A characteristics based curse-of-dimensionality-free approach for approximating control Lyapunov functions and feedback stabilization

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Yegorov, Ivan ; Dower, Peter ; Grüne, Lars:
A characteristics based curse-of-dimensionality-free approach for approximating control Lyapunov functions and feedback stabilization.
Bayreuth ; Melbourne , 2018 . - 8 S.

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Abstract

This paper develops a curse-of-dimensionality-free numerical approach to construct control Lyapunov functions (CLFs) and stabilizing feedback strategies for deterministic con- trol systems described by systems of ODEs. An extension of the Zubov method is used to represent a CLF as the value function for an appropriate infinite-horizon optimal control problem. The infinite-horizon stabilization problem is approximated by an exit time problem, with target set given by a sufficiently small closed neighborhood of the origin in the state space. In order to compute the related value function and optimal feedback control law separately at different initial states and thereby to attenuate the curse of dimensionality, an extension of a recently developed characteristics based framework is proposed. Theoretical foundations of the developed approach are given together with practical discussions regarding its implementation, and numerical examples are also provided. In particular, it is pointed out that the curse of complexity may remain a significant issue even if the curse of dimensionality is avoided.