Optimal sampled-data control, and generalizations on time scales Lo¨ıc Bourdin∗ Emmanuel Tr´elat† 5 1 0 Abstract 2 In this paper, we derive a version of the Pontryagin maximum principle for general finite- c dimensional nonlinear optimal sampled-data control problems. Our framework is actually e muchmoregeneral,andwetreatoptimalcontrolproblemsforwhichthestatevariableevolves D onagiventimescale(arbitrarynon-emptyclosedsubsetofR),andthecontrolvariableevolves 8 onasmallertimescale. Sampled-datasystemsarethenaparticularcase. Ourproofisbasedon theconstructionofappropriateneedle-likevariationsandontheEkelandvariationalprinciple. ] C O Keywords: optimal control; sampled-data; Pontryagin maximum principle; time scale. . h AMS Classification: 49J15; 93C57; 34N99; 39A12. t a m 1 Introduction [ 2 Optimal control theory is concerned with the analysis of controlled dynamical systems, where one v aims at steering such a system from a given configuration to some desired target by minimizing 1 some criterion. The Pontryagin maximum principle (in short, PMP), established at the end of the 6 50’s for general finite-dimensional nonlinear continuous-time dynamics (see [46], and see [30] for 3 7 the history of this discovery), is certainly the milestone of the classical optimal control theory. It 0 provides a first-order necessary condition for optimality, by asserting that any optimal trajectory 1. must be the projection of an extremal. The PMP then reduces the search of optimal trajectories 0 to a boundary value problem posed on extremals. Optimal control theory, and in particular the 5 PMP, has an immense field of applications in various domains, and it is not our aim here to list 1 them. : v We speak of a purely continuous-time optimal control problem, when both the state q and the i X control u evolve continuously in time, and the control system under consideration has the form r a q˙(t)=f(t,q(t),u(t)), for a.e. t∈R+, where q(t) ∈ Rn and u(t) ∈ Ω ⊂ Rm. Such models assume that the control is permanent, that is, the value of u(t) can be chosen at each time t ∈ R+. We refer the reader to textbooks on continuous optimal control theory such as [4, 13, 14, 18, 20, 21, 33, 42, 43, 46, 47, 49, 50] for many examples of theoretical or practical applications. ∗Universit´e de Limoges, Institut de recherche XLIM, D´epartement de Math´ematiques et d’Informatique. UMR CNRS7252. Limoges,France([email protected]). †Sorbonne Universit´es, UPMC Univ Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, Institut UniversitairedeFrance,F-75005,Paris,France([email protected]). 1 We speak of a purely discrete-time optimal control problem, when both the state q and the controluevolveinadiscretewayintime,andthecontrolsystemunderconsiderationhastheform q −q =f(k,q ,u ), k ∈N, k+1 k k k where q ∈ Rn and u ∈ Ω ⊂ Rm. As in the continuous case, such models assume that the k k control is permanent, that is, the value of u can be chosen at each time k ∈ N. A version of the k PMP for such discrete-time control systems has been established in [32, 39, 41] under appropriate convexity assumptions. The considerable development of the discrete-time control theory was in particular motivated by the need of considering digital systems or discrete approximations in numerical simulations of differential control systems (see the textbooks [12, 24, 45, 49]). It can be noted that some early works devoted to the discrete-time PMP (like [27]) are mathematically incorrect. Some counterexamples were provided in [12] (see also [45]), showing that, as is now well known, the exact analogous of the continuous-time PMP does not hold at the discrete level. More precisely, the maximization condition of the continuous-time PMP cannot be expected