A simulation study on spatial exponential decay of perturbations in a two-dimensional wave equation with optimal boundary/line control
Recent results have shown that domain-uniform stabilizability and detectability imply spatial exponential decay of perturbations in optimally controlled hyperbolic PDEs on one-dimensional domains. This domain-uniform stabilizability and detectability can be achieved only if the control domain is distributed over the whole spatial domain such that the distance between neighboring control intervals is bounded from above. Motivated by these insights we investigate whether analogous effects can be observed for optimal boundary control in higher dimensions. For this purpose we conduct a simulation study on a two-dimensional wave equation on expanding square domains which is driven by a localized perturbation of the initial displacement. We compare two control geometries: a control acting only on the outer boundary and a control acting on a regular grid of interior line interfaces combined with the boundary. The problem is discretized by conforming P1 finite elements in space and the implicit midpoint rule in time. Instead of assembling the full space-time KKT system, we use a condensed formulation and solve the reduced optimality system by preconditioned conjugate gradients. We show that domain-uniform spatial decay of the perturbation can only be observed in the scenario with a regular grid of line controls. This is because finite propagation velocity requires a uniform bound on the distance that a wave can travel without reaching the control domain.