Stabilization and tracking of the trajectory of a linear system with jump drift

For a controlled linear stochastic differential system, we consider the problem of tracking the jumping state of an additive input action determining the current stabilization direction (drift). The tracking objective, which is to stabilize the system near the varying drift, is formalized by a quadratic performance functional. The input action defines a continuous-time Markov chain. The problem is considered in the cases of complete and incomplete information. Dynamic programming is used to solve it in both cases. The solution of the Bellman equation in the first case is obtained based on the properties of a finite-dimensional chain, and in the second case, based on the principle of separation of control and state estimation problems provided by the Wonham filter estimate and the properties of the quadratic performance criterion. A numerical experiment uses an applied model describing the position of a simple mechanical drive. The results of calculations confirming the applicability of the solutions obtained, as well as ways to overcome the difficulties of their numerical implementation, are presented and discussed in detail.