Asymptotically stable control problems by infinite horizon optimal control with negative discounting

TL;DR

This paper develops a method for designing asymptotically stable controls for systems with attractors and fixed points using infinite horizon optimal control with negative discounting, verified through numerical examples.

Contribution

It introduces a new stable control formulation for systems with attractors, utilizing infinite horizon optimal control with negative discounts, and classifies initial conditions for robust stability.

Findings

Successfully derived stable control functions for a 2D system.

Phase space can be partitioned based on orbit behavior.

Numerical results demonstrate effective control on the Bonhoeffer-van der Pol model.

Abstract

In the paper, for the system which possesses both an attractor and a stable fixed point, we first formulate new stable control problems to find the asymptotically stable control function which realizes to transit a state moving around the attractor to the stable fixed point. Then by using the ordinary differential equation based on the infinite horizon optimal control model with negative discounts, we give one of answers for the stable control problem in a two-dimensional case. Furthermore, under some conditions, we verify that the phase space can be separated to some open connected components depending on the asymptotic behavior of the orbit starting from the initial point in their components. This classification of initial points suggests that it is enable to robustly achieve a stable control. Moreover, we illustrate some numerical results for the stable control obtained by applying our focused system for the Bonhoeffer-van der Pol model.