Second-order Dynamical Systems with Fixed-time Convergence for Fixed Points of Lipschitz Operators

Abstract This paper is devoted to developing novel second-order time-varying dynamical systems with fixed-time convergence for nonsmooth optimization problems. We first propose a second-order dynamical system to locate a fixed point of Lipschitz continuous operators within a fixed time. The existence and uniqueness of solutions for this dynamical system are established, followed by a convergence analysis through the construction of a Lyapunov function. We then present an alternative second-order fixed-time convergent dynamical system for finding the fixed point of contraction operators, which not only features a simpler structure but also offers more flexible parameters. Finally, we employ the proposed dynamical systems to solve both generalized monotone inclusion problems and nonsmooth additive composite optimization problems.