Risk-sensitive control, differential games, and limiting problems in infinite dimensions

In this paper we present the solutions of the stochastic finite and infinite horizon risk-sensitive control problems in infinite dimensions, with μ, ε>0, respectively, representing the risk-sensitivity and small noise parameters. Invoking a logarithmic transformation, a stochastic differential game equivalent to the risk-sensitive problem is obtained. In the limit as ε↓0, the deterministic differential game associated with the H_∞-disturbance attenuation control problem of distributed parameter systems is recovered. In the limit as μ↓0 (resp. μ↓0, ε↓0) the usual stochastic (resp. deterministic) control problem with integral cost is recovered. Both finite and infinite horizon cases are treated. This article extends the recent relations given by James (1992) and Fleming et al. (1992) between risk-sensitive and H_∞-robust control from finite to infinite dimensional spaces