Optimisation and asymptotic stability. Issue 11 (2nd November 2018)

• Record Type: Journal Article
• Authors: Polyak, B. T.; Shcherbakov, P. S.
• Is Part Of: International journal of control. Volume 91:Issue 11(2018)
• Page Start: 2404
• Page End: 2410
• Publication Date: 2018-11-02
• Subjects: Optimisation -- Lyapunov functions -- asymptotic stability -- gradient method -- heavy-ball method -- pendulum equation -- synchronous motor -- basin of attraction; Automatic control -- Periodicals; Electronic journals; 629.8
• DOI: 10.1080/00207179.2016.1257154
• Languages: English
• ISSNs: 0020-7179

Abstract

ABSTRACT: The problems of unconstrained optimisation 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 analysing 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 minimisation 'in parallel' with the problem of asymptotic stability for the synchronous motor equation.