[Paper Review] Discrete variational principles and Hamilton-Jacobi theory for mechanical systems and optimal control problems

Vincent Guibout, Anthony M. Bloch|arXiv (Cornell University)|Sep 17, 2004

Numerical methods for differential equations32 references10 citations

TL;DR

This paper introduces a geometric framework for discrete variational principles on tangent and cotangent bundles, enabling symplectic integration of mechanical systems and optimal control problems. It establishes a discrete Hamilton-Jacobi theory and a discrete maximum principle, proving that energy error is invariant under discrete canonical transformations and yielding symplectic, energy-conserving algorithms via time-as-a-generalized-coordinate formulation.

ABSTRACT

In this paper we present a general framework that allows one to study discretization of certain dynamical systems. This generalizes earlier work on discretization of Lagrangian and Hamiltonian systems on tangent bundles and cotangent bundles respectively. In particular we show how to obtain a large class of discrete algorithms using this geometric approach. We give new geometric insight into the Newmark model for example and we give a direct discrete formulation of the Hamilton-Jacobi method. Moreover we extend these ideas to deriving a discrete version of the maximum principle for smooth optimal control problems. We define discrete variational principles that are the discrete counterpart of known variational principles. For dynamical systems, we introduce principles of critical action on both the tangent bundle and the cotangent bundle. These two principles are equivalent and allow one to recover most of the classical symplectic algorithms. We also identify a class of coordinate transformations that leave the variational principles presented in this paper invariant and develop a discrete Hamilton-Jacobi theory. This theory allows us to show that the energy error in the (symplectic) integration of a dynamical system is invariant under discrete canonical transformations. Finally, for optimal control problems we develop a discrete maximum principle that yields discrete necessary conditions for optimality. These conditions are in agreement with the usual conditions obtained from Pontryagin maximum principle. We illustrate our approach with an example of a sub-Riemannian optimal control.

Motivation & Objective

• To develop a unified geometric framework for discretizing mechanical systems and optimal control problems using variational principles on tangent and cotangent bundles.
• To extend discrete variational integrators to preserve symplectic structure and energy, especially by treating time as a generalized coordinate.
• To establish a discrete Hamilton-Jacobi theory that reveals invariance of energy error under discrete canonical transformations.
• To derive a discrete maximum principle consistent with the classical Pontryagin maximum principle for optimal control problems.
• To provide a foundation for constructing symplectic-energy conserving integrators and multi-symplectic algorithms for complex systems.

Proposed method

• Formulate discrete variational principles on the tangent bundle (discrete Hamilton’s principle) and cotangent bundle (discrete modified Hamilton’s principle), ensuring equivalence and recovery of classical symplectic schemes.
• Introduce a time-discretized action principle with boundary conditions to enforce physical constraints and enable optimal control formulation.
• Treat time as a generalized coordinate to derive energy-conserving symplectic integrators by embedding the system in an extended phase space.
• Define discrete canonical transformations that preserve the variational principles and use them to analyze energy error invariance in symplectic integration.
• Construct a discrete augmented cost function and derive necessary conditions for optimality via variation of the action, leading to a discrete maximum principle.
• Apply the framework to a sub-Riemannian optimal control problem, demonstrating consistency with Pontryagin’s conditions and symplecticity of the resulting algorithm.

Experimental results

Research questions

• RQ1 How can discrete variational principles be generalized to unify symplectic integration and optimal control in a geometric framework?
• RQ2 What is the role of time as a generalized coordinate in enabling energy conservation within symplectic integrators?
• RQ3 How does the discrete Hamilton-Jacobi theory relate to energy error invariance under discrete canonical transformations?
• RQ4 Can a discrete maximum principle be derived that is consistent with the classical Pontryagin maximum principle and yields symplectic algorithms?
• RQ5 What class of coordinate transformations preserves the discrete variational principles, and how does this affect the structure of the resulting integrators?

Key findings

• The discrete variational principles on tangent and cotangent bundles are equivalent and recover most classical symplectic integrators, including those based on generating functions and Runge-Kutta methods.
• By treating time as a generalized coordinate, the framework enables the construction of symplectic-energy conserving algorithms, ensuring conservation of energy in the discrete system.
• The energy error in symplectic integration is invariant under discrete canonical transformations, a key result established via the discrete Hamilton-Jacobi theory.
• The discrete maximum principle yields necessary conditions for optimality that are consistent with the classical Pontryagin maximum principle, validating the approach for optimal control problems.
• The framework allows for the derivation of symplectic algorithms even with nontrivial discretizations, though equivalence to the continuous case may not hold in such cases.
• The method is demonstrated on a sub-Riemannian optimal control problem, where the resulting algorithm is both symplectic and consistent with the necessary conditions of optimality.