Linear quadratic optimal control of time-varying systems with indefinite costs on Hilbert spaces: the finite horizon problem

In this paper we consider time{varying linear systems on Hilbert spaces and study the optimal control problem with indefinite performance criteria over a finite horizon interval. Due to the indefiniteness of the cost function, the associated integral Riccati equation in general does not process a solution on the whole interval. Applying an operator theoretic approach due to Hinrichsen and Pritchard [10] equivalent conditions are arrived for the unique solvability of the linear quadratic optimization problem and for the existence of solutions to the integral Riccati equation. Contrary to the finite
dimensional situation these problems are not generally equivalent. The results are applied to a parameterized Riccati equation which plays an important role in robustness analysis.