Pontryagin's original book takes forty page, the whole second chapter.  
> **username_2**: As for $m$ in $(4.1)$ and $(4.2)$ are you sure that it is not $n$ as in $(3.1)$ ? I see a gradient for $m>1$ and a partial derivative for $m=1$.  
> **username_2**: The equation $ \lambda_f = \frac{\partial \mathcal{H}}{\partial x_f} S(x_f)$ in $(3.2)$ arises in the so-called Bolza Problem, where the functional to be minimized is the sum of an integral and a function depending on the boundary, in your case the final value $x_f$. It is a problem of Calculus of Variation, that has been dealt with the Pontryagin’s Maximum Principle,  see https://encyclopediaofmath.org/wiki/Bolza_problem  
> **username_2**: I synthetise my comments in an answer, for clarity, and because this way your question has an answer.  
> **username_0**: @username_2, yes, it seems multiplicity refers to the number of constraint functions. Thank you very much for your help, I am now aware that my note's and those provided by my lecturer contain many errors.  
> **username_2**: You are welcome!  

---

## 1 answers

### Answer by username_2 on 2024-01-25. Score: 2 | Accepted answer

I give below just an outline to orient oneself in the context of Pontryagin’s Principle, answering briefly your questions.

First of all, it must be noticed that what is called the ***Pontryagin's Maximum Principle*** is actually **not a single theorem, but a collection of theorems,** that differ according to different assumptions.

In particular, various cases of the Maximum Principle exist according to **the interval of time under consideration,** which can be *finite* or *infinite* (the second extreme of integration becoming $\infty$), and the **characteristics of the boundary conditions** we have.

In your case you have a ***finite time problem***, and as for boundary conditions a distinction must be made between ***fixed end* problems** and ***moving end* problems**.
$$\;$$

> In the Hamiltonian equations for the state variables **3.1**, where
> did the second equation come from?

Are you sure there aren’t typos in your formulas? In $(3.1)$, the second equation should be indeed $\dot{x}\_i = \frac{\partial \mathcal{H}}{\partial \lambda\_i}\; i=1,...,n$.

In $(3.1)$ the first equation relates to the case there is only one constraint $g$, that is $n=1$, the second equation to the case we have $n>1$ constraints $g\_i$.
$$\;$$

> In equation **5.1**, what is the *transversality condition*, and
> when/how is it applied? Why is it not necessary when boundary
> conditions are present, as shown in **5.2**?

**Transversality conditions arise only in moving ends problems (and also in infinite time problems) and are absent in fixed end problems:** they are present in the so-called *moving-end problems*, that is **when the value of your $x(t)$ at the final time $t\_f$ is not fixed**, and they are not present in the *fixed-end problems*, when **you know the final value of $x(t)$**, as in $(5.2)$.

Transversality conditions are **mathematical conditions for a maximum,** necessary conditions in the case of moving ends.

Their explanation is not trivial, as all proofs of the maximum principle. They have a geometrical meaning, they arise when the arrival of the trajectory is a so called *manifold*, think of it as a surface in $\mathbb{R}^n$, and not a point.

In a sense, yes, from a practical point of view, they are used as one uses boundary conditions, but they are actual optimum conditions.

Proofs of the Maximum Principle are complex, consider that the proof of the main theorem in Pontryagin's original book$^1$ takes forty pages, the whole second chapter.
$$\;$$

> In the Hamiltonian equations for the costate variables **3.2**, where
> does $\lambda\_f$ come from? When is it used?

The equation $\lambda\_f = \frac{\partial \mathcal{H}}{\partial x\_f} S(x\_f)$
in $(3.2)$ arises in the so-called ***Bolza Problem***, where the functional to be minimized is the sum of an integral and a function depending on the boundary, in your case (eq.$1.1$) a function $S(x\_f)$ of the final value $x\_f$ (remember that in this problem the final value $x\_f$ is a variable, it is not fixed).

It is a problem of Calculus of Variation, that has been dealt with the Pontryagin’s Maximum Principle, see  
<https://encyclopediaofmath.org/wiki/Bolza_problem>

$$\;$$

> In equations **4.1** and **4.2**, what does $m$ stand for? I
> understand from my professors lectures that it means *multiplicity*

As for $m$ in $(4.1)$ and $(4.2$) are you sure that it is not $n$ as in $(3.1)$ ? I see a gradient for $m>1$ and a partial derivative for $m=1$.

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$^1$  Pontryagin L.S. and others, *The Mathematical Theory of Optimal Processes*, Pergamon Press, 1964.

> **Comments:**
> **username_0**: thank you! Your help has been crucial in terms of understanding this part of my course's curriculum.  
> **username_2**: I'm happy having ben useful! Pontryagin maximum principle is not an easy subject.  
> **username_1**: @username_2 :Would you mind mentioning ( should I start a new thread ? ) what texts you used that helped lead to your knowledge. Your answer above is MUCH, MUCH appreciated. I've never seen such an eloquent "seeing the forest through the trees" explanation of this material. Dorfman also wrote a nice, intro paper on the hamiltonian from an economic point of view. If I can find a free version, I'll send a link.  
> **username_1**: There was a free version of this somewhere at some point but now I can only find it on JSTOR. https://www.jstor.org/stable/1810679   Note that I mis-stated the title. It's "optimal control theory from an economic point of view" rather than the Hamiltonian. But there is material regarding the Hamiltonian in the paper.  
> **username_1**: username_2 Note that I have the reference by L.S. Pontryagin but I found it not readable ( for me anyway ) so I was hoping that you'd mention something else. Thanks.  
> **username_2**: @username_1,   I didn't mention Pontryagin as refererence, but only to say, indeed,  that  the proofs are complicated. A good illustration of the Principle in economics is in Takayama, *Analitical Methods in Economics*. If you want a pure mathematical approach, there are many books of control theory, many are advanced. A not advanced book, readable, is Liberzon, *Calculus of Variations and Optimal Control Theory: A Concise Introduction*., Priceton Univ. Press, 2012  
> **username_2**: @username_1 Thank you very much for your appreciation of my answer! And thank you for the reference of Dorfman, I will read it!  
> **username_1**: Thanks username_2. I have Liberzon in storage so I will someday read that. I don't have Takayama so I will check it out. Yes, Pontryagin's text is DIFFICULT. Thank you again for your great answer and I hope you like Dorfman's paper. It's basic but I found it to be a nice intro and way of looking at things.  
> **username_2**: You are welcome, and thank you again for your interesting reference to Dorfman.  
> **username_1**: you're welcome username_2. all of your nice answers are appreciated.  
> **username_2**: You are really very kind username_1.  

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