  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.
