Finite-time and fixed-time stable dynamical systems for solving inverse quasi-variational inequality problems

In this paper, we propose two projected dynamical systems for solving inverse quasi-variational inequality problems in finite-dimensional Hilbert spaces–one ensuring finite-time stability and the other guaranteeing fixed-time stability. We first establish the connection between these dynamical systems and the solutions of inverse quasi-variational problems. Then, under mild conditions on the operators and parameters, we analyze the finite-time and fixed-time stability of the proposed systems. Both approaches offer accelerated convergence; however, while the settling time of a finite-time stable dynamical system depends on initial conditions, the fixed-time stable system achieves convergence within a predefined time, independent of initial conditions. Furthermore, we consider an explicit forward Euler discretization of the dynamical system, ensuring a consistent discretization of the fixed-time stable dynamics. This leads to a novel forward-backward algorithm, for which we present a detailed convergence analysis. To demonstrate their effectiveness, we provide numerical experiments, including an application to the traffic assignment problem.