ABSTRACT
Given an abstract hyperbolic equation, we consider two optimal control problems with quadratic cost: the first over an infinite horizon, the second over a finite horizon with suitable finite state penalization. We then provide a new proof of the interesting relationship between them. In the simpler case (say, distributed control), where both Algebraic Riccati Equation (for the first problem) and Differential Riccati Equation (for the second problem) are available, such relationship is an immediate consequence of comparing these two equations. In the general case of boundary control and non-smoothing finite state penalization, however, no Differential Riccati Equation is available for the second problem. Our proof is Riccati equation-independent. Instead, it relies on intrinsic optimization properties of the two problems. Said relationship, besides being of interest in itself, plays a critical role in the numerical analysis of algebraic Riccati equations in the abstract hyperbolic case [L.1], [L-T.3].
