Control problems with differential constraints of higher order

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 uo(t) and a set of initial conditions σo 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),σ(s)) of control curves and initial data starting from uo(t)=u(t,0) and σo=σ(0). 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.