Control of a Wave Equation with a Dynamic Boundary Condition

Abstract

The general problem of this paper is the analysis of wave propagation in a bounded medium where the uncontrolled boundary obeys a coupled differential equation. More precisely, we study a one-dimensional wave equation with a nonlinear second-order dynamic boundary condition and a Neuman-type boundary control acting on the other extremity. A generic class of nonlinear collocated feedback laws is considered. Hadamard well-posedness is established for the closed-loop system, with initial data lying in the natural energy space of the problem. Moreover, we investigate an example of stabilization through a proportional controller. t 0 g(s) ds for a.e. t in (0, T). Such class f is identified with its continuous

Velocity Stabilization of a Wave Equation With a Nonlinear Dynamic Boundary Condition

“…t ∈ R + , with initial data in the domain D(A g ) of the generator; • Weak solutions are limits of strong solutions with respect to the topology of C([0, T ], H) for a given T > 0, with initial data in the energy space H. The well-posedness properties of the closed-loop system are summarized in the following theorem. We refer the reader to [21] for the proof. (1).…”

“…In the preliminary conference version of this work[21], only linear velocity feedback is investigated for the stabilization of (1a)-(1b).…”

Identification of source terms in wave equation with dynamic boundary conditions

“…Interested readers can be directed to [8,[26][27][28] for inverse problems in the static case. Further results on controllability for wave-like systems with dynamic boundary conditions have been investigated in [2,5,12,32,33,38,44]. More recently, the authors have numerically studied an inverse source problem for a one-dimensional wave equation with dynamic boundary conditions [11].…”