Optimal Stabilization of Linear Stochastic System with Statistically Uncertain Piecewise Constant Drift

The paper presents an optimal control problem for the partially observable stochastic differential system driven by an external Markov jump process. The available controlled observations are indirect and corrupted by some Wiener noise. The goal is to optimize a linear function of the state (output) given a general quadratic criterion. The separation principle, verified for the system at hand, allows examination of the control problem apart from the filter optimization. The solution to the latter problem is provided by the Wonham filter. The solution to the former control problem is obtained by formulating an equivalent control problem with a linear drift/nonlinear diffusion stochastic process and with complete information. This problem, in turn, is immediately solved by the application of the dynamic programming method. The applicability of the obtained theoretical results is illustrated by a numerical example, where an optimal amplification/stabilization problem is solved for an unstable externally controlled step-wise mechanical actuator.