Abstract
In this paper we present the solutions of the stochastic finite and infinite horizon risk-sensitive control problems in infinite dimensions, with /spl mu/, /spl epsiv/>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 /spl epsiv//spl darr/0, the deterministic differential game associated with the H/sub /spl infin//-disturbance attenuation control problem of distributed parameter systems is recovered. In the limit as /spl mu//spl darr/0 (resp. /spl mu//spl darr/0, /spl epsiv//spl darr/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/sub /spl infin//-robust control from finite to infinite dimensional spaces. >
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