Dynamical behavior of a class of nonsmooth gradient-like systems

In this paper, we consider a class of nonsmooth gradient-like systems, which is a generalization of existing neural-network models. Under several assumptions, by virtue of Lyapunov function and topological degree theory, we investigate the dynamical behaviors of this system, such as the existence of global solution and equilibrium point, the global asymptotic stability of the global solution. Then, we apply these results into optimization problem, such as the problem of minimizing a convex objective function over the discrete set {0,1}n and nonlinear programming problems. Besides, we also investigate the existence and global exponential stability of periodic solution of this system with a periodic input vector. Some examples are given to illustrate our results.