Optimal control of port-Hamiltonian descriptor systems with minimal energy supply

TL;DR

This paper develops an optimal control framework for port-Hamiltonian descriptor systems, demonstrating a global turnpike property towards a conservative subspace and characterizing dissipative Hamiltonian matrices.

Contribution

It introduces a reduction to an ODE with feed-through, derives a turnpike property for optimal control, and characterizes dissipative Hamiltonian matrices and pencils.

Findings

Optimal states exhibit a turnpike behavior towards a conservative subspace.

The turnpike property is global in the initial state despite control constraints.

Characterization of dissipative Hamiltonian matrices and pencils.

Abstract

We consider the singular optimal control problem of minimizing the energy supply of linear dissipative port-Hamiltonian descriptor systems subject to control and terminal state constraints. To this end, after reducing the problem to an ODE with feed-through term, we derive an input-state turnpike towards a subspace for optimal control of generalized port-Hamiltonian ordinary differential equations. We study the reachability properties of the system and prove that optimal states exhibit a turnpike behavior with respect to the conservative subspace. By means of the port-Hamiltonian structure, we show that, despite control constraints, this turnpike property is global in the initial state. Further, we characterize the class of dissipative Hamiltonian matrices and pencils.