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

Nicolas Vanspranghe (1) , Francesco Ferrante (1, 2) , Christophe Prieur (1)

1 GIPSA-INFINITY - GIPSA - Infinite Dimensional Dynamics
2 UNIPG - Università degli Studi di Perugia = University of Perugia

This paper deals with a one-dimensional wave equation with a nonlinear dynamic boundary condition and a Neumann-type boundary control acting on the other extremity. We consider a class of nonlinear stabilizing feedbacks that only depend on the velocity at the controlled extremity. The uncontrolled boundary is subject to a nonlinear first-order term, which may represent nonlinear boundary anti-damping. Initial data is taken in the optimal energy space associated with the problem. Exponential decay of the mechanical energy is investigated in different cases. Stability and attractivity of suitable invariant sets are established.

Mots clés: Boundary conditions; Stability analysis; Nonlinear dynamical systems; Propagation; Mathematical models; Drilling; Aerospace electronics

Domaines: Optimisation et contrôle [math.OC]; Equations aux dérivées partielles [math.AP]; Automatique / Robotique

Nicolas Vanspranghe, Francesco Ferrante, Christophe Prieur. Velocity Stabilization of a Wave Equation with a Nonlinear Dynamic Boundary Condition. IEEE Transactions on Automatic Control, 2022, 67 (12), pp.6786-6793. ⟨10.1109/TAC.2021.3136086⟩. ⟨hal-03349947v2⟩