A Problem in Control of Elastodynamics with Piezoelectric Effects

We consider an optimal control problem where the state equations are a coupled hyperbolic-elliptic system. This system arises in elastodynamics with piezoelectric effects -- the elastic stress tensor is a function of elastic displacement and electric potential. The electric flux acts as the control variable and, in addition to the state constraints, the bound constraints on the control are considered. We develop a complete analysis for the state equations and the control problem. The requisite regularity on the control, to show the well-posedness of state equations, is enforced using the cost functional. We rigorously derive the first order necessary and sufficient conditions using adjoint equations and further study their well-posedness. For spatially discrete (time continuous) problem, we show the convergence of our numerical scheme. Three dimensional numerical experiments are provided showing convergence properties of a fully discrete method and the practical applicability of our approach.