First passage risk-sensitive criterion for stochastic evolutions

The purpose of this paper is to investigate in an infinite dimensional space, the first passage problem with a risk-sensitive performance criterion, and to illustrate the asymptotic behavior of the associated value function, as related to differential games arising in robust control theory. The model of interest is described by a controlled stochastic evolution with small Wiener noise intensity. The Wiener and state processes take values in infinite dimensional Hilbert spaces. The objective is to control the evolution of the state process, so as to keep it in some compact set G. By using a logarithmic transformation, it is shown that in the limit as the small noise parameter, ε→0, the risk-sensitive value function converges to the value of a deterministic differential game. In the limit as the risk parameter, θ→0, the risk-sensitive value function converges to the value function corresponding to the mean escape time problem. In addition, a lower bound on the first escape time is derived which is slightly different from the known bound for finite dimensional systems. The magnitude of the lower bound derived here, increases as θ increases, thus robustness is achieved