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Discrete-time markovian-jump linear quadratic optimal control. (English) Zbl 0591.93067

Discrete-time jump linear systems X K+1 =A K (r K )X K +B K (r K )u K , K=K 0 ,···,N, P{r K+1 =j/r K =i}=P K+1 (i,j) with initial state X(K 0 )=X 0 , r(K 0 )=r 0 are considered. It is assumed that the x-process, and the control vector u are m-dimensional and that the form process {r|K=K 0 ,···,N} is a finite-state Markov chain taking values in ={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 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
MSC:
93E20Optimal stochastic control (systems)
60J10Markov chains (discrete-time Markov processes on discrete state spaces)
93C55Discrete-time control systems
60J75Jump processes
93C05Linear control systems