Riccati Equations for Hyperbolic Partial Differential Equations with L₂ (0,T;L₂ (Γ ))—Dirichlet Boundary Terms

This paper studies the quadratic optimal control problem for second order (linear) hyperbolic partial differential equations defined on a bounded domain Ω ⊂ Rⁿ with boundary Γ. Both the finite interval case $[0,T]$, T < ∞, and the infinite interval case T = ∞ (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 L₂ (0,T;L₂ (Γ ))-functions which act in the Dirichlet B.C. and that the corresponding solutions are penalized in the L₂ (0,T;L₂ (Ω ))-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 shown to satisfy in a suitable sense a Riccati differential equation for T < ∞ and a Riccati algebraic equation for T = ∞. Unlike most, if not all, of the literature on quadratic control problems (for different dynamics!), the algebraic Riccati equation is not derived as a limit process as T ↑ ∞ of the Riccati differential (or integral) equation on $[0,T]$. This has a special advantage in the case of hyperbolic dynamics. Rather, the approach followed for the control problems is direct, in the sense that first an operator is defined in terms of the hyperbolic dynamics and only subsequently shown to satisfy a Riccati equation (differential for T < ∞, algebraic for T = ∞). Regularity results of the optimal pair are also included. A functional analytic model, based on cosine operator theory and introduced by the authors in [L - T.1], [L - T.3], is used throughout to describe the hyperbolic dynamics.