On second order conditions for singular optimal control of port-Hamiltonian systems
M. Soledad Aronna, Volker Mehrmann
TL;DR
The paper addresses singular optimal control problems for port-Hamiltonian systems with energy-based costs by applying the Goh and generalized Legendre-Clebsch second-order conditions and exploiting the pH structure. It derives concise second-order conditions for ordinary nonlinear pH systems, nonlinear pH systems with control bounds, and nonlinear pH descriptor systems, then extends the analysis to linear pH systems (with and without control bounds) and to linear descriptor forms via index reduction. A central outcome is that the LC Hessian $W$ is negative semidefinite in general (and definite in favorable cases), enabling potential feedback representations for singular controls, with particularly elegant expressions arising in the linear, energy-minimizing setups. The results provide a rigorous, structure-aware framework for analyzing and synthesizing singular optimal controls in port-Hamiltonian networks, with implications for energy-based engineering applications and model-reduction contexts.
Abstract
We study nonlinear singular optimal control problems of port-Hamil-tonian (descriptor) systems. We employ general control-affine cost functionals that include as a special case the energy supplied to the system. We first derive optimality conditions for the case of ordinary differential equations with and without control bounds by applying the general theory to the specially structured port-Hamiltonian case, and show that this leads to elegant optimality conditions, in particular in the linear case. We then extend these results to classes of nonlinear port-Hamiltonian descriptor systems.