An explicit formula for the solution of certain optimal control problems on domains with corners. (English) Zbl 0990.60043

Dupuis, P.; Ramanan, K.

Theory Probab. Math. Stat. 63, 33-49 (2001) and Teor. Jmovirn. Mat. Stat. 63, 32-48 (2000).

The authors consider a dynamical model of the form \(\dot\varphi(t)=\pi(\varphi(t),\beta(t))\), \(\varphi(0)=x\). The deterministic control process is \(\beta(t)\), and the state space of the controlled process is \(R^{n}_{+}\). The function \(\pi(x,\beta)\) is discontinuous when \(x\) is on the boundary of the state space. The form of \(\pi\) is determined by certain directions of constraint that are used on \(\partial R^{N}_{+}\). The cost to be minimized is \(\int_{0}^{\tau_{x}}c(\beta(t))dt\), where \(\tau_{x}\) is the first time \(t\) such that \(\varphi(t)=0\). Here \(c:R^{N}\to[0,\infty)\) is a proper convex function. The main result of the paper is the following one. Let \(V(x)\) be the minimal cost function for the considered problem and let \(C\) be the negative of the convex cone generated by the associated directions of constraint. Then, under some conditions, \(V\) is convex and has the representation \(V(x)=\inf\{\sigma c(\beta):x+\sigma\beta\in C\}\).

Reviewer: A.D.Borisenko (Kyïv)

MSC:

60G35

Signal detection and filtering (aspects of stochastic processes)

Keywords:

explicit formula; solution; optimal control; domains with corners; Skorokhod problem