Risk-sensitive and robust escape criteria

The problem of controlling a noisy process so as to prevent it from leaving a prescribed set has a number of interesting applications. In this paper, new criteria for this problem are considered. First, a risk-sensitive criterion for a stochastic diffusion process model is examined, and it is shown that the value is a classical solution of a related PDE. The qualitative properties of this criteria are favorably contrasted with those of existing criteria in the riskaverse limit. It is proved that in the risk-averse limit the value of the risk-sensitive criterion converges to a viscosity solution of a first-order PDE. It is then demonstrated that the value function of a deterministic differential game is also a viscosity solution to the PDE. This game represents a robust control problem which appears to be analogous to H°° control. In particular, the opposing player attempts to push the process out of the prescribed set, and suffers an L cost for his efforts. Lower bounds on the escape time as a function of this cost are obtained. "The work of this author was supported in part by NSF grant DMS-94Q3820. ^ h e work of this author was supported in part by the Army Research Office and the National Science Foundation through the Center for Nonlinear Analysis