Relatório Técnico do Núcleo de Investigação Officina Mathematica.Standard necessary conditions of optimality (NCO) for constrained optimal control problems -- Maximum Principle type onditions -- may fail to provide useful information to select candidates to minimizers among the overall set of admissible solutions.  This
phenomenon is known as the degeneracy phenomenon and there has been
continuing interest in the literature in proposing stronger forms of NCO that can be informative in such cases: the so-called nondegenerate NCO. The nondegenerate NCO proposed here are validunder a different set of hypothesis and under a constraint qualification of an integral-type that, in relation to some previous literature, can be verified for more problems.Fundação para a Ciência e a Tecnologia (FCT)
Projecto FCT POSC/EEA-SRI/61831/200

Fontes, Fernando A. C. C.

Lopes, Sofia Oliveira

Pinho, Maria do Rosário de

Universidade do Minho: RepositoriUM

An Integral-type ConstraintQualification for Optimal ControlProblems with State Constraints ∗S. Lopes, F. A. C. C. Fontes†and M. d. R. de Pinho‡Officina Mathematica report, April 4, 2007AbstractStandard necessary conditions of optimality (NCO) for constrainedoptimal control problems – Maximum Principle type conditions – may failto provide useful information to select candidates to minimizers amongthe overall set of admissible solutions. This phenomenon is known as thedegeneracy phenomenon and there has been continuing interest in theliterature in proposing stronger forms of NCO that can be informative insuch cases: the so-called nondegenerate NCO. The nondegenerate NCOproposed here are valid under a different set of hypothesis and undera constraint qualification of an integral-type that, in relation to someprevious literature, can be verified for more problems.1 IntroductionIn this report we are interested in Necessary Conditions of Optimality – Max-imum Principle type conditions – for optimal control problem with pathwisestate constraints. We consider the following problemMinimize g(x(1)) (1)subject tox˙(t) = f (t, x(t), u(t)) a.e. t ∈ [0, 1] (2)(P ) x(0) = x0x(1) ∈ Cu(t) ∈ Ω(t) a.e. t ∈ [0, 1]h (t, x(t)) ≤ 0 for all t ∈ [0, 1], (3)∗The financial support from Projecto FCT POSC/EEA-SRI/61831/2004 is gratefully ac-knowledged.†S. Lopes and F. A. C. C. Fontes are with Officina Mathematica, Departamento deMatema´tica para a Cieˆncia e Tecnologia, Universidade do Minho, 4800-058 Guimara˜es, Por-tugal. sofialopes@mct.uminho.pt, ffontes@mct.uminho.pt,‡M. d. R. de Pinho is with ISR and DEEC-FEUP, University of Porto, Portugal.mrpinho@fe.up.pt,1 INTRODUCTION 2for which the data comprises functions g : IRn 7→ IR, f : [0, 1]×IRn×IRm 7→ IRn,h : [0, 1]× IRn 7→ IR, a set C and a multifunction Ω : [0, 1]⇒ IRm.The set of control functions for (P ) isU := {u : [0, 1] 7→ IRm : u is a measurable function, u(t) ∈ Ω(t) a.e. t ∈ [0, 1]} .The state trajectory is an absolutely continuous function which satisfies (2). Thedomain of the above optimization problem is the set