to hold in general in the discrete-time case. Nevertheless, a weaker condition can be derived, in terms of nonpositive gradient condition (see [12, Theorem 42.1]). We speak of an optimal sampled-data control problem, when the state q evolves continuously in time, whereas the control u evolves in a discrete way in time. This hybrid situation is often considered in practice for problems in which the evolution of the state is very quick (and thus can be considered continuous) with respect to that of the control. We often speak, in that case, of digital control. This refers to a situation where, due for instance to hardware limitations or to technicaldifficulties,thevalueu(t)ofthecontrolcanbechosenonlyattimest=kT,whereT >0 is fixed and k ∈N. This means that, once the value u(kT) is fixed, u(t) remains constant over the time interval [kT,(k+1)T). Hence the trajectory q evolves according to q˙(t)=f(t,q(t),u(kT)), for a.e. t∈[kT,(k+1)T), k ∈N. In other words, this sample-and-hold procedure consists of “freezing” the value of u at each con- trolling time t=kT on the corresponding sampling time interval [kT,(k+1)T), where T is called the sampling period. In this situation, the control of the system is clearly nonpermanent. Tothebestofourknowledge,theclassicaloptimalcontroltheorydoesnottreatgeneralnonlin- ear optimal sampled-data control problems, but concerns either purely continuous-time, or purely discrete-time optimal control problems. It is one of our objectives to derive, in this paper, a PMP which can be applied to general nonlinear optimal sampled-data control problems. Actually, we will be able to establish a PMP in the much more general framework of time scales, which unifies and extends continuous-time and discrete-time issues. But, before coming to that point, we feel that it is of interest to enunciate a PMP in the particular case of sampled-data systems and where the set Ω of pointwise constraints on the controls is convex. PMP for optimal sampled-data control problems and Ω convex. Let n, m and j be nonzerointegers. LetT >0beanarbitrarysamplingperiod. Inwhatfollows,foranyrealnumber t,wedenotebyE(t)theintegerpartoft,definedastheuniqueintegersuchthatE(t)≤t<E(t)+1. Note that k = E(t/T) whenever kT ≤ t < (k+1)T. We consider the general nonlinear optimal sampled-data control problem  (cid:90) tf min 0 f0(τ,q(τ),u(k(τ)T))dτ, with k(τ)=E(τ/T), R+ (OSDCP) q˙(t)=f(t,q(t),u(k(t)T)), with k(t)=E(t/T), NT u(kT)∈Ω, g(q(0),q(t ))∈S. f 2 Here, f :R×Rn×Rm →Rn and f0 :R×Rn×Rm →R are continuous, and of class C1 in (q,u), g :Rn×Rn →Rj isofclassC1,andΩ(resp.,S)isanon-emptyclosedconvexsubsetofRm (resp., of Rj). The final time t ≥0 can be fixed or not. f Note that, under appropriate (usual) compactness and convexity assumptions, the optimal R+ control problem (OSDCP) has at least one solution (see Theorem 2 in Section 2.2). NT Recall that g is said to be submersive at a point (q ,q ) ∈ Rn×Rn if the differential of g at 1 2 this point is surjective. We define the Hamiltonian H :R×Rn×Rn×R×Rm →R, as usual, by H(t,q,p,p0,u)=(cid:104)p,f(t,q,u)(cid:105)Rn +p0f0(t,q,u). Theorem 1 (Pontryagin maximum principle for (OSDCP)R+). If a trajectory q∗, defined on NT [0,t∗] and associated with a sampled-data control u∗, is an optimal solution of (OSDCP)R+, then f NT there exists a nontrivial couple (p,p0), where p:[0,t∗]→Rn is an absolutely continuous mapping f (called adjoint vector) and p0 ≤0, such that the following conditions hold: • Extremal equations: ∂H ∂H q˙∗(t)= (t,q∗(t),p(t),p0,u∗(k(t)T)), p˙(t)=− (t,q∗(t),p(t),p0,u∗(k(t)T)), ∂p ∂q for almost every t∈[0,t∗), with k(t)=E(t/T). f • Maximization condition: For every controlling time kT ∈[0,t∗) such that (k+1)T ≤t∗, we have f f (cid:42) (cid:43) (cid:90) (k+1)T ∂H (τ,q∗(τ),p(τ),p0,u∗(kT))dτ , y−u∗(kT) ≤0, (1) ∂u kT Rm for every y ∈ Ω. In the case where kT ∈ [0,t∗) with (k+1)T > t∗, the above maximization f f condition is still valid provided (k+1)T is replaced with t∗. f • Transversality conditions on the adjoint vector: Ifg issubmersiveat(q∗(0),q∗(t∗)), thenthenontrivialcouple(p,p0)canbeselectedtosatisfy f (cid:18) ∂g (cid:19)(cid:62) (cid:18) ∂g (cid:19)(cid:62) p(0)=− (q∗(0),q∗(t∗)) ψ, p(t∗)= (q∗(0),q∗(t∗)) ψ, ∂q f f ∂q f 1 2 where −ψ belongs to the orthogonal of S at the point g(q∗(0),q∗(t∗))∈S. f • Transversality condition on the final time: If the final time is left free in the optimal control problem (OSDCP)R+, if t∗ > 0 and if f NT f and f0 are of class C1 with respect to t in a neighborhood of t∗, then the nontrivial couple f (p,p0) can be moreover selected to satisfy H(t∗,q∗(t∗),p(t∗),p0,u∗(k∗T))=0, f f f f where k∗ =E(t∗/T) whenever t∗ ∈/ NT, and k∗ =E(t∗/T)−1 whenever t∗ ∈NT. f f f f f f Note that the only difference with the usual statement of the PMP for purely continuous- time optimal control problems is in the maximization condition. Here, for sampled-data control systems, the usual pointwise maximization condition of the Hamiltonian is replaced with the in- equality (1). This is not a surprise, because already in the purely discrete case, as mentioned 3 earlier,thepointwisemaximizationconditionfailstobetrueingeneral,andmustbereplacedwith a weaker condition. The condition (1), which is satisfied for every y ∈ Ω, gives a necessary condition allowing to compute u∗(kT) in general, and this, for all controlling times kT ∈ [0,t∗). We will provide in f Section3.1asimpleoptimalconsumptionproblemwithsampled-datacontrol,andshowhowthese computations can be done in a simple way. R+ Note that the optimal sampled-data control problem (OSDCP) can of course be seen as NT a finite-dimensional optimization problem where the unknowns are u∗(kT), with k ∈ N such that kT ∈ [0,t∗). The same remark holds, by the way, for purely discrete-time optimal control f problems. OnecouldthenapplyclassicalLagrangemultiplier(orKKT)rulestosuchoptimization problems with constraints (numerically, this leads to direct methods). The Pontryagin maximum principleisafar-reachingversionoftheLagrangemultiplierrule,yieldingmorepreciseinformation and reducing the initial optimal control problem to a shooting problem (see, e.g., [51] for such a discussion). Extension to the time scale framework. In this paper, we actually establish a version of Theorem1inamuchmoregeneralframework,allowingforexampletostudysampled-datacontrol systems where the control can be permanent on a first time interval, then sampled on a finite set, then permanent again, etc. More precisely, Theorem 1 can be extended to a general framework in which the set of controlling times is not NT but some arbitrary non-empty closed subset of R (i.e., a time scale), and also the state may evolve on another time scale. We will state a PMP for such general systems in Section 2.3 (see Theorem 3). Since such systems, where we have two time scales(oneforthestateandoneforthecontrol),canbeviewedasageneralizationofsampled-data control systems, we will refer to them as sampled-data control systems on time scales. Let us first recall and motivate the notion of time scale. The time scale theory was introduced in [34] in order to unify discrete and continuous analysis. By definition, a time scale T is an arbitrary non-empty closed subset of R, and a dynamical system is said to be posed on the time scale T whenever the time variable evolves along this set T. The time scale theory aims at closing the gap between continuous and discrete cases, and allows one to treat general processes involving