Bochner integrable solutions to Riccati partial differential equations and optimal sensor placement

In this paper we provide sufficient conditions to ensure that solutions to the time varying Riccati partial differential equations are Bochner integrable with range in the space of trace class operators. The fact that Bochner integrals can be uniformly approximated by simple functions provides a basis for obtaining bounds on integration errors. These bounds can then be used for rigorous numerical analysis and to ensure the convergence of algorithms used to compute approximate solutions. We demonstrate how this result can be employed to develop convergent computational methods for a sensor placement problem based on optimal filtering. Theoretical results are presented and numerical examples are given to illustrate the ideas.