Risk-sensitive optimal control for jump systems with application to sampled-data systems

Abstract

The partially observed optimal stochastic control of jump systems with sampled inputs and sampled observations, which minimizes the expected value of an exponential cost criterion, is considered. A jump system is a very general representation and covers a sampled-data system as its special case. The change-in-measure technique is employed to obtain the conditional expectation of the cost criterion given the measurement history, which is called the information state. It is the sufficient statistics for the problem and thus helps to convert the partially observed problem into an equivalent fully observed problem. The risk-sensitive optimal controller for jump systems is derived through a combination of the continuous-time and discrete-time Riccati equations. The result for jump systems is extended to obtain the risk-sensitive optimal controllers for sampled-data systems. The infinite-time problems for both jump and sampled-data systems are also considered. Furthermore, asymptotic behaviours of small-noise and small-risk limits of the problems, which correspond to a deterministic game and the standard stochastic optimal control problem, respectively, are analysed.