bothcontinuous-timeanddiscrete-timevariables. Thepurelycontinuous-timecasecorrespondsto T = R+ and the purely discrete-time case corresponds to T = N. But a time scale can be much more general (see, e.g., [29, 44] for a study of a seasonally breeding population whose generations do not overlap, and see [6] for applications to economics), and can even be a Cantor set. Many notions of standard calculus have been extended to the time scale framework, and we refer the reader to [1, 2, 10, 11] for details on that theory. The theory of the calculus of variations on time scales, initiated in [8], has been well studied in the existing literature (see, e.g., [7, 9, 17, 28, 35, 38]). In [36, 37], the authors establish a weak version of the PMP (with a nonpositive gradient condition) for control systems defined on general time scales. In [16], we derived a strong version of the PMP, in a very general time scale setting, encompassingboththepurelycontinuous-timePMP(withamaximizationcondition)and the purely discrete-time PMP (with a nonpositive gradient condition). All these works are concerned with control systems defined on general time scales with per- manent control. The main objective of the present paper is to handle control systems defined on general time scales with nonpermanent control, that we refer to as sampled-data control systems on time scales, and for which we assume that the state and the control are allowed to evolve on different time scales (the time scale of the control being a subset of the time scale of the state). This framework is the natural extension of the classical sampled-data setting, and allows to treat simultaneously many sampling-data control situations. OurmainresultisaPMPforgeneralfinite-dimensionalnonlinearoptimalsampled-datacontrol 4 problems on time scales. Note that our result will be derived without any convexity assumption on the set Ω of pointwise constraints on the controls. Our proof is based on the construction of appropriate needle-like variations and on the Ekeland variational principle. In the case of a permanentcontrol,ourstatementencompassesthetimescaleversionofthePMPobtainedin[16], and a fortiori it also encompasses the classical continuous and discrete versions of the PMP. Organization of the paper. InSection2, afterhavingrecalledseveralbasicfactsintimescale calculus, we define a general nonlinear optimal sampled-data control problem defined on time scales, and we state a Pontryagin maximum principle (Theorem 3) for such problems. Section 3 is devoted to some applications of Theorem 3 and further comments. Section 4 is devoted to the proof of Theorem 3. 2 Main result LetTbeatimescale,thatis,anarbitrarynon-emptyclosedsubsetofR. Withoutlossofgenerality, we assume that T is bounded below, denoting by a=minT, and unbounded above.1 Throughout the paper, T will be the time scale on which the state of the control system evolves. We start the section by recalling some useful notations and basic results of time scale calculus, in particular the notion of Lebesgue ∆-measure and of absolutely continuous function within the time scale setting. The reader already acquainted with time scale calculus may jump directly to Section 2.2. 