Optimal Control and Filtering of Systems Governed by Partial Differential Equations: A Tutorial

The problems of optimal control and optimal filtering are of central importance in engineering. An optimal control problem consists of searching for a control function u and a corresponding state function x=x(u) in order to optimize a quantity that depends on both u and x=x(u). An optimal filtering problem consists of searching for a state estimate function x̂ for the state function x from a stochastic observation function y=y(x) in order to minimize the difference of x̂ and x. They turn out to be dual problems and have been well-understood for linear finite-dimensional systems.

Their extensions to evolution systems governed by linear partial differential equations have been treated in detail in the seminal works by J. L. Lions and A. Bensoussan, respectively. The final forms of their solutions are formally similar to those of the finite-dimensional case. However, these results are sometimes referred to in a sloppy way, and as a result, one may run into trouble when trying to establish mathematically rigorous stability conclusions which are necessarily related to the well-posedness and regularity of the optimal solutions. The first difficulty is due to the suitable interpretation of the operator Riccati equation involved in both cases. The second difficulty lies in the interpretation of the stochastic process involved in the filtering problem. In fact, if one naively extends the Wiener process to a Hilbert space, then the resultant covariance operator is necessarily non-invertible, which is incompatible with the finite-dimensional case.

This tutorial aims to provide a complete introduction to the optimal control and filtering problems of systems governed by linear partial differential equations based on the results of Lions and Bensoussan. First, we review the classic results of these two problems in their finite-dimensional counterparts. Next, we introduce some preliminaries of evolution systems to prepare us for the infinite-dimensional case. Then, we present Lions' theory for the optimal control problem. This framework shows that the operator Riccati equation arises by first applying the calculus of variations to the optimal control problem and then decoupling the resultant two-point boundary value problem characterizing the optimal control, thus providing a clear characterization of the Riccati equation. Finally, we present Bensoussan's ingenious application of Lions’ framework and the theory of generalized random variables to the optimal filtering problem. Instead of defining Wiener processes in Hilbert spaces, this treatment uses linear random functionals to define the filtering problem, converts it into an optimal control problem lying in Lions' framework, and shows that the resultant operator Riccati equation possesses statistical interpretations similar to the finite-dimensional case.