of admissible processes,namely pairs (x, u) comprising a control function u and a corresponding statetrajectory x which satisfy the constraints of (P ). We say that an admissibleprocess (x¯, u¯) is a strong local minimizer if there exists δ > 0 such thatg(x¯(1)) ≤ g(x(1))for all admissible processes (x, u) satisfying‖x(t)− x¯(t)‖L∞ ≤ δ.It is well known that, the necessary conditions for such problems appears inthe form of Maximum Principle (MP). The original formulation of the MP (see[1]) applied to problems with very regular properties for the involved functions.However, over the last three decades continuous development allow to reformu-late the MP for ”nonsmooth” data (data that can be non differentiable), freeendtime constraints, a broad description of endpoint constraints, and other re-finements. Se for example [2], [3], [4], [5], [6]. We consider here a nonsmoothversion of the MP, as in [6].We assume that problem (P) satisfies the following set of hypothesis:There exists a positive scalar δ′ such that:H1 The function (t, u) 7→ f(t, x, u) is L × B measurable for each x. (L × Bdenotes the product σ-algebra generated by the Lebesgue subsets L of[0, 1] and the Borel subsets of IRm.)H2 There exists a L×B measurable function k(t, u) such that t 7→ k(t, u¯(t)) isintegrable and‖f(t, x, u)− f(t, x′, u)‖ ≤ k(t, u)‖x− x′‖for x, x′ ∈ x¯(t) + δ′B, u ∈ Ω(t) a.e. t ∈ [0, 1]. Furthermore there existscalars Kf > 0 and ′ > 0 such that‖f(t, x, u)− f(t, x′, u)‖ ≤ Kf‖x− x′‖for x, x′ ∈ x¯(0) + δ′B, u ∈ Ω(t) a.e. t ∈ [0, ′].H3 The function g is Lipschitz continuous on x¯(1) + δ′B.H4 The end-point constraint set C is closed.H5 The graph of Ω is L × B measurable.H6 The function h is upper semicontinuous and there exists a scalar Kh > 0such that the function x 7→ h(t, x) is Lipschitz of rank Kh for all t ∈ [0, 1].1 INTRODUCTION 3Define the HamiltonianH(t, x, p, u) = p · f(t, x, u).The Necessary Conditions assert existence of an absolutely continuous func-tion p, a measurable function γ, a non-negative measure µ representing anelement in C∗([0, 1] : IR) and λ ≥ 0 such that−p˙(t) ∈ co ∂xH(t, x¯(t), q(t), u¯(t)) a.e. t ∈ [0, 1], (4)−q(1) ∈ NC(x¯(1)) + λ∂g(x¯(1)), (5)γ(t) ∈ ∂>x h(t, x¯(t)) µ a.e. , (6)supp{µ} ⊂ {t ∈ [0, 1] : h (t, x¯(t)) = 0} , (7)for almost every t ∈ [0, 1], u¯(t) maximizes over Ω(t)u 7→ H(t, x¯(t), q(t), u) (8)and,µ{[0, 1]}+ ‖p‖L∞ + λ > 0, (9)whereq(t) =p(t) +∫[0,t)γ(s)µ(ds) t ∈ [0, 1)p(t) +∫[0,1]γ(s)µ(ds) t = 1.Here, ∂f denotes the limiting subdifferential of f , NC denotes the limitingnormal cone and ∂>x h denotes the hybrid partial subdifferential . (The defini-tions are in Section 2)Stronger forms of NCO for optimal control problems that guarantee nonde-generacy and/or normality were discussed in [7]. In this report we are interested,in the strong form of the MP introduced in [8], which ensures that the MP canbe written with the nontriviality conditionµ{(0, 1]}+ ‖q‖L∞ + λ > 0,if the problem satisfies the following additional condition - constraint qualifica-tion:(CQ) If h(0, x0) = 