A Higher-Order Maximum Principle for Impulsive Optimal Control Problems

Abstract

We consider a nonlinear system, affine with respect to an unbounded control u which is allowed to range in a closed cone. To this system we associate a Bolza type minimum problem, with a Lagrangian having sublinear growth with respect to u. This lack of coercivity gives the problem an impulsive character, meaning that minimizing sequences of trajectories happen to converge towards discontinuous paths. As is known, a distributional approach does not make sense in such a nonlinear setting, where, instead, a suitable embedding in the graph-space is needed. We provide higher order necessary optimality conditions for properly defined impulsive minima, in the form of equalities and inequalities involving iterated Lie brackets of the dynamical vector fields. These conditions are derived under very weak regularity assumptions and without any constant rank conditions. Date: July 11, 2019.

Necessary conditions involving Lie brackets for impulsive optimal control problems

“…In particular, this result generalizes to impulsive trajectories a result that, for minimum time problems, has been established for absolutely continuous processes with unbounded controls (see [9]). The detailed proof of the result's main point is rather long and technical, and is provided -under much weaker regularity hypotheses-in [1]. Instead, here we just give some hints of the idea lying behind the stated higher order conditions.…”

“…At this point, we derive assertion (a) as soon as we choose δ such that 2δ + (1 + L)δ < δ ′ , since Ψ(T ,x(T )) = Ψ((ȳ 0 ,ȳ)(S)) ≤ Ψ((y 0 , y)(S)) = Ψ(T, x(T )), for all strict sense feasible processes verifying (11) for such δ. where W is as in (1). For any continuous vector field F : R n → R n , let us introduce the classical F -Hamiltonian…”

“…Instead, the fact that the same multiplier verifies the higher order relations in (vi) needs a proof that exceeds the space limits of the present paper. A proof of a stronger version of this theorem, including very low regularity assumptions on both the vector fields and the target, can be found in [1].…”

“…, g m . Actually, in the result proved in [1] we do not even assume that the vector fields g 1 , . .…”

“…under the 7th Framework Program -Grant agreement SADCO, by CAPES, CNPq, and FAPERJ (Brazil) through the "Jovem Cientista de Nosso Estado" Program and by the von Humboldt Foundation (Germany). 1 regularity hypotheses and replacing the state space R n with a manifold, as done in [1]. Let us point out that problem (P ) has an impulsive character, i.e.…”