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The stochastic control problem with linear stochastic differential equations driven by Brownian motion processes and as cost functional the exponential of a quadratic form, is considered. The solution is shown to exist of a linear control law, and of a linear stochastic differential equation which has the same structure as the Kalman filter but depends explicitly on the cost functional. The separation property does not hold in general for the solution to this problem.

The paper presents realization theory of discrete-time linear switched systems (abbreviated by DTLSSs). We present necessary and sufficient conditions for an input-output map to admit a discrete-time linear switched state-space realization. In addition, we present a characterization of minimality of discrete-time linear switched systems in terms of reachability and observability. Further, we prove that minimal realizations are unique up to isomorphism. We also discuss algorithms for converting a linear switched system to a minimal one and for constructing a state-space representation from input-output data. The paper uses the theory of rational formal power series in non-commutative variables.

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