0, then there exist positive constants ,1, δ, and a controlu˜ ∈ Ω(t) such that for a.e. t ∈ [0, )ζ · [f(t, x0, u˜(t))− f(t, x0, u¯(t))] < −δfor all ζ ∈ ∂>x h(s, x), s ∈ [0, ), x ∈ {x0}+ 1B.In this report, the main result consist of a stronger MP, as above, but re-quiring a weaker constraint qualification of an integral-type:2 PRELIMINARIES 4(CQ∗) If h(0, x0) = 0, then there exist positive constants Ku, 1, δ, τ¯∗ ∈ (0, 1]and a control u˜ ∈ Ω(t) such that for all τ ∈ [0, τ¯∗]∫ τ0ζ · [f(t, x0, u˜(t))− f(t, x0, u¯(t))]dt < −δτfor all ζ ∈ ∂>x h(s, x), s ∈ [0, τ¯∗), x ∈ {x0}+ 1BNote that CQ implies CQ∗, so the new constraint qualification CQ∗ is moreinteresting as a condition since it can be verified for more problems.However, the NCO given here (valid under CQ∗) require a convex velocityset as an additional hypothesis.2 PreliminariesThroughout, B will denote the closed unit ball in Euclidean space and coS theconvex hull of a set S.The limiting normal cone of a closed set C ⊂ IRn at x¯ ∈ C, denoted byNC(x¯), is the setNC(x¯) := {η ∈ IRn : ∃ sequences {Mi} ∈ IR+, xi → x¯, ηi → η such that xi ∈ Cand ηi · (y − xi) ≤Mi|y − xi|2 for all y ∈ IRn, i = 1, 2, ...}.The limiting subdifferential of a lower semicontinuous function f : IRn →IR∪{∞} at a point x¯ ∈ IRn such that f(x¯) < +∞, denoted by ∂f(x¯), is definedto be∂f(x¯) = {η ∈ Rn : (η,−1) ∈ Nepif (x¯, f(x))};where epif = {(x, α) ∈ Rn+1 : α ≥ f(x)} denotes the epigraph of a function f .We define also ∂>x h(t, x), to be the following the hybrid partial subdifferentialof h in the x-variable∂>x h(t, x) := co{ξ : there exist (ti, xi)→ (t, x) s.t. h(ti, xi) > 0,h(ti, xi)→ h(t, x), and ∇xh(ti, xi)→ ξ}See [6] for a review of Nonsmooth Analysis and related concepts using asimilar notation.3 Main ResultIn addition to (H1)-(H6), we assume thatH7 There exists positive constants τ¯∗ and 1 such that f(t, x,Ω(t)) is convexfor all t ∈ [0, τ¯∗) and for all x ∈ {x0}+ 1B,and(CQ∗) (constraint qualification) If h(0, x0) = 0 then there exist positive con-stants Ku, 1, δ, τ¯∗ ∈ (0, 1] and a control u˜ ∈ Ω(t) such that for allτ ∈ [0, τ¯∗]‖f(t, x0, u¯(t))‖ ≤ Ku, ‖f(t, x0, u˜(t))‖ ≤ Ku,4 PROOF OF THE MAIN RESULT 5and ∫ τ0ζ · [f(t, x0, u˜(t))− f(t, x0, u¯(t))]dt < −δτfor all ζ ∈ ∂>x h(s, x), s ∈ [0, τ¯∗), x ∈ {x0}+ 1B.Proposition 3.1 Assume hypotheses (H1)-(H7) and (CQ∗). Then the nontriv-iality condition (9) of the Maximum Principle can be replaced by the strongerconditionµ{(0, 1]}+ ‖q‖L∞ + λ > 0. (10)As in ([8]), the proof consists of making an appropriate modification of thedata ”near” to the left endpoint and then apply the standard Maximum Prin-ciple.4 Proof of the Main ResultWe assume that h(0, x0) = 0, since otherwise the Maximum Principle cannotbe satisfied by the trivial multipliers.Step 1: Consider, for α ∈ (0, 1], the system of equations (S)(S)x˙(t) = f(t, x(t), u¯(t)) + y(t) ·∆f(t, x(t)) a.e. t ∈ [0, α]x(0) = x0,y˙(t) = 0 a.e. t ∈ [0, α]y(0) ∈ [0, 1](11)where we define∆f(t, x) := f(t, x, u˜(t))− f(t, x, u¯(t)). (12)Here u˜ is the control function featuring