Fixed-Time Stable Proximal Dynamical System for Solving MVIPs

• Published in: IEEE transactions on automatic control Vol. 68; no. 8; pp. 1 - 8
• Main Authors: Garg, Kunal, Baranwal, Mayank, Gupta, Rohit, Benosman, Mouhacine
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.08.2023; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Asymptotic stability; Convergence; Convex functions; Discretization; Dynamic stability; Dynamical systems; Fixed-Time stability; Initial conditions; Mixed variational inequality problem; Optimization; Proximal dynamical system; Stability criteria; Trajectory
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2022.3214795

In this paper, a novel modified proximal dynamical system is proposed to compute the solution of a mixed variational inequality problem (MVIP) within a fixed time, where the time of convergence is finite and is uniformly bounded for all initial conditions. Under the assumptions of strong monotonicity and Lipschitz continuity, it is shown that a solution of the modified proximal dynamical system exists, is uniquely determined, and converges to the unique solution of the associated MVIP within a fixed time. Furthermore, the fixed-time stability of the modified projected dynamical system continues to hold, even if the assumption of strong monotonicity is relaxed to that of strong pseudomonotonicity. Finally, it is shown that the solution obtained using the forward-Euler discretization of the proposed modified proximal dynamical system converges to an arbitrarily small neighborhood of the solution of the associated MVIP within a fixed number of time steps, independent of the initial conditions.