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 by the first and the third author, we prove that a curve of controls $u_o(t)$ and a set of initial conditions $σ_o$ gives 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 $u_o(t) = u(t, 0)$ and $σ_o = σ(0)$. 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.

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