in the constraint qualification CQ∗.Since y˙ = 0 and the function y is absolutely continuous, it will be constant.We will here denote the value of the function simply by y in [0, α] instead ofy(t), when it is not ambiguous.Step 2: By reducing the size of α, we can ensure thath(t, x(t)) ≤ 0 for all t ∈ [0, α], (13)for all trajectories x solving system (S).For that, we start by introducing the following lemma, which is a simpleconsequence of the hypotheses imposed on the data and standard Gronwall-type estimates. A similar proof can be found in [9].Lemma 4.1 Consider the functions x, y solving the system of equations (S),and x¯ solving (P ). There exist positive constants A, and B such that for α smallenough‖x(t)− x0‖ ≤ At‖x(t)− x¯(t)‖ ≤ Bytfor all t ∈ [0, α].4 PROOF OF THE MAIN RESULT 6Choose an α satisfyingα < min{2δKhKf (2A+B),1A, τ¯∗}. (14)Suppose, in contradiction, that for some fixed t ∈ [0, α]h(t, x(t)) > 0. (15)Define for β ∈ [0, 1]r(β) := h(t, x¯(t) + β(x(t)− x¯(t))).In view of the properties of h as a function of x, r is continuous. We havealso thatr(0) = h(t, x¯(t)) ≤ 0r(1) = h(t, x(t)) > 0It follows that the setD := {β ∈ [0, 1] : r(β) = 0}is non-empty, closed and bounded. We can therefore defineβm := maxβ∈Dβ.Since r(1) > 0, we have βm < 1.Take any β ∈ (βm, 1].Applying the Lebourg Mean-Value Theorem ([2]), we obtainh(t, x(t))− r(β) = ζt · [x(t)− x¯(t)− β(x(t)− x¯(t))]= (1− β)ζt · [x(t)− x¯(t)]for some ζt ∈ co ∂xh(t, xˆ), and xˆ in the segment (x(t), x¯(t) + β[x(t)− x¯(t)]).As r(β) > 0 for all β ∈ (βm, 1], we have that h(t, xˆ) > 0, which implies thatco∂xh(t, xˆ) ⊂ ∂>x h(t, xˆ). It follows that ζt ∈ ∂>x h(t, xˆ).Expanding the expression above yieldsh(t, x(t))− r(β)= (1− β) ζt ·∫ t0[f(s, x(s), u¯(s)) + y∆f(s, x(s))− f(s, x¯(s), u¯(s))] ds≤ (1− β)(ζt ·∫ t0y∆f(s, x(s)) ds +‖ζt‖Kf∫ t0‖x(s)− x¯(s)‖ ds)≤ (1− β)(∫ t0ζt · y∆f(s, x0) ds + 2Kf‖ζt‖y∫ t0‖x(s)− x0‖ ds+KhKf∫ t0‖x(s)− x¯(s)‖ ds)≤ (1− β) ∫ t0ζt · y∆f(s, x0)ds+ 2KfKhy∫ t0‖x(s)− x0‖ds+KhKf∫ t0‖x(s)− x¯(s)‖ds≤ (1− β) (−yδt+KhKfy(A+B/2)t2)≤ 0 for all β ∈ (βm, 1].4 PROOF OF THE MAIN RESULT 7Here we have used the fact that the norm of every element of the subdifferen-tial is bounded by the Lipschitz rank of the function. In the last two inequalitieswe have used CQ∗ and (14).Since r is continuous and r(βm) = 0, it follows thath(t, x(t)) ≤ 0.Step 3: Take a decreasing sequence {αi} on (0, α), converging to zero.Associate with each αi the following problem (Pi), in which satisfaction of thestate constraint is enforced only on the subinterval [αi, 1].Minimize g(x(1))subject tox˙(t) = f(t, x(t), u¯(t)) + y(t) ·∆f(t, x(t))a.e. t ∈ [0, αi)x˙(t) = f (t, x(t), u(t))a.e. t ∈ [αi, 1]y˙(t) = 0 a.e. t ∈ [0, αi)(Pi) x(0) = x0x(1) ∈ Cy(0) ∈ [0, 1]u(t) ∈ Ω(t) a.e. t ∈ [αi, 1]h (t, x(t)) ≤ 0for all t ∈ [αi, 1].By convexity hypotheses (H7) all possible components x trajectories of (Pi)are contained in the set of possible trajectories of (P ). Moreover the trajectoryfor (Pi) y ≡ 0 and x ≡ x¯ will lead to a cost g(x¯(1)) identical to the optimal