Maximum Principle for the Optimal Control of a Hyperbolic Equation in Two Space Dimensions

Abstract

A maximum principle is developed for a class of problems involving the optimal control of a damped parameter system governed by a not-necessarily separable linear hyperbolic equation in two space dimensions. An index of performance is formulated, which consists of functions of the state variable, its first and second order space derivatives and first order time derivative, and a penalty function involving the open-loop control force. The solution of the optimal control problem is shown to be unique using convexity arguments. The maximum principle given involves a Hamiltonian, which contains an adjoint variable as well as an admissible control function. The state and adjoint variables are linked by terminal conditions leading to a boundary/initial/terminal value problem. The maximum principle can be used to compute the optimal control function and is particularly suitable for problems involving the active control of two-dimensional structural elements for vibration suppression.