The algebraic Riccati equation and singular optimal control

The paper links the class of nonnegative definite linear-quadratic optimal control problems to a subset of the set of real symmetric matrices that satisfy the dissipation inequality. This important subset is characterized by means of a certain algebraic Riccati equation and a linear condition. Moreover, we attach every positive semi-definite element of the subset in a one-to-one way to a certain subspace of a properly defined factor space. If the input weighting matrix in the cost functional is positive semi-definite, then the known results on bijective relations between positive semi-definite solutions of the algebraic Riccati equation and certain subspaces of the state space are recovered. Keywords: Linear-quadratic control problems, dissipation inequality, algebraic Riccati equation, strongly reachable subspace, induced map.