Second-Order Necessary Conditions in Pontryagin Form for Optimal Control Problems

Abstract

sn this reportD we stte nd prove (rstE nd seondEorder neessry onditions in ontrygin form for optiml ontrol prolems with pure stte nd mixed ontrolEstte onstrintsF e sy tht vgrnge multiplier of n optiml ontrol prolem is ontrygin multiplier if it is suh tht ontrygin9s minimum priniple holdsD nd we ll optimlity onditions in ontrygin form those whih only involve ontrygin multipliersF yur onditions rely on tehnique of prtil relxtionD nd pply to ontrygin lol minimF Key-words: yptiml ontrolY pure stte nd mixed ontrolEstte onstrintsY ontrygin9s prinipleY ontrygin multipliersY seondEorder neessry onditionsY prtil relxtionFThe research leading to these results has received funding from the EU 7th Framework Programme (FP7-PEOPLE-2010-ITN), under GA number 264735-SADCO, and from the Gaspard Monge Program for Optimization and operations research (PGMO). * Inria-Saclay and CMAP, Ecole Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France.Emails: frederic.bonnans@inria.fr, xavier.dupuis@cmap.polytechnique.fr, laurent.pfeier@polytechnique.edu. Conditions nécessaires du second ordre sous formePontryaguine pour des problèmes de commande optimale Résumé : hns e rpportD nous énonçons et prouvons des onditions néessires du premier et seond ordre sous forme ontryguine pour des prolèmes de ommnde optimle ve onE trintes pures sur l9étt et mixtes sur l9étt et l ommndeF xous ppelons multipliteur de ontryguine tout multipliteur de vgrnge pour lequel le prinipe de ontryguine est stisE fit et prlons de onditions d9optimlité sous forme ontryguine si elles ne font intervenir que des multipliteurs de ontryguineF xos onditions s9ppuient sur une tehnique de relxtion prtielle et sont vlles pour des minim de ontryguineF

Second Order Analysis for Strong Solutions in the Optimal Control of Parabolic Equations

“…When we consider only control constraints, the equivalence between the two notions of solutions holds true without any qualification condition (see Theorem 5.4). Of course, these arguments provide also the proof of the analogous statements in the case of optimal control problems of ordinary differential equations (see [8,9]) and of semilinear elliptic equations (see [3]). …”

Second order conditions for a control problem with discontinuous cost

“…To express optimality conditions, it is convenient to write U and K as sub-level sets of given functions satisfying qualification conditions. We fix for the rest of the article a solutionū ∈ U to (8), with associated trajectoryx := xū, satisfying Assumption (H1). (H1) There is a function c : R m → R l of class C 2 such that…”

Jacobson type necessary optimality conditions for general control systems

“…Theorem 5: Let (x,ū) ∈ P 0 (K 0 ) be a strong local minimizer, Assumptions 1-3 hold true andp solve (3). Then for all (u, v) ∈ M (2) (ū) and all (y 0 , w 0 ) ∈ J 2 K0 (x(0)) − p(0), w 0 + 1 2 ϕ (x(1))y(1)y(1)…”

“…In the literature dealing with general (smooth) dynamics and control constraints, second order necessary optimality conditions are usually stated by requiring a certain quadratic functional to be nonnegative on a set of critical control maps, see for instance [3], [6], [8], [11], [16]- [18] and the references therein. Such conditions are not simple to check because they have to be satisfied on a set of functions.…”

“…This creates an important gap between the optimality conditions in integral form that hold under quite general assumptions, see for instance [3], [8], [18] and the references therein, and pointwise conditions. For instance, the Goh type conditions, despite some advances in recent years, are available only under some structural assumptions on U , supposed to be time independent.…”