[Paper Review] High-order convergent Finite-Elements Direct Transcription Method for Constrained Optimal Control Problems

Martin Neuenhofen|arXiv (Cornell University)|Dec 21, 2017

Spacecraft Dynamics and Control19 references3 citations

TL;DR

This paper presents a high-order convergent finite element method for solving constrained nonlinear optimal control problems with general differential-algebraic equations (DAEs), pointwise constraints, and inequality constraints. By using a regularized penalty-barrier functional and leveraging local strict convexity and Lipschitz continuity in the finite-element space, the method achieves high-order convergence under mild assumptions, enabling flexible, component-wise mesh refinement and efficient sparsity-aware NLP solution via general-purpose solvers.

ABSTRACT

In this paper we present a finite element method for the direct transcription of constrained non-linear optimal control problems. We prove that our method converges of high order under mild assumptions. Our analysis uses a regularized penalty-barrier functional. The convergence result is obtained from local strict convexity and Lipschitz-continuity of this functional in the finite-element space. The method is very flexible. Each component of the numerical solution can be discretized with a different mesh. General differential-algebraic constraints of arbitrary index can be treated easily with this new method. From the discretization results an unconstrained non-linear programming problem (NLP) with penalty- and barrier-terms. The derivatives of the NLP functions have a sparsity pattern that can be analysed and tailored in terms of the chosen finite-element bases in an easy way. We discuss how to treat the resulting NLP in a practical way with general-purpose software for constrained non-linear programming.

Motivation & Objective

• To develop a high-order convergent numerical method for constrained optimal control problems with general DAEs, pointwise constraints, and inequality constraints.
• To establish rigorous convergence guarantees for the proposed method under mild assumptions, particularly through the use of a regularized penalty-barrier functional.
• To enable flexible spatial discretization by allowing different meshes for each component of the solution vector.
• To ensure that the resulting nonlinear programming (NLP) problem preserves a structured sparsity pattern amenable to efficient solution with general-purpose solvers.
• To provide a practical framework for solving the discrete NLP using interior-point methods with logarithmic barriers, ensuring equivalence to the original problem under appropriate parameter settings.

Proposed method

• The method employs a finite element discretization of the state and control variables over time, with independent mesh refinement for each component of the solution vector.
• A regularized penalty-barrier functional is used to handle constraints, combining equality constraints (via penalty terms) and inequality constraints (via barrier terms) into a single unconstrained NLP formulation.
• The cost functional and constraint functions are discretized using high-order finite elements, ensuring high-order convergence of the numerical solution to the true optimal control.
• The resulting NLP has a sparsity pattern that is explicitly tied to the chosen finite-element basis functions, enabling efficient computation and exploitation in linear algebra kernels.
• The NLP is solved using interior-point methods such as IPOPT or Knitro, with logarithmic barriers and a parameter continuation strategy to maintain equivalence to the original problem.
• The method allows for the use of a small, fixed penalty parameter ω and barrier parameter τ, with the option to set τ = ϑ in the solver to ensure equivalence to the original problem.

Experimental results

Research questions

• RQ1 Can a finite element method be developed that achieves high-order convergence for constrained optimal control problems with general DAEs and inequality constraints?
• RQ2 Does the use of a regularized penalty-barrier functional ensure convergence to locally optimal solutions under mild assumptions?
• RQ3 Can the method support independent, adaptive mesh refinement for each component of the solution vector without compromising convergence or sparsity?
• RQ4 How can the resulting NLP be structured to preserve a sparsity pattern that enables efficient solution with general-purpose nonlinear programming solvers?
• RQ5 Is it possible to maintain equivalence to the original optimal control problem when reformulating the NLP using penalty and barrier terms, especially when using interior-point methods?

Key findings

• The proposed finite element method achieves high-order convergence for constrained optimal control problems under mild assumptions, including local strict convexity and Lipschitz continuity of the regularized penalty-barrier functional in the finite-element space.
• The method allows for independent, component-wise mesh refinement, enabling efficient resolution of different solution components with varying regularity or dynamics.
• The resulting NLP inherits a structured sparsity pattern that is directly determined by the finite-element basis functions, enabling efficient exploitation in linear algebra operations.
• The NLP formulation is designed to be compatible with general-purpose interior-point solvers like IPOPT and Knitro, with the option to set the barrier parameter ϑ = τ to ensure equivalence to the original problem.
• Numerical experiments suggest that the method remains feasible even when the number of quadrature points exceeds the number of degrees of freedom, unlike alternative formulations that may become infeasible.
• The method provides a robust and flexible framework for solving complex optimal control problems with DAEs, pointwise constraints, and inequality constraints, with convergence guarantees and practical solvability.