# The Inverse Problem of the Calculus of Variations and the Stabilization of Controlled Lagrangian Systems

@article{Puiggal2016TheIP, title={The Inverse Problem of the Calculus of Variations and the Stabilization of Controlled Lagrangian Systems}, author={Marta Farr{\'e} Puiggal{\'i} and Tom Mestdag}, journal={SIAM J. Control. Optim.}, year={2016}, volume={54}, pages={3297-3318} }

We apply methods of the so-called `inverse problem of the calculus of variations' to the stabilization of an equilibrium of a class of two-dimensional controlled mechanical systems. The class is general enough to include, among others, the inverted pendulum on a cart and the inertia wheel pendulum. By making use of a condition that follows from Douglas' classification, we derive feedback controls for which the control system is variational. We then use the energy of a suitable controlled…

## 3 Citations

### An Extension to the Theory of Controlled Lagrangians Using the Helmholtz Conditions

- MathematicsJ. Nonlinear Sci.
- 2019

Using the Helmholtz conditions, this work is able to recover the matching conditions from Bloch et al. (IEEE Trans Autom Control 45(12):2253–2270, 2000) and derive new matching conditions for a particular class of mechanical systems that only depend on the configuration variables.

### Robust output‐feedback control for the cart–pole system: a coupled super‐twisting sliding‐mode approach

- Mathematics, EngineeringIET Control Theory & Applications
- 2019

This study presents an output-feedback controller for a cart-pole system via coupled sliding-mode algorithms. Due to the fact that this system is underactuated, a special type of sliding surface is…

### An Extension to the Theory of Controlled Lagrangians Using the Helmholtz Conditions

- MathematicsJournal of Nonlinear Science
- 2018

The Helmholtz conditions are necessary and sufficient conditions for a system of second-order differential equations to be variational, that is, equivalent to a system of Euler–Lagrange equations for…

## References

SHOWING 1-10 OF 20 REFERENCES

### The Helmholtz Conditions and the Method of Controlled Lagrangians

- Mathematics
- 2015

In this chapter we consider the relationship between the classical inverse problem of the calculus of variations and the method of controlled Lagrangians. The latter is a technique for deriving…

### The Integrability Conditions in the Inverse Problem of the Calculus of Variations for Second-Order Ordinary Differential Equations

- Mathematics
- 1998

A novel approach to a coordinate-free analysis of the multiplier question in the inverseproblem of the calculus of variations, initiated in a previous publication, is completed in thefollowing sense:…

### Controlled Lagrangians and the stabilization of mechanical systems. II. Potential shaping

- MathematicsIEEE Trans. Autom. Control.
- 2001

The method of controlled Lagrangians is extended to include potential shaping, which achieves complete state-space asymptotic stabilization of mechanical systems and extends the method to include a class of mechanical system without symmetry such as the inverted pendulum on a cart that travels along an incline.

### Stabilizability of Controlled Lagrangian Systems of Two Degrees of Freedom and One Degree of Under-Actuation by the Energy-Shaping Method

- MathematicsIEEE Transactions on Automatic Control
- 2010

Criteria for stabilizability by the energy-shaping method for the class of all controlled Lagrangian systems of two degrees of freedom and one degree of under-actuation are provided and it is shown that some of the asymptotically stabilizing controllers that were designed in old literatures with theEnergy-Shaping method are actually exponentially stabilizer controllers.

### The Inverse Problem of the Calculus of Variations: Separable Systems

- Mathematics
- 1999

This paper deals with the inverse problem of the calculus of variations for systems of second-order ordinary differential equations. The case of the problem which Douglas, in his classification of…

### Solution of the Inverse Problem of the Calculus of Variations.

- MathematicsProceedings of the National Academy of Sciences of the United States of America
- 1939

The procedure consists in an application of the Riquier theory of systems of partial differential equations to a certain linear differential system e on which the inverse problem can be made to depend.