Dynamical systems with finite stopping times. Part 1: Relaxation, oscillation and their application to diffusion and wave dissipation

TL;DR

This paper develops theorems for controlling differential equations to ensure solutions cease at finite times, applying these to model diffusion with finite front speeds and dissipative waves with finite oscillation durations.

Contribution

It introduces new control methods for differential equations to achieve finite stopping times, linking these to physical diffusion and wave dissipation models.

Findings

Finite stopping times can be achieved in relaxation and oscillation models.

Relations between control functions and PDE models are established.

Application of Paley-Wiener-Schwartz Theorem in the analysis.

Abstract

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.