Risk-sensitive control of Markov jump linear systems: Caveats and difficulties |
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Authors: | Moon Jun Başar Tamer |
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Affiliation: | 1.School of Electrical and Computer Engineering, Ulsan National Institute of Science and Technology (UNIST), Ulsan, South Korea ;2.Coordinated Science Laboratory, University of Illinois at Urbana and Champaign, Illinois, 61801, USA ; |
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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. |
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