  In this paper, we derive general theorems for controlling (vector-valued)
first order ordinary differential equations such that its solutions stop at a
finite time $T>0$ and apply them to relaxation and dissipative oscillation
processes. We discuss several interesting examples for relaxation processes
with finite stopping time and their energy behaviour. Our results on relaxation
and dissipative oscillations enable us to model diffusion processes with finite
front speeds and dissipative waves that cause in each space point $x$ an
oscillation with a finite stopping time $T(x)$. In the latter case, we derive
the relation between $T(0)$ and $T(x)$. Moreover, the relations beteween the
control functions in the ode model and the respective pde model are derived.In
particular, we present an application of the Paley-Wiener-Schwartz Theorem that
is used in our analysis. A complementary approach for dissipative oscillations
and its application to dissipative waves is presented in [Ko19b], where the
finite stopping time is achieved due to nonconstant coefficients in second
order odes.

Analysis of PDEs;Dynamical Systems