EXISTENCE OF STOCHASTIC-CONTROL UNDER STATE CONSTRAINTS

In this Note, we give a necessary and sufficient condition for the exi stence of control that keeps the corresponding trajectory of the relat ed stochastic control system within a prescribed closed subset of the state space. The problem of existence of stochastic control under a st ate-constraint is also called the viability property of the underlying control system. Our result is: the square of the distance function of this constraint is a viscosity supersolution of a Hamilton-Jacobi-Bel lman equation if and only if the system enjoys the viability property. (C) Academie des Sciences/Elsevier, Paris.