## 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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