# American Institute of Mathematical Sciences

December  2017, 9(4): 459-486. doi: 10.3934/jgm.2017018

## On the relationship between the energy shaping and the Lyapunov constraint based methods

 1 Centro Atómico Bariloche and Instituto Balseiro, 8400 S.C. de Bariloche, and CONICET, Argentina 2 Departamento de Matemática, Facultad de Ciencias Exactas, UNLP, 1900 La Plata, and CONICET, Argentina 3 Departamento de Matemática, Facultad de Ciencias Exactas, UNLP, 1900 La Plata, Argentina

S. Grillo and L. Salomone thank CONICET for its financial support. The authors also thank the referees and the editor for their useful remarks.

Received  December 2016 Revised  April 2017 Published  October 2017

In this paper, we make a review of the controlled Hamiltonians (CH) method and its related matching conditions, focusing on an improved version recently developed by D.E. Chang. Also, we review the general ideas around the Lyapunov constraint based (LCB) method, whose related partial differential equations (PDEs) were originally studied for underactuated systems with only one actuator, and then we study its PDEs for an arbitrary number of actuators. We analyze and compare these methods within the framework of Differential Geometry, and from a purely theoretical point of view. We show, in the context of control systems defined by simple Hamiltonian functions, that the LCB method and the Chang's version of the CH method are equivalent stabilization methods (i.e. they give rise to the same set of control laws). In other words, we show that the Chang's improvement of the energy shaping method is precisely the LCB method. As a by-product, coordinate-free and connection-free expressions of Chang's matching conditions are obtained.

Citation: Sergio Grillo, Leandro Salomone, Marcela Zuccalli. On the relationship between the energy shaping and the Lyapunov constraint based methods. Journal of Geometric Mechanics, 2017, 9 (4) : 459-486. doi: 10.3934/jgm.2017018
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