Perturbation of linear quadratic systems with jump parameters and hybrid controls

We consider the problem of the perturbation of a class of linear-quadratic differential games with piecewise deterministic dynamics, where the changes from one structure (for the dynamics) to another are governed by a finite-state Markov process. Player 1 controls the continuous dynamics, whereas Player 2 controls the rate of transition for the finite-state Markov process; both have access to the states of both processes. Player 1 wishes to minimize a given quadratic performance index, while player 2 wishes to maximize or minimize the same quantity. The problem above leads to the analysis of some linearly coupled set of quadratic equations (Riccati equations). We obtain a Taylor expansion in the perturbation for the solution of the equation for a fixed stationary policy of the player 2. This allows us to solve the game or team problem as a function of the perturbation.