2.1 Preliminaries on time scale calculus The forward jump operator σ : T → T is defined by σ(t) = inf{s ∈ T | s > t} for every t ∈ T. A point t∈T is said to be right-scattered whenever σ(t)>t. A point t∈T is said to be right-dense whenever σ(t)=t. We denote by RS the set of all right-scattered points of T, and by RD the set of all right-dense points of T. Note that RS is at most countable (see [23, Lemma 3.1]) and that RDisthecomplementofRSinT. Thegraininessfunctionµ:T→R+ isdefinedbyµ(t)=σ(t)−t for every t∈T. For every subset A of R, we denote by AT =A∩T. An interval of T is defined by IT where I is an interval of R. For every b∈T\{a} and every s∈[a,b)T∩RD, we set Vs,b ={β ≥0|s+β ∈[s,b]T}. (2) Note that 0 is not isolated in Vs,b. ∆-differentiability. Letn∈N∗. Thenotations(cid:107)·(cid:107)Rn and(cid:104)·,·(cid:105)Rn respectivelystandfortheusual Euclidean norm and scalar product of Rn. A function q :T→Rn is said to be ∆-differentiable at t∈T if the limit qσ(t)−q(s) q∆(t)= lim s→t σ(t)−s s∈T exists in Rn, where qσ =q◦σ. Recall that, if t∈RD, then q is ∆-differentiable at t if and only if the limit of q(t)−q(s) as s→t, s∈T, exists; in that case it is equal to q∆(t). If t∈RS and if q is t−s continuous at t, then q is ∆-differentiable at t, and q∆(t)= qσ(t)−q(t) (see [10]). µ(t) 1In this paper we only work on a bounded subinterval of type [a,b]∩T with a, b ∈ T. It is not restrictive to assumethata=minTandthatTisunboundedabove. Ontheotherhand,thesetwoassumptionswidelysimplify thenotationsintroducedinSection2.1(otherwisewewouldhave,inallfurtherstatements,todistinguishbetween pointsofT\{maxT}andmaxT). 5 If q, q(cid:48) : T → Rn are both ∆-differentiable at t ∈ T, then the scalar product (cid:104)q,q(cid:48)(cid:105)Rn is ∆-differentiable at t and (cid:104)q,q(cid:48)(cid:105)∆Rn(t)=(cid:104)q∆(t),q(cid:48)σ(t)(cid:105)Rn +(cid:104)q(t),q(cid:48)∆(t)(cid:105)Rn =(cid:104)q∆(t),q(cid:48)(t)(cid:105)Rn +(cid:104)qσ(t),q(cid:48)∆(t)(cid:105)Rn. (3) These equalities are usually called Leibniz formulas (see [10, Theorem 1.20]). Lebesgue ∆-measure and Lebesgue ∆-integrability. Let µ be the Lebesgue ∆-measure ∆ on T defined in terms of Carath´eodory extension in [11, Chapter 5]. We also refer the reader to [3, 5, 23, 31] for more details on the µ -measure theory. For all (c,d) ∈ T2 such that c ≤ d, one ∆ has µ∆([c,d)T)=d−c. Recall that A⊂T is a µ∆-measurable set of T if and only if A is an usual µ -measurable set of R, where µ denotes the usual Lebesgue measure (see [23, Proposition 3.1]). L L Moreover, if A⊂T, then (cid:88) µ (A)=µ (A)+ µ(r). ∆ L r∈A∩RS Let A ⊂ T. A property is said to hold ∆-almost everywhere (in short, ∆-a.e.) on A if it holds for every t ∈ A\A(cid:48), where A(cid:48) ⊂ A is some µ -measurable subset of T satisfying µ (A(cid:48)) = 0. In ∆ ∆ particular,sinceµ ({r})=µ(r)>0foreveryr ∈RS,weconcludethat,ifapropertyholds∆-a.e. ∆ on A, then it holds for every r ∈A∩RS. Similarly, if A⊂T is such that µ (A)=0, then A⊂RD. ∆ Let n ∈ N∗ and let A ⊂ T be a µ -measurable subset of T. Consider a function q defined ∆ ∆-a.e.onAwithvaluesinRn. LetA˜=A∪(r,σ(r)) ,andletq˜betheextensionofq defined r∈A∩RS µ -a.e. on A˜ by q˜(t) = q(t) whenever t ∈ A, and by q˜(t) = q(r) whenever t ∈ (r,σ(r)), for every L r ∈A∩RS. Recall that q is µ -measurable on A if and only if q˜is µ -measurable on A˜ (see [23, ∆ L Proposition 4.1]). Let n ∈ N∗ and let A ⊂ T be a µ -measurable subset of T. The functional space L∞(A,Rn) ∆ T is the set of all functions q defined ∆-a.e. on A, with values in Rn, that are µ -measurable on A ∆ and bounded ∆-almost everywhere. Endowed with the norm (cid:107)q(cid:107)L∞T (A,Rn) = supessτ∈A(cid:107)q(τ)(cid:107)Rn, it is a Banach space (see [3, Theorem 2.5]). The functional space L1(A,Rn) is the set of all T functions q defined ∆-a.e. on A, with values in Rn, that are µ -measurable on A and such that ∆ (cid:82) (cid:82) A(cid:107)q(τ)(cid:107)Rn∆τ < +∞. Endowed with the norm (cid:107)q(cid:107)L1T(A,Rn) = A(cid:107)q(τ)(cid:107)Rn∆τ, it is a Banach space (see [3, Theorem 2.5]). We recall here that if q ∈L1(A,Rn) then T (cid:90) (cid:90) (cid:90) (cid:88) q(τ)∆τ = q˜(τ)dτ = q(τ)dτ + µ(r)q(r), A A˜ A r∈A∩RS see [23, Theorems 5.1 and 5.2]. Note that if A is bounded then L∞(A,Rn)⊂L1(A,Rn). T T