Unbounded Solutions to the Linear Quadratic Control Problem

Examples are presented to show that the solution of the operational algebraic Riccati equation can be an unbounded operator for infinite dimensional systems in a Hilbert space even with bounded control and observation operators. This phenomenon is connected to the presence of a continuous spectrum in one of the operators. The object of this paper is to fill up the gap in the classical linear quadratic theory. The key step is the introduction of the set of stabilizable initial conditions. Then a new simple approach to the linear-quadratic problem is presented that provides the connection with the notion of approximate stabilizability for the triplet (A, B, C).