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 [1] and [2] between risk-sensitive and H_∞-robust control from finite to infinite dimensional spaces.