Multiple-objective risk-sensitive control and its small noise limit

This paper is concerned with a (minimizing) multiple-objective risk-sensitive control problem. Asymptotic analysis leads to the introduction of a new class of two-player, zero-sum, deterministic differential games. The distinguishing feature of this class of games is that the cost functional is multiple-objective in nature, being composed of the risk-neutral integral costs associated with the original risk-sensitive problem. More precisely, the opposing player in such a game seeks to maximize the most 'vulnerable' member of a given set of cost functionals while the original controller seeks to minimize the worst 'damage' that the opponent can do over this set. It is then shown that the problem of finding an efficient risk-sensitive controller is equivalent, asymptotically, to solving this differential game. Surprisingly, this differential game is proved to be independent of the weights on the different objectives in the original multiple-objective risk-sensitive problem. As a by-product, our results generalize the existing results for the single-objective risk-sensitive control problem to a substantially larger class of nonlinear systems, including those with control-dependent diffusion terms.