The paper considers periodic selector-linear difference inclusions. A class of time-periodic quasi-quadratic Lyapunov functions is distinguished, as well as parametric classes of piecewise-quadratic and piecewise-linear Lyapunov functions. These functions establish necessary and sufficient conditions for asymptotic stability. An example leading to periodic selector-linear differential and difference inclusions are given. The results can find applications in the stability analysis of control systems with periodic parameters, in particular, servomechanisms whose elements operate on alternating current, control systems with amplitude-frequency modulation.