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Optimal control for linear discrete-time systems with Markov perturbations in Hilbert spaces. (English) Zbl 1159.93036

Summary: We discuss a quadratic control problem for linear discrete-time systems with Markov perturbations in Hilbert spaces, which is linked to a discrete-time Riccati equation defined on certain infinite-dimensional ordered Banach space. We prove that under stabilizability and stochastic uniform observability conditions, the Riccati equation has a unique, uniformly positive, bounded on \(\mathbb N\) and stabilizing solution. Based on this result, we solve the proposed optimal control problem. An example illustrates the theory.

MSC:

93E20 Optimal stochastic control
49N05 Linear optimal control problems
60J75 Jump processes (MSC2010)
93C05 Linear systems in control theory
93C55 Discrete-time control/observation systems
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