Absolutely continuous functions. Let n ∈ N∗ and let (c,d) ∈ T2 such that c < d. Let C([c,d]T,Rn)denotethespaceofcontinuousfunctionsdefinedon[c,d]TwithvaluesinRn. Endowed with its usual uniform norm (cid:107)·(cid:107)∞, it is a Banach space. Let AC([c,d]T,Rn) denote the subspace of absolutely continuous functions. Let t0 ∈ [c,d]T and q : [c,d]T → Rn. It is easily derived from [22, Theorem 4.1] that q ∈ AC([c,d]T,Rn)ifandonlyifq is∆-differentiable∆-a.e.on[c,d)T andsatisfiesq∆ ∈L1T([c,d)T,Rn), qa(ntd)f−or(cid:82)everyqt∆(∈τ)[c∆,τd]Twhoenneevhearstq≤(t)t =. q(t0) + (cid:82)[t0,t)Tq∆(τ)∆τ whenever t ≥ t0, and q(t) = 0 [t,t0)T 0 Assume that q ∈ L1T([c,d)T,Rn), and let Q be the function defined on [c,d]T by Q(t) = (cid:82) (cid:82) q(τ)∆τ whenever t ≥ t , and by Q(t) = − q(τ)∆τ whenever t ≤ t . Then Q ∈ [t0,t)T 0 [t,t0)T 0 AC([c,d]T) and Q∆ =q ∆-a.e. on [c,d)T. 6 Note that, if q ∈ AC([c,d]T,Rn) is such that q∆ = 0 ∆-a.e. on [c,d)T, then q is constant on [c,d]T,andthat,ifq,q(cid:48) ∈AC([c,d]T,Rn),then(cid:104)q,q(cid:48)(cid:105)Rn ∈AC([c,d]T,R)andtheLeibnizformula(3) is available ∆-a.e. on [c,d)T. For every q ∈ L1T([c,d)T,Rn), we denote by L[c,d)T(q) the set of points t ∈ [c,d)T that are ∆-Lebesgue points of q. It holds µ∆(L[c,d)T(q))=µ∆([c,d)T)=d−c, and 1 (cid:90) lim q(τ)∆τ =q(s), β→0 β [s,s+β)T β∈Vs,d for every s∈L (q)∩RD, where Vs,d is defined by (2). [c,d)T 2.2 Optimal sampled-data control problems on time scales LetT beanothertimescale. Throughoutthepaper,T willbethetimescaleonwhichthecontrol 1 1 evolves. We assume that T ⊂T.2 1 SimilarlytoT,weassumethatminT =aandthatT isunboundedabove. Asintheprevious 1 1 paragraph, weintroducethenotationsσ , RS , RD , Vs,b, ∆ , etc., associatedwiththetimescale 1 1 1 1 1 T . Since T ⊂T, note that RS⊂RS and RD ⊂RD. We define the map 1 1 1 1 Φ: T −→ T 1 t (cid:55)−→ Φ(t)=sup{s∈T |s≤t}. 1 For every t ∈ T , we have Φ(t) = t. For every t ∈ T\T , we have Φ(t) ∈ RS and Φ(t) < t < 1 1 1 σ (Φ(t)). Note that, if t∈T is such that Φ(t)∈RD , then t∈T . 1 1 1 In what follows, given a function u:T →R, we denote by uΦ the composition u◦Φ:T→R. 1 Of course, when dealing with functions having multiple components, this composition is applied to each component. Let us mention, at this step, that if u ∈ L∞(T ,R) then uΦ ∈ L∞(T,R) (see T 1 T 1 Proposition 1 and more properties in Section 4.1.1). Let n, m and j be nonzero integers. We consider the general nonlinear optimal sampled-data control problem on time scales  (cid:90) min [a,b)Tf0(τ,q(τ),uΦ(τ))∆τ, (OSDCP)T q∆(t)=f(t,q(t),uΦ(t)), (4) T 1 ug(∈q(aL)∞T,1q((Tb1)),Ω∈),S. Here, the trajectory of the system is q : T → Rn, the mappings f : T×Rn ×Rm → Rn and f0 : T×Rn×Rm → R are continuous, of class C1 in (q,u), the mapping g : Rn×Rn → Rj is of class C1, Ω is a non-empty closed subset of Rm, and S is a non-empty closed convex subset of Rj. The final time b∈T can be fixed or not. Remark 1. We recall that, given u∈L∞T (T1,Rm), we say that q is a solution of (4) on IT if: 1 1. IT is an interval of T satisfying a∈IT and IT\{a}=(cid:54) ∅; 2Indeed,itisnotnaturaltoconsidercontrollingtimest∈T1 atwhichthedynamicsdoesnotevolve,thatis,at whicht∈/T. Thevalueofthecontrolatsuchtimest∈T1\Twouldnotinfluencethedynamics,or,maybe,onlyon [t∗,+∞[T wheret∗=inf{s∈T|s≥t}. Inthislastcase,notethatt∗∈TandwecanreplaceT1 by(T1∪{t∗})\{t} withoutlossofgenerality. 7 2. For every c∈IT\{a}, q ∈AC([a,c]T,Rn) and (4) holds for ∆-a.e. t∈[a,c)T. Existence and uniqueness of solutions (Cauchy-Lipschitz theorem on time scales) have been estab- lished in [15], and useful results are recalled in Section 4.1.2. Remark 2. The time scale T stands for the set of controlling times of the control system (4). 