Impulse control of stochastic Navier–Stokes equations

Abstract

Optimal control theory of 2uid dynamics has numerous applications such as aero= hydrodynamic control, combustion control, Tokomak magnetic fusion as well as ocean and atmospheric prediction. During the past decade several fundamental advances have been made by a number of researchers as documented in Sritharan [21,22]. In this paper we develop a new direction to this subject, namely we mathematically formulate and resolve impulse and stopping time problems. Impulse control of Navier–Stokes equations has signi7cance beyond control theory. In fact, in optimal weather prediction the task of updating the initial data optimally at strategic times can be reformulated precisely as an impulse control problem for the primitive cloud equations (which consist of the Navier–Stokes equation coupled with temperature and species evolution equations, cf. [11]), see [1,9,15]. For the study of optimal stopping problem alone, it is possible to impose regularity assumptions on the stopping cost. However, in our case, optimal stopping problems are used as intermediate steps to treat the impulse control problem through an iteration process. This dictates that we must work with stopping costs which have only continuity property. Optimal stopping and impulse control problems are very well known, particularly for di;usion processes (e.g., see the books of Bensoussan and Lions [3,4]), for degenerate