We investigate optimal control of linear port-Hamiltonian systems with control constraints, in which one aims to perform a state transition with minimal energy supply. Decomposing the state space into dissipative and non-dissipative (i.e. conservative) subspaces, we show that the set of reachable states is bounded w.r.t. the dissipative subspace. We prove that the optimal control problem exhibits the turnpike property with respect to the non-dissipative subspace, i.e., for varying initial conditions and time horizons optimal state trajectories evolve close to the conservative subspace most of the time. We analyze the corresponding steady-state optimization problem and prove that all optimal steady states lie in the non-dissipative subspace. We conclude this paper by illustrating these results by a numerical example from mechanics.

我们研究了具有控制约束的线性 port-Hamiltonian 系统的最优控制，其中一个目标是用最少的能量供应来执行状态转换。将状态空间分解为耗散和非耗散（即保守）子空间，我们表明可到达状态集是有界的，与耗散子空间有关。我们证明了最优控制问题在非耗散子空间方面表现出收费公路的特性，即对于不同的初始条件和时间范围，大多数时候最优状态轨迹演化接近保守子空间。我们分析了相应的稳态优化问题，并证明所有最优稳态都位于非耗散子空间中。我们通过力学的数值例子说明这些结果来结束本文。