Lyapunov Functions: An Optimization Theory Perspective*. Issue 1 (July 2017)

• Record Type: Journal Article
• Authors: Polyak, Boris; Shcherbakov, Pavel
• Is Part Of: IFAC-PapersOnLine. Volume 50:Issue 1(2017)
• Page Start: 7456
• Page End: 7461
• Publication Date: 2017-07
• Subjects: optimization -- Lyapunov functions -- asymptotic stability -- gradient method -- heavy-ball method -- pendulum equation -- synchronous motor -- basin of attraction; Automatic control -- Periodicals; 629.805
• DOI: 10.1016/j.ifacol.2017.08.1513
• Languages: English
• ISSNs: 2405-8963

Abstract

Abstract: The problems of unconstrained optimization and establishing asymptotic stability have much in common. Understanding the analogy between these two sheds light on their interconnection and may lead to a number of new results. For instance, in this paper, we provide estimates of the rate of convergence when analyzing asymptotic stability of differential equations, rather than just ascertain the very fact of stability. Also, standard methods for the design of Lyapunov functions (e.g., those having the meaning of the full energy of the system) turn out to be unsatisfactory from this point of view and have to be modified. These claims are exemplified in the paper by considering the heavy-ball method for function minimization "in parallel" with the problem of asymptotic stability for the synchronous motor equation.