Stabilization and control of the nonlinear plate equation

TL;DR

This paper establishes new stabilization and control results for nonlinear plate equations with hinged boundary conditions, leveraging unique continuation properties and observability from linear Schrödinger equations.

Contribution

It introduces a novel unique continuation property for nonlinear plate equations, relaxing geometric conditions and enabling stabilization and control results.

Findings

Exponential decay of nonlinear energy under defocusing nonlinearity.

Semiglobal exact controllability within the attractor of the damped system.

New unique continuation property for nonlinear plate equations.

Abstract

In this article we prove semiglobal stabilization and exact controllability results for nonlinear plate equations with hinged boundary conditions and analytic nonlinearity. These results hold when the damping or control is localized in a region where observability for the linear Schr\"odinger equation is known to hold. At the core of these results lies a new unique continuation property for the nonlinear plate equation, which significantly relaxes the geometric conditions required for such property to hold. This property is obtained by combining recent results on propagation of analyticity in time and unique continuation for linear plate operators. More broadly, our approach exploits the linear observability of the plate equation to establish both stabilization and control results. First, we prove exponential decay of the nonlinear energy under a defocusing assumption on the nonlinearity. Second, under a weaker asymptotic assumption on the nonlinearity, we prove semiglobal exact control by analyzing control properties inside the compact attractor provided by the dynamics of the damped equation.