We consider cost minimising control problems, in which the dynamical system is constrained by higher order differential equations of Euler–Lagrange type. Following ideas from a previous paper, we prove that a curve of controls $u_o\left(t\right)$ and a set of initial conditions ${\sigma }_o$ give an optimal solution for a control problem of the considered type if and only if an appropriate double integral is greater than or equal to zero along any homotopy $(u(t,s),\sigma \left(s\right))$ of control curves and initial data starting from $u_o\left(t\right)=u(t,0)$ and ${\sigma }_o=\sigma \left(0\right)$. This property is called Principle of Minimal Labour. From this principle we derive a generalisation of the classical Pontryagin Maximum Principle that holds under higher order differential constraints of Euler–Lagrange type and without the hypothesis of fixed initial data.

我们考虑成本最小化的控制问题，其中动力学系统受到Euler-Lagrange类型的高阶微分方程的约束。根据先前论文的想法，我们证明了控制曲线${ü}_{Ø}（Ť）$ 和一组初始条件 ${\sigma }_{Ø}$ 当且仅当沿着任何同伦的适当对偶积分大于或等于零时，才可以为所考虑类型的控制问题提供最优解 $（ü（Ť，s），\sigma （s））$ 控制曲线和初始数据从 ${ü}_{Ø}（Ť）=ü（Ť，0）$ 和 ${\sigma }_{Ø}=\sigma （0）$。此属性称为“最小劳动原理”。从这一原理中，我们得出了经典庞特里亚金最大原理的推广，该原理在欧拉-拉格朗日类型的高阶微分约束下且没有固定初始数据的假设下成立。