Linear quadratic optimal control of time-varying systems with indefinite costs on Hilbert spaces (Q1293214)

For a time-varying, infinite-dimensional, linear system the author investigates the linear quadratic optimal control problem for an indefinite cost criterium which is given over an infinite-time interval. The solution of this optimal control is described in terms of solutions of the integral Riccati equation. This Riccati equation is equivalent to the algebraic Riccati equation for time-invariant, finite-dimensional systems. It is shown that the integral Riccati equation has a bounded solution if and only if the cost-criterium is bounded from below. Under this condition the infimum of the cost equals \(\langle x_0,P(t_0)x_0\rangle\), where \(x_0\) is the state at time initial \(t_0\) to \(P(t_0)\) is the solution of the integral Riccati equation. By means of a simple example it is shown that in general this infimum will not be a minimum.