The regulator problem with indefinite quadratic cost for boundary control systems: the finite horizon case

In this paper we study the finite horizon non-standard LQ-problem for an abstract dynamics, which models a large class of hyperbolic-like partial differential equations. We provide necessary/sufficient conditions for finiteness of the value function corresponding to the control problem. Sharpness of sufficient conditions is shown by means of counterexamples. The specific features of the finite, in contrast to infinite, horizon case are illustrated.