On the value functions of the optimal quadratic regulation problem for discrete-time switched linear systems

In this paper, we study the value functions associated with the discrete-time LQR problem for switched linear systems (DLQRS). Some important properties of the value functions and value iterations are derived. In particular, we will show that under some mild conditions, the family of the finite-horizon value functions of the DLQRS problem is homogeneous (of degree 2), uniformly bounded over the unit ball, and converges exponentially fast to the corresponding infinite-horizon value function. More importantly, the exponential convergence rate of the value iteration is characterized analytically in terms of the subsystem matrices. The properties derived in this paper are not only of theoretical importance, but also crucial in the analysis and design of various optimal and suboptimal control strategies for DLQRS problems.