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Discrete variational integrators and optimal control theory

M. de Leon, D. Martin de Diego, A. Santamaria-Merino

TL;DR

The paper proposes a geometric framework for constructing structure-preserving numerical integrators for time-dependent optimal control by merging discrete variational mechanics with OCT. It uses generating functions of the first and second kinds to encode discrete symplectic (and cosymplectic) flows, derives discrete OCT optimality conditions via a Lagrange-multiplier/Hamiltonian approach, and shows how to build symplectic integrators from discrete generating functions. Higher-order schemes are discussed through Hamilton–Jacobi–based expansions, and the discrete Hamiltonian system is shown to admit symplectic maps via simple generating-function constructions. Overall, the work provides a cohesive, geometric method for discretizing OCT that preserves fundamental structure and accommodates explicit time dependence, with implications for long-time accuracy and stability.

Abstract

A geometric derivation of numerical integrators for optimal control problems is proposed. It is based in the classical technique of generating functions adapted to the special features of optimal control problems.

Discrete variational integrators and optimal control theory

TL;DR

The paper proposes a geometric framework for constructing structure-preserving numerical integrators for time-dependent optimal control by merging discrete variational mechanics with OCT. It uses generating functions of the first and second kinds to encode discrete symplectic (and cosymplectic) flows, derives discrete OCT optimality conditions via a Lagrange-multiplier/Hamiltonian approach, and shows how to build symplectic integrators from discrete generating functions. Higher-order schemes are discussed through Hamilton–Jacobi–based expansions, and the discrete Hamiltonian system is shown to admit symplectic maps via simple generating-function constructions. Overall, the work provides a cohesive, geometric method for discretizing OCT that preserves fundamental structure and accommodates explicit time dependence, with implications for long-time accuracy and stability.

Abstract

A geometric derivation of numerical integrators for optimal control problems is proposed. It is based in the classical technique of generating functions adapted to the special features of optimal control problems.

Paper Structure

This paper contains 9 sections, 5 theorems, 82 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Optimal control theory
  3. Generating functions
  4. Generating functions of the first kind
  5. Generating functions of the second kind
  6. Optimal control of Discrete-time systems
  7. Problem solution
  8. Generating functions of the second kind and discrete optimal control problems
  9. Discrete Hamiltonian systems

Key Result

Theorem 3.2

Let $S_1^{(t_N, t_0)}$ be a function defined by where $q_k\in M_{t_k}$, $1\leq k\leq N-1$, are stationary points of the right-hand side, that is If $S^{(t_k, t_{k-1})}_1$ are generating functions of the first kind for $g_{(t_{k}, t_{k-1})}$, then $S^{t_N,t_0)}_1$ is a generating function of the first kind for $g_{(t_N, t_0)}: M_{t_0}\rightarrow M_{t_N}$.

Figures (1)

  • Figure :

Theorems & Definitions (9)

  • Definition 3.1
  • Theorem 3.2
  • Proposition 3.3
  • Remark 3.4
  • Definition 3.5
  • Theorem 3.6
  • Proposition 3.7
  • Theorem 3.8: Hamilton-Jacobi equation for $S_2$
  • Remark 4.1