RICCATI-EQUATIONS FOR HYPERBOLIC PARTIAL-DIFFERENTIAL EQUATIONS WITH L2(0, T, L2(GAMMA)) - DIRICHLET BOUNDARY TERMS

This paper studies the quadratic optimal control problem for second order (linear) hyperbolic partial differential equations defined on a bounded domain in R**n with boundary GAMMA . Both the finite interval case left bracket 0, T right bracket , T less than infinity , and the infinite interval case T equals infinity (regulator problem) are considered. The distinguishing feature of the paper is that the controls are only L//2(0, T; L//2( GAMMA ))-functions which act in the Dirichlet B. C. and that the corresponding solutions are penalized in the L//2(0, T; L//2( OMEGA ))-norm. Under minimal assumptions, the optimal control is synthesized, in a pointwise feedback form, through an operator which is shown to satisfy in a suitable sense a Riccati differential equation for T less than infinity and a Riccati algebraic equation for T equals infinity . Regularity results of the optimal pair are also included. A functional analytic model, based on cosine operator theory, is used throughout to describe the hyperbolic dynamics.