Riccati Equations for Hyperbolic Partial Differential Equations with $L_2 (0,T;L_2 (\Gamma ))$—Dirichlet Boundary Terms

Abstract

This paper studies the quadratic optimal control problem for second order (linear) hyperbolic partial differential equations defined on a bounded domain $\Omega \subset R^n $ with boundary $\Gamma $. Both the finite interval case $[0,T]$, $T < \infty $, and the infinite interval case $T = \infty $ (regulator problem) are considered. The distinguishing feature of the paper, which differentiates it from previous (scarce!) literature on the subject, 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 (smoother controls, particularly in space, were taken in the few previous works on this subject). The well-posedness of this formulation stems from recent results by the authors about regularity of second order hyperbolic mixed problems [L - T.1], [L - T.3]. Under minimal assumptions, the optimal control is synthesized, in a pointwise feedback form, through an operator which is s...