Bernt Øksendal: Optimal control of SPDEs driven by time-space Brownian sheet

Abstract

 In this talk we study the optimal control of systems where the state dynamics satisfy a stochastic partial differential equation (SPDE) driven by a two-parameter (time-space) Brownian motion (also called Brownian sheet). Such equations can for example model the growth of an ecosystem under uncertainty in time and space.

We first discuss some properties of linear SPDEs of this type.

Further, applying time-space white noise calculus we derive Pontryagin type sufficient conditions and necessary conditions of optimality of the control.

Finally, we illustrate our results by solving a linear quadratic control problem and studying an optimal harvesting problem in the plane. We also study possible applications to machine learning.

The talk is based on joint works with Nacira Agram (KTH), Frank Proske (UiO) and Olena Tymoshenko (UiO).