cost(P ). We have proved the following Lemma.Lemma 4.2 The trajectory y ≡ 0 and x ≡ x¯ solves all problems (Pi).The necessary conditions for problem (Pi) assert the existence of an arc(pi, ci) : [0, 1] 7→ IRn × IR, a measurable function γi, a nonnegative Radonmeasure µi ∈ C∗([αi, 1], IR), and a scalar λi ≥ 0 such thatµi{[αi, 1]}+ ‖(pi, ci)‖+ λi > 0, (16)−p˙i(t) ∈pi(t) · co∂xf(t, x¯(t), u¯(t)),a.e. t ∈ [0, αi);(pi(t) +∫[αi,t)γi(s)µi(ds))· co∂xf(t, x¯(t), u¯(t)),a.e. t ∈ [αi, 1].(17)−c˙i(t) ={pi(t) ·∆f(t, x¯(t)), a.e. t ∈ [0, αi);0, a.e. t ∈ [αi, 1]. (18)−(pi(1) +∫[αi,1]γi(s)µi(ds) + λiξi)∈ NC(x¯(1)) (19)4 PROOF OF THE MAIN RESULT 8where ξi ∈ ∂xg(x¯(1))−ci(1) = 0{0 = λi∂yg(x¯(1))} (20)ci(0) ∈ N[0,1](0)γi(t) ∈ ∂>x h(t, x¯(t)) µ a.e. ,supp{µi} ⊂ {t ∈ [αi, 1] : h (t, x¯(t)) = 0} , (21)and for almost every t ∈ [αi, 1], u¯(t) maximizes over Ω(t)u 7→(pi(t) +∫[αi,t)γi(s)µi(ds))· f (t, x¯(t), u) . (22)It remains to pass to the limit as i → ∞ and thereby to obtain a set ofnondegenerate multipliers for the original problem.Without changing the notation, we extend µi as a regular Borel measure on[0, 1]µi(B) = µi(B ∩ [αi, 1]) for all Borel set B ⊂ [0, 1].Extend also γi, originally defined on [αi, 1], arbitrarily to the interval [0, 1] asa Borel measurable function. With theses extensions, noting that µ([0, αi)) = 0,we can write−p˙i(t) ∈(pi(t) +∫[0,t)γi(s)µi(ds))· co∂xf(t, x¯(t), u¯(t))a.e. t ∈ [0, 1].It can be easily seen that ci can be omitted from (16), since pi ≡ 0 impliesci ≡ 0. By scaling the multipliers we can then ensure that‖µi{[αi, 1]}‖T.V. + ‖pi‖L∞ + λi = 1.The multifunction ∂>x is uniformly bounded, compact, convex, and has aclosed graph. As {pi} is uniformly bounded and {p˙i} is uniformly integrallybounded, we can arrange by means of subsequence extraction [2, Thm 3.1.7,Prop. 3.1.8] thatpi → p uniformly, γidµi → γdµ weak∗, λi → λ, ξi → ξ,where µ is the weak∗ limit of µi in the space of nonnegative-valued functions inC∗([0, 1], IR), γ is a measurable selection of ∂>x h(t, x¯(t)) µ a.e., and ξ ∈ ∂g(x¯(1)).To obtain ξ we have used the fact that ∂g(x¯(1)) is a compact set.It follows that the conditions (7), (9), (4) for problem (P) are satisfied andas NC(x¯(1)) is closed, (5) also holds.Consider the set Si = [αi, 1]\Ωi where Ωi is a null Lebesgue measure set in[αi, 1] containing all times where the maximization of (22) is not achieved at u¯.We can then write(pi(t) +∫[αi,t)γi(s)µi(ds))· f (t, x¯(t), u) ≤(pi(t) +∫[αi,t)γi(s)µi(ds))· f (t, x¯(t), u¯(t)) ,4 PROOF OF THE MAIN RESULT 9for all t ∈ Si and for all u ∈ Ω(t).Now consider the full measure set S = (0, 1]\⋃i Ωi. Fix some t in S. Thenfor all i > N , where N is such that αN ≤ t we have(pi(t) +∫[0,t)γi(s)µi(ds))· f (t, x¯(t), u) ≤(pi(t) +∫[0,t)γi(s)µi(ds))· f (t, x¯(t), u¯(t)) .for all u ∈ Ω(t). Applying limits to both sides of this inequality we obtain (8).At this point we have established that the set of multipliers (p, µ, λ), ob-tained as limit of (pi, µi, λi) satisfy the necessary conditions of optimality forthe original problem (P ).Step 4: It remains to verifyµ{(0, 1]}+ ||q||L∞ + λ > 0. (23)We start by proof the following lemma:Lemma 4.3 The adjoint vector pi in the necessary conditions of optimality forproblem (Pi) satisfies ∫ αi0pi(t) ·∆f(t, x¯(t))dt ≤ 0. (24)As the cost function g does not depend on y, ci(1) = 0. The set N[0,1](0) is(−∞, 0], so ci(0) ≤ 0. Now, by integrating the differential equation involving ci(18) we getci(1) = ci(0) +∫ αi0−pi(t) ·∆f(t, x¯(t)) = 0.The result easily follows.In view of the constraint qualification, there exists a constant δ > 0 andτ¯∗ ∈ (0, 1] such that, ∀τ ∈ [0, τ¯∗]∫ τ0ζ · [f(t, x0, u˜(t))− f(t, x0, u¯(t))]dt < −δτfor all ζ ∈ ∂>x h(s, x), s ∈ [0, τ¯∗), x ∈ {x0}+ 1BSuppose, in contradiction, thatµ{(0, 1]}+ ||q||L∞ + λ = 0. (25)Since (λ, µ, p) 6= 0, we must haveλ = 0,µ = βδ{0},p(t) = −βζ for some β > 0 and ζ ∈ ∂>x h(0, x0).(26)The constraint qualification (CQ∗) implies∫ τ0−p(t) ·∆f(t, x0)dt =∫ τ0βζ ·∆f(t, x0)dt < −δβτ ∀τ ∈ [0, τ¯∗].REFERENCES 10But expanding this last expression we can write∫ αi0pi(t) ·∆f(t, x¯(t))dt=∫ αi0p(t) ·∆f(t, x0) + (pi(t)− p(t))∆f(t, x0) + pi(t)[∆f(t, x¯(t))−∆f(t, x0)]dt≥ δβαi −∫ αi02Ku‖pi(t)− p(t)‖+ 2Kf‖x¯(t)− x0‖‖pi(t)‖dt≥ δβαi −∫ αi02Ku‖pi(t)− p(t)‖+ 2KfAt‖pi(t)‖dtBy the uniform convergence of pi, we can make ‖pi − p‖ < ¯ for any ¯ > 0 ofour choice provided we choose a sufficient large i. Moreover ‖pi‖ ≤ 1.It follows that∫ αi0pi(t) ·∆f(t, x¯(t))dt ≥ δβαi − (2Ku¯αi +KfAα2i ) > δβ/2αi > 0if ¯ <δβ8Kuand αi <δβ4KfA.So, we would have∫ αi0pi(t) ·∆f(t, x¯(t))dt > 0 contradicting Lemma (4.3).We deduce (23).References[1] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F.Mishchenko, The Mathematical Theory of Optimal Processes. Wiley In-terscience, New York, 1962.[2] F. H. Clarke, Optimization and Nonsmooth Analysis. New York: Wiley-Interscience, 1983.[3] L. W. Neustadt, “A general theory of extremals,” Journal of Computer andSystem Sciences, vol. 3, pp. 57–92, 1969.[4] J. Warga, Optimal Control of Differential and Functional Equations. Aca-demic Press, New York, 1972.[5] A. V. Arutyunov and S. M. Aseev, “State constraints in optimal control:The degeneracy phenomenon,” Systems and Control Letters, vol. 26, pp.267–273, 1995.[6] R. Vinter, Optimal control. Boston: Birkhauser, 2000.[7] S. O. Lopes and F. A. C. C. Fontes, “On the degeneracy phenomenon inoptimization problems with inequality constraints,” in ORP3 conference,Guimaraes, Portugal (submitted), 2007.[8] M. M. A. Ferreira, F. A. C. C. Fontes, and R. B. Vinter, “Nondegeneratenecessary conditions for nonconvex optimal control problems with state con-straints,” Journal of Mathematical Analysis and Applications, vol. 233, pp.116–129, 1999.[9] F. A. C. C. Fontes, “Optimisation-based control of constrained nonlinearsystems,” Ph.D. dissertation, Imperial College of Science Technology andMedicine, University of London, London SW7 2BY, U.K., 1999.

An integral-type constraint qualification for optimal control problems with state constraints

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