On the Value Functions of the Discrete-Time Switched LQR Problem

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> In this paper, we derive some important properties for the finite-horizon and the infinite-horizon value functions associated with the discrete-time switched LQR (DSLQR) problem. It is proved that any finite-horizon value function of the DSLQR problem is the pointwise minimum of a finite number of quadratic functions that can be obtained recursively using the so-called <emphasis emphasistype="italic">switched Riccati mapping</emphasis>. It is also shown that under some mild conditions, the family of the finite-horizon value functions is homogeneous (of degree 2), is uniformly bounded over the unit ball, and converges exponentially fast to the infinite-horizon value function. The exponential convergence rate of the value iterations is characterized analytically in terms of the subsystem matrices. </para>