Well-Posedness and Strong Convergence of Infinite-Dimensional Port-Hamiltonian Primal-Dual Gradient Dynamics
Abstract
We establish a functional-analytic framework that justifies the LaSalle convergence argument outlined in [van der Schaft, 2026] on port-Hamiltonian primal-dual gradient dynamics for finite-horizon optimal control of linear systems. In addition to strict convexity of the running cost as in \cite{vanderSchaft2026PDGJuly}, our analysis requires global Lipschitz continuity of the cost gradient. This ensures that the nonlinear gradient operator is well-defined and Lipschitz on the entire underlying space. We establish global well-posedness of the optimizer dynamics: every initial value determines a unique global mild solution, and the associated solution operators form a contraction semigroup. Similar to [Gernandtand Schaller, 2025], this is proved using tools from maximal monotone operator theory. Assuming existence of an optimal solution, we combine a dissipation inequality with an infinite-dimensional LaSalle invariance principle to prove strong convergence of every trajectory with a relatively compact positive orbit to the unique optimal state, control, and costate. Moreover, we present a sufficient condition for compactness through uniform control regularity and show that convergence holds for all initial states. Finally, we present an example with a strictly but not strongly convex running cost for which every mild solution converges strongly to the unique equilibrium, while no global exponential estimate uniformly over initial values can be obtained.
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BibTeX
@misc{iftime2026wellposedness,
title = {{Well-Posedness and Strong Convergence of Infinite-Dimensional Port-Hamiltonian Primal-Dual Gradient Dynamics}},
author = {O. V. Iftime and A. J. van der Schaft},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16263},
eprint = {2610.16263},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16263},
}
RIS
TY - UNPB
TI - Well-Posedness and Strong Convergence of Infinite-Dimensional Port-Hamiltonian Primal-Dual Gradient Dynamics
AU - O. V. Iftime
AU - A. J. van der Schaft
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16263
UR - https://arxiv.org/abs/2610.16263
AB - We establish a functional-analytic framework that justifies the LaSalle convergence argument outlined in [van der Schaft, 2026] on port-Hamiltonian primal-dual gradient dynamics for finite-horizon optimal control of linear systems. In addition to strict convexity of the running cost as in \cite{vanderSchaft2026PDGJuly}, our analysis requires global Lipschitz continuity of the cost gradient. This ensures that the nonlinear gradient operator is well-defined and Lipschitz on the entire underlying space. We establish global well-posedness of the optimizer dynamics: every initial value determines a unique global mild solution, and the associated solution operators form a contraction semigroup. Similar to [Gernandtand Schaller, 2025], this is proved using tools from maximal monotone operator theory. Assuming existence of an optimal solution, we combine a dissipation inequality with an infinite-dimensional LaSalle invariance principle to prove strong convergence of every trajectory with a relatively compact positive orbit to the unique optimal state, control, and costate. Moreover, we present a sufficient condition for compactness through uniform control regularity and show that convergence holds for all initial states. Finally, we present an example with a strictly but not strongly convex running cost for which every mild solution converges strongly to the unique equilibrium, while no global exponential estimate uniformly over initial values can be obtained.
ER -