Invariants in Optimal Control: An Exact Solution of the Optimal Stabilization Problem

Gennadii V Kondratiev

doi:10.18052/www.scipress.com/bmsa.21.9

Abstract

The stabilizing optimal feedback is a function onthe separatrix of stable points of the associated Hamiltoniansystem. Three geometric objects - the symplectic form, Hamiltonianvector field, and Lyapunov function, generating the separatrix - are %intrinsic to the optimal control system. They areinvariantly attached to the optimal control system under canonicaltransformations of the phase space. The separatrix equations can be writtenin terms of these invariants through invariant operations.There is a computable representative of the equivalence class,containing the original system. It is its linear approximation system at the stable point.

Highlights

The problem of optimal stabilization is normally reformulated in terms of the associated Hamilto nian system.The separatrix of stable points with respect to the origin of the Hamiltonian system is a Lagrangian manifold of the maximal dimension, which consists of all integral curves going to the origin from a maximal neighbourhood
Invariants are just functions constructed from the base intrinsic objects using operations, invariant under admissible maps, such as the canonical transformations in Hamiltonian Mechanics
A canonical representative whose invariants are com putable is the linear approximation of the system at the stable point

Summary

Introduction

The problem of optimal stabilization is normally reformulated in terms of the associated Hamilto nian system. The Cauchy charac teristics method fails due to the incorrect initial conditions — only one point, the origin of the phase space, is known. In such cases, a powerful method of invariants associated to the object can be successfully used. The paper proposes to use the following invariants — the symplectic form, Hamiltonian vector field, separatrix of stable points, generating Lyapunov function of the separatrix — to obtain exact formulas of the separatrix and optimal feedback. There is only one known point of the separatrix — the origin of the phase space R2n — the method of Cauchy characteristics does not work.

The Method of Invariants

Conclusion