Optimisation and asymptotic stability

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.

KEYWORDS:

• Optimisation
• Lyapunov functions
• asymptotic stability
• gradient method
• heavy-ball method
• pendulum equation
• synchronous motor
• basin of attraction

Acknowledgment

The authors would like to thank Denis Efimov for his interest in the subject of the paper, suggestions and references to the literature. Help of Yana Kvinto in performing numerical experiments is also appreciated.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The financial support for this work was provided by the Russian Science Foundation [grant number 16-11-10015].