  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.
