In this PhD thesis, several problems of stability analysis and control design of discrete-time switched nonlinear systems are addressed. As main contribution, a new class of Lyapu- nov functions which takes the nonlinearity into account has been proposed. We show that these functions are suitable to solve the classical stability analysis problem of linear systems connected to a cone bounded nonlinearity. Instead of the original Lyapunov Lur'e function, the assumptions about the nonlinearity variation are not required. Furthermore, the local stability analysis and control synthesis problems of Lur'e systems subject to control satura- tion are tackled by considering the level set of our function as an estimate of the basin of at- traction. We expose that this estimate, which is given by non-convex and disconnected sets, is less conservative than ellipsoidal sets. We extend these results in order to deal with the problems of stability analysis and stabilization of discrete-time switched nonlinear systems. On one hand, we consider the case of arbitrary switching such that our sufficient conditions assure the properties of stability for all possible switching rules. In this framework, we high- light that our function is able to provide a suitable estimate of the basin of attraction. On the other hand, we tackle the problem of switching rule design aiming at the stabilization of discrete-time switched systems with nonlinear modes. We propose a switching strategy de- pending on the minimum of our switched Lyapunov Lur'e function. Hence, our framework leads to state space partitions, related to the mode activation, which are not restricted to conic sets, commonly exhibited by the switched quadratic functions approaches.
