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Discrete-time markovian-jump linear quadratic optimal control. (English) Zbl 0591.93067
Discrete-time jump linear systems $X\sb{K+1}=A\sb K(r\sb K)X\sb K+B\sb K(r\sb K)u\sb K$, $K=K\sb 0,...,N$, $P\{r\sb{K+1}=j/r\sb K=i\}=P\sb{K+1}(i,j)$ with initial state $X(K\sb 0)=X\sb 0$, $r(K\sb 0)=r\sb 0$ are considered. It is assumed that the x-process, and the control vector u are m-dimensional and that the form process $\{r\vert K=K\sb 0,...,N\}$ is a finite-state Markov chain taking values in ${\cal M}=\{1,2,...,M\}$. Further, it is assumed that the cost criterion is quadratic. First, the optimal control law is presented. This optimal control law is linear in $X\sb K$ at each time K, and it is different (in general) for each possible set of parameter values. Further, necessary and sufficient conditions for the existence of a steady-state optimal controller are given. The results are illustrated by examples.
Reviewer: V.Kankova

93E20Optimal stochastic control (systems)
60J10Markov chains (discrete-time Markov processes on discrete state spaces)
93C55Discrete-time control systems
60J75Jump processes
93C05Linear control systems
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