On the optimal controllability time for linear hyperbolic systems with time-dependent coefficients

TL;DR

This paper investigates how the optimal controllability time for linear hyperbolic systems with time-dependent coefficients can differ from the time-independent case, highlighting the impact of zero-order term regularity.

Contribution

It extends previous results by analyzing the effects of time-varying zero-order terms on controllability time, including cases with analyticity.

Findings

Optimal time can be significantly larger with time-varying zero-order terms.

Analyticity of the zero-order term ensures the optimal time matches the time-independent case.

Zero-order term regularity critically influences controllability time.

Abstract

The optimal time for the controllability of linear hyperbolic systems in one dimensional space with one-side controls has been obtained recently for time-independent coefficients in our previous works. In this paper, we consider linear hyperbolic systems with time-varying zero-order terms. We show the possibility that the optimal time for the null-controllability becomes significantly larger than the one of the time-invariant setting even when the zero-order term is indefinitely differentiable. When the analyticity with respect to time is imposed for the zero-order term, we also establish that the optimal time is the same as in the time-independent setting.