Null-controllability, exact controllability, and stabilization of hyperbolic systems for the optimal time

TL;DR

This paper investigates the controllability and stabilization of one-dimensional linear hyperbolic systems using boundary controls, identifying the optimal time for control and designing feedbacks for stabilization, with extensions to nonlinear systems.

Contribution

It establishes the optimal time for null and exact controllability of hyperbolic systems and designs feedback controls for stabilization at this optimal time, including nonlinear extensions.

Findings

Optimal control time identified for hyperbolic systems.

Null and exact controllability achieved for any time greater than optimal.

Feedback controls designed for stabilization at the optimal time.

Abstract

In this paper, we discuss our recent works on the null-controllability, the exact controllability, and the stabilization of linear hyperbolic systems in one dimensional space using boundary controls on one side for the optimal time. Under precise and generic assumptions on the boundary conditions on the other side, we first obtain the optimal time for the null and the exact controllability for these systems for a generic source term. We then prove the null-controllability and the exact controllability for any time greater than the optimal time and for any source term. Finally, for homogeneous systems, we design feedbacks which stabilize the systems and bring them to the zero state at the optimal time. Extensions for the non-linear homogeneous system are also discussed