Controllability of the semilinear wave equation governed by a multiplicative control

Abstract

In this paper we establish several results on approximate controllability of a semilinear wave equation by making use of a single multiplicative control. These results are then applied to discuss the exact controllability properties for the one dimensional version of the system at hand. The proof relies on linear semigroup theory and the results on the additive controllability of the semilinear wave equation. The approaches are constructive and provide explicit steering controls. Moreover, in the context of undamped wave equation, the exact controllability is established for a time which is uniform for all initial states.

Approximate controllability of the semilinear reaction-diffusion equation governed by a multiplicative control

“…• Based on the previous controllability results and inspired by the idea of [39], we can show that similar controllability results are still valid for a system of reaction-diffusion equations of the form (1), where the bilinear control is acting on all the equations.…”

Problem of optimal control for bilinear systems with endpoint constraint

“…According to [25], there exists a control v ∈ L 2 (0, T; L 2 (Ω) such that the corresponding solution z v of the system (1) verifies z v (T) = z d . Then, according to Theorem 5 there exists a control u * ∈ L 2 (0, T, R), which guarantees the exact attainability of z d at time T, and is a solution of the problem (P) with U ad = {u ∈ L 2 (0, T, L 2 (Ω)) /z(T) = z d }.…”

Boundary controllability for a degenerate wave equation in non divergence form with drift

“…In [36] the authors investigate the problem of maximizing the observability constant, or its asymptotic average in time, over all possible subsets ω of [0, π] of Lebesgue measure Lπ. In [35] the author considers the controllability problem of a semilinear wave equation with a control multiplying the nonlinear term.…”