The regulator problem for the wave equation with high internal damping controlled on the boundary: a new look via systems with memory

TL;DR

This paper addresses the quadratic regulator problem for the wave equation with high internal damping by transforming it into a system with memory, introducing a time-dependent Hilbert space, and deriving a feedback control via a Riccati operator.

Contribution

It introduces a novel approach transforming the wave equation with damping into a memory system and develops a Riccati-based feedback control in a time-dependent Hilbert space.

Findings

Representation of optimal control as a Riccati-based feedback.

Introduction of a time-dependent Hilbert space framework.

Explicit form for the derivative of the Riccati operator.

Abstract

We study the quadratic regulator problem on a finite time horizon for the wave equation with high internal damping controlled on the boundary by square integrable controls. The approach in this paper transforms the wave equation with high internal damping to an equation with persistent memory controlled on the boundary. One of the results of this paper is the introduction of a state space which is an extended Hilbert space, so a time dependent Hilbert space. We prove that the unique optimal control can be represented as a feedback control via a Riccati operator which solves a suitable version of the Riccati equation. Both the feedback operator and the Riccati equation acts on such time dependent space. The derivation of these main results requires a very precise analysis of the properties of the derivatives of the value function and we find an explicit form for the derivative of the Riccati operator.