Globally Finite time and Globally Fixed-time stable Dynamical Systems for solving Inverse Quasi-variational inequality problems
Nam Van Tran, Le Thi Thanh Hai
TL;DR
This work addresses inverse quasi-variational inequality problems (IQVIPs) by formulating two projection-based continuous-time dynamical systems: one guaranteeing global finite-time stability and the other global fixed-time stability, with equilibrium points coinciding with IQVIP solutions under mild Lipschitz/monotonicity and moving-set conditions. The authors provide Lyapunov-based analyses that yield explicit settling-time bounds and prove that a consistent forward-Euler discretization preserves fixed-time convergence. They also demonstrate the approach on numerical examples, including a traffic-assignment IQVIP, showing faster convergence compared to existing projection-based methods and highlighting practical applicability to constrained equilibrium problems. Overall, the results offer provably fast and robust continuous-time and discrete-time schemes for solving IQVIPs in finite dimensions, with potential extensions to broader function spaces.
Abstract
In this paper, we propose two projection 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 global finite-time and global 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. To demonstrate their effectiveness, we provide numerical experiments, including an application to the traffic assignment problem.