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Risk-sensitive Control of Markov Jump Linear Systems: Caveats and Difficulties

Jun Moon* and Tamer Basar
International Journal of Control, Automation, and Systems, vol. 15, no. 1, pp.462-467, 2017

Abstract : "In this technical note, we revisit the risk-sensitive optimal control problem for Markov jump linear systems (MJLSs). We first demonstrate the inherent difficulty in solving the risk-sensitive optimal control problem even if the system is linear and the cost function is quadratic. This is due to the nonlinear nature of the coupled set of Hamilton-Jacobi-Bellman (HJB) equations, stemming from the presence of the jump process. It thus follows that the standard quadratic form of the value function with a set of coupled Riccati differential equations cannot be a candidate solution to the coupled HJB equations. We subsequently show that there is no equivalence relationship between the problems of risk-sensitive control and H¥ control of MJLSs, which are shown to be equivalent in the absence of any jumps. Finally, we show that there does not exist a large deviation limit as well as a risk-neutral limit of the risk-sensitive optimal control problem due to the presence of a nonlinear coupling term in the HJB equations."

Keyword : Markov jump linear systems, risk-sensitive control, stochastic zero-sum differential games.

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