1 If T = T , then the control is permanent. The case T = T = R+ corresponds to the classical 1 1 continuous case, whereas T = T = N coincides with the classical discrete case. If T (cid:32) T, the 1 1 control is nonpermanent and sampled. In that case, the sampling times are given by t∈RS such 1 that σ(t)<σ1(t) and the corresponding sampling time intervals are given by [t,σ1(t))T. The clas- R+ sical optimal sampled-data control problem (OSDCP) investigated in Theorem 1 corresponds NT to T=R+ and T =NT, with T >0. 1 T T Remark 3. Let us consider two optimal control problems (OSDCP) and (OSDCP) , posed T T onthesamegeneraltimescaleTforthetrajectories,butwithtwodiffere1ntsetsofcontrollin2gtimes T andT , andletusassumethatT ⊂T . WedenotebyΦ andΦ thecorrespondingmappings 1 2 2 1 1 2 from T to T and T respectively. If u ∈ L∞(T ,Ω) is an optimal control for (OSDCP)T and 1 2 1 T 1 T 1 1 if there exists u ∈L∞(T ,Ω) such that uΦ2(t)=uΦ1(t) for ∆-a.e. t∈T, it is clear that u is an 2 T2 2 T 2 1 2 optimal control for (OSDCP) . We refer to Section 3.1 for examples. T 2 T Remark 4. The framework of (OSDCP) encompasses optimal parameter problems. Indeed, T 1 let us consider the parametrized dynamical system q∆(t)=f(t,q(t),λ), ∆-a.e. t∈T, (5) with λ ∈ Ω. Then, considering T1 = {a}∪[b,+∞[T, (4) coincides with (5) where u(a) plays the role of λ. In this situation, Theorem 3 (stated in Section 2.3) provides necessary conditions for optimal parameters λ. We refer to Section 3.1 for examples. Remark 5. A possible extension is to study dynamical systems on time scales with several sampled-data controls but with different sets of controlling times: q∆(t)=f(t,q(t),uΦ1(t),uΦ2(t)), ∆-a.e. t∈T, 1 2 where T and T are general time scales contained in T, and Φ and Φ are the corresponding 1 2 1 2 mappings from T to T and T . Our main result (Theorem 3) can be easily extended to this 1 2 framework. Actually, this multiscale version will be useful in order to derive the transversality condition on the final time (see Remark 30). Remark6. Anotherpossibleextensionistostudydynamicalsystemsontimescaleswithsampled- data control where the state q and the constraint function f are also sampled: q∆(t)=f(Φ (t),qΦ2(t),uΦ3(t)), ∆-a.e. t∈T, 1 whereT ,T andT aregeneraltimescalescontainedinT,andΦ ,Φ andΦ arethecorrespond- 1 2 3 1 2 3 ing mappings from T to T , T and T respectively. In particular, the setting of [16] corresponds 1 2 3 to the above framework with T=R+ and T =T =T a general time scale. 1 2 3 Although this is not the main objective of our paper, we provide hereafter a result stating T the existence of optimal solutions for (OSDCP) , under some appropriate compactness and T 1 convexity assumptions. Actually, if the existence of solutions is stated, the necessary conditions provided in Theorem 3, allowing to compute explicitly optimal sampled-data controls, may prove the uniqueness of the optimal solution. We refer to Section 3.1 for examples. LetMstandforthesetoftrajectoriesq,associatedwithb∈Tandwithasampled-datacontrol u∈L∞T (T1,Ω), satisfying (4)∆-a.e. on[a,b)T andg(q(a),q(b))∈S. Wedefinethesetofextended 1 velocities W(t,q)={(f(t,q,u),f0(t,q,u))(cid:62) |u∈Ω} for every (t,q)∈T×Rn. 8 Theorem 2. If Ω is compact, M is non-empty, (cid:107)q(cid:107) +b ≤ M for every q ∈ M and for some ∞ M ≥ 0, and if W(t,q) is convex for every (t,q) ∈ T×Rn, then (OSDCP)T has at least one T 1 optimal solution. TheproofofTheorem2isdoneinSection4.4. Notethat, inthistheorem, itsufficestoassume thatg iscontinuous. Besides,theassumptionontheboundednessoftrajectoriescanbeweakened, by assuming, for instance, that the extended dynamics have a sublinear growth at infinity (see, e.g., [25]; many other easy and standard extensions are possible). T 2.3 Pontryagin maximum principle for (OSDCP) T 1 2.3.1 Preliminaries on convexity and stable Ω-dense directions The orthogonal of the closed convex set S at a point x∈S is defined by OS[x]={x(cid:48) ∈Rj |∀x(cid:48)(cid:48) ∈S, (cid:104)x(cid:48),x(cid:48)(cid:48)−x(cid:105)Rj ≤0}. It is a closed convex cone containing 0. We denote by dS the distance function to S defined by dS(x) = infx(cid:48)∈S(cid:107)x−x(cid:48)(cid:107)Rj, for every x∈Rj. Recallthat,foreveryx∈Rj,thereexistsauniqueelementP (x)∈S(projectionofxonto S S)suchthatdS(x)=(cid:107)x−PS(x)(cid:107)Rj. Itischaracterizedbytheproperty(cid:104)x−PS(x),x(cid:48)−PS(x)(cid:105)Rj ≤0 for every x(cid:48) ∈S. In particular, x−P (x)∈O [P (x)]. The function P is 1-Lipschitz continuous. S S S S We recall the following obvious lemmas. Lemma 1. Let (xk)k∈N be a sequence of points of Rj and (ζk)k∈N be a sequence of nonnegative real numbers such that x →x∈S and ζ (x −P (x ))→x(cid:48) ∈Rj as k →+∞. Then x(cid:48) ∈O [x]. k k k S k S Lemma 2. The function d2 : x (cid:55)→ d (x)2 is differentiable on Rj, with dd2(x)(x(cid:48)) = 2(cid:104)x − S S S PS(x),x(cid:48)(cid:105)Rj. Hereafter we recall the notion of stable Ω-dense directions and we state an obvious lemma. We refer to [16, Section 2.2] for more details. Definition 1. Letv ∈Ω. Adirectiony ∈ΩissaidtobeastableΩ-densedirectionfromv ifthere existsε>0suchthat0isnotisolatedin{α∈[0,1], v(cid:48)+α(y−v(cid:48))∈Ω}foreveryv(cid:48) ∈BRm(v,ε)∩Ω. The set of all stable Ω-dense directions from v is denoted by DΩ (v). stab Lemma 3. If Ω is convex, then DΩ (v)=Ω for every v ∈Ω. stab 2.3.2 Main result Recall that g is said to be submersive at a point (q ,q ) ∈ Rn×Rn if the differential of g at this 1 2 point is surjective. We define the Hamiltonian H : T×Rn×Rn×R×Rm → R of (OSDCP)T T 1 by H(t,q,p,p0,u)=(cid:104)p,f(t,q,u)(cid:105)Rn +p0f0(t,q,u). Theorem 3 (Pontryagin maximum principle for (OSDCP)T ). If a trajectory q∗, defined on T [a,b∗]T and associated with a sampled-data control u∗ ∈ L∞T (1T1,Ω), is an optimal solution of (OSDCP)TT ,thenthereexistsanontrivialcouple(p,p0),where1p∈AC([a,b∗]T,Rn)(calledadjoint 1 vector) and p0 ≤0, such that the following conditions hold: • Extremal equations: ∂H ∂H q∗∆(t)= (t,q∗(t),pσ(t),p0,u∗Φ(t)), p∆(t)=− (t,q∗(t),pσ(t),p0,u∗Φ(t)), (6) ∂p ∂q for ∆-a.e. t∈[a,b∗)T. 9 • Maximization condition: – For ∆1-a.e. s∈[a,b∗)T1 ∩RD1, we have u∗(s)∈argmaxH(s,q∗(s),p(s),p0,z). z∈Ω – For every r ∈[a,b∗)T1 ∩RS1 such that σ1(r)≤b∗, we have (cid:42) (cid:43) (cid:90) ∂H (τ,q∗(τ),pσ(τ),p0,u∗(r))∆τ , y−u∗(r) ≤0, (7) ∂u [r,σ1(r))T Rm for every y ∈ DΩstab(u∗(r)). In the case where r ∈ [a,b∗)T1 ∩RS1 with σ1(r) > b∗, the above maximization condition is still valid provided σ (r) is replaced with b∗. 1 • Transversality conditions on the adjoint vector: Ifg issubmersiveat(q∗(a),q∗(b∗)), thenthenontrivialcouple(p,p0)canbeselectedtosatisfy (cid:18) ∂g (cid:19)(cid:62) (cid:18) ∂g (cid:19)(cid:62) p(a)=− (q∗(a),q∗(b∗)) ψ, p(b∗)= (q∗(a),q∗(b∗)) ψ, (8) ∂q ∂q 1 2 where −ψ ∈O [g(q∗(a),q∗(b∗))]. S • Transversality condition on the final time: If the final time is left free in the optimal control problem (OSDCP)T , if b∗ belongs to the T 1 interior of T (for the topology of R), and if f and f0 are of class C1 with respect to t in a neighborhood of b∗, then the nontrivial couple (p,p0) can be moreover selected such that the Hamiltonian function t (cid:55)→ H(t,q∗(t),p(t),p0,u∗Φ(t)) coincides almost everywhere, in some neighborhood of b∗, with a continuous function vanishing at t=b∗. In particular, if u∗Φ(t) has a left-limit at t=b∗ (denoted by u∗Φ(b∗)), then the transversality − condition can be written as H(b∗,q∗(b∗),p(b∗),p0,u∗Φ(b∗))=0. − Theorem 3 is proved in Section 4. Several remarks are in order. Remark 7. As is well known, