Necessary conditions involving Lie brackets for impulsive optimal control problems

Abstract

We obtain higher order necessary conditions for a minimum of a Mayer optimal control problem connected with a nonlinear, control-affine system, where the controls range on an m-dimensional Euclidean space. Since the allowed velocities are unbounded and the absence of coercivity assumptions makes big speeds quite likely, minimizing sequences happen to converge toward impulsive, namely discontinuous, trajectories. 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 will illustrate how the chance of using impulse perturbations makes it possible to derive a Higher Order Maximum Principle which includes both the usual needle variations (in space-time) and conditions involving iterated Lie brackets. An example, where a third order necessary condition rules out the optimality of a given extremal, concludes the paper.

A Higher-Order Maximum Principle for Impulsive Optimal Control Problems

“…3 Through minor changes one might generalize this hypothesis with the fact that, for every (x, u), the function a → (f (x, a), l(x, u, a)) is bounded. 4 Hypothesis (i) on C is by no means restrictive, since it can be recovered by replacing the single vector fields gi with suitable linear combinations of {g1, . .…”

“…The necessary conditions established in Theorems 3.1 and 4.1 can be used to get information on the structure of optimal trajectories: for instance, one can wonder under which conditions an optimal trajectory is a finite concatenation of impulsive and non impulsive paths (as it occurs e.g. in the example in [4]). Though an accurate investigation in this direction goes beyond the objectives of this paper, let us highlight some rank conditions that happen to force an optimal process ( S, w0 , w, ᾱ, ȳ0 , ȳ, β) to be fully impulsive.…”

“…In particular, the jump x( t+) − x( t−) = y(s 2 ) − y(s 1 ) depends on the restriction w | [s 1 ,s 2 ] . Our necessary conditions concern a minimum for the extended, space-time, problem (3)- (4). We begin by exploiting the rate-independence of the extended problem in order to establish a First Order Maximum Principle, in Theorem 3.1.…”

“…. , m 1 , and C 2 ⊂ R m 2 is a closed cone which does not contain straight lines;4 (ii) the drift dynamics f : R n ×A → R n is continuous and has continuous partial the vector fields g 1 , . .…”

Unbounded Control, Infimum Gaps, and Higher Order Normality

“…This linear separability translates into the abnormality of the minimizer. On the other hand, in the first order Maximum Principle the approximating cones to the reachable set are built as convex hulls of the so-called needle variations, while several higher order Maximum Principles are basically obtained by enlarging these first order approximating cones by means of new higher order (in the time-scale) variations involving iterated Lie brackets (see, e.g., [2,3,10,19,18,30,6,7,33] and references therein). So, the question arises whether we can prove a gap-abnormality relation involving iterated Lie brackets.…”