Generalized Operator Riccati Equations

Abstract

The Riccati equation \[\frac{{dU}}{{dt}} = AU + UB^* + UCU + D\]on the space $\mathcal {L}(X)$ of bounded linear operators on a reflexive Banach space X arises in control theory and transport theory. A more general problem is the following. Let A and B be closed linear operators in X and $X^* $ respectively and let $\mathcal {F}$ be a map from $[0,T) \times \mathcal {L}(X)$ into $\mathcal {L}(X)$ and consider the initial value problem \[\frac{{dU}}{{dt}} = \mathcal{L}\ell U[AU + UB^* ] + \mathcal{F}(t,U),\quad U(0) = U_0 ,\]on $\mathcal {L}(X)$. It is shown that for a certain class of initial conditions, determined by A, B and the geometry of X, there exist continuously differentiable solutions with respect to the uniform operator topology. It is also shown that if A and B have compact inverses, then there exist solutions with respect to the strong operator topology for arbitrary initial conditions.