Hidden Regularity in Singular Optimal Control of port-Hamiltonian Systems

Timm Faulwasser; Jonas Kirchhoff; Volker Mehrmann; Friedrich Philipp; Manuel Schaller; Karl Worthmann

doi:10.52825/dae-p.v3i.960

Authors

Timm Faulwasser Hamburg University of Technology https://orcid.org/0000-0002-6892-7406
Jonas Kirchhoff Technische Universität Ilmenau https://orcid.org/0000-0001-6451-4500
Volker Mehrmann Technische Universität Berlin https://orcid.org/0000-0001-5051-2870
Friedrich Philipp Technische Universität Ilmenau https://orcid.org/0000-0002-4670-8894
Manuel Schaller Chemnitz University of Technology https://orcid.org/0000-0002-8081-5108
Karl Worthmann Technische Universität Ilmenau https://orcid.org/0000-0002-1450-2373

DOI:

https://doi.org/10.52825/dae-p.v3i.960

Keywords:

Singular Optimal Control, port-Hamiltonian Systems, Regular Pencil, Hidden Regularity, Minimal Energy, Differential-Algebraic Equation, Index

Abstract

We study the problem of state transition on a finite time interval with minimal energy supply for linear port-Hamiltonian systems. While the cost functional of minimal energy supply is intrinsic to the port-Hamiltonian structure, the necessary conditions of optimality resulting from Pontryagin's maximum principle may yield singular arcs. The underlying reason is the linear dependence on the control, which makes the problem of determining the optimal control as a function of the state and the adjoint more complicated or even impossible. To resolve this issue, we fully characterize regularity of the (differential-algebraic) optimality system by using the interplay of the cost functional and the dynamics. In case of the optimality DAE being characterized by a regular matrix pencil, we fully determine the control on the singular arc. In case of singular matrix pencils of the optimality system, we propose an approach to compute rank-minimal quadratic perturbations of the objective such that the optimal control problem becomes regular. We illustrate the applicability of our results by a general second-order mechanical system and a discretized boundary-controlled heat equation.

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Hidden Regularity in Singular Optimal Control of port-Hamiltonian Systems

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