The paper addresses the problem of stabilizing discrete-time systems subject to time varying polytopic uncertainty. Non stationary quadratic Lyapunov functions are derived for synthesis in a Poly-Quadratic Lyapunov function concept which avoids in a large extent the conservatism linked with the classical single Lyapunov function quadratic approach. The state space feedback synthesis problem is addressed. The results are extended to cope with two particular problems: H_∞ infinity performance analysis and synthesis problem as well as state feedback design while maximizing the size of the uncertainty domain.