Achieving Convex Optimization Within Prescribed Time for Networked Euler–Lagrange Systems: A Novel Adaptive Distributed Approach With Small-Gain Conditions

In this article, we address the problem of prescribed-time distributed convex optimization (DCO) for a class of networked Euler–Lagrange systems (NELSs) operating over undirected connected graphs. By utilizing position-dependent measured gradient values of local objective functions and facilitating local information exchanges among neighboring agents, we construct a set of auxiliary systems that collaboratively seek the optimal solution. The prescribed-time DCO problem is then reformulated as a prescribed-time stabilization challenge of an interconnected error system. We propose a prescribed-time small-gain criterion to characterize the prescribed-time stabilization of the system, presenting a novel approach that enhances effectiveness beyond existing asymptotic or finite-time stabilization methods for interconnected systems. Based on this criterion and the auxiliary systems, we design innovative adaptive prescribed-time local tracking controllers for the subsystems. The prescribed-time convergence is achieved through the introduction of time-varying gains that increase to infinity as time approaches the prescribed deadline. The Lyapunov function, along with prescribed-time mapping, is employed to establish the prescribed-time stability of the closed-loop system and the boundedness of internal signals. Finally, the theoretical results are validated through a numerical example.