Well-Posedness and Strong Convergence of Infinite-Dimensional Port-Hamiltonian Primal-Dual Gradient Dynamics
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.