Generalized operator Riccati equations. (English) Zbl 0577.34053

Kuiper, Hendrik J.

SIAM J. Math. Anal. 16, 675-694 (1985).

Summary: The Riccati equation \(dU/dt=AU+UB^*+UCU+D\) 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 \[ dU/dt={\mathcal C}\ell [AU+UB^*]+{\mathcal F}(t,U),\quad U(0)=U_ 0, \] 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.

MSC:

34G20

Nonlinear differential equations in abstract spaces

Keywords:

Riccati equation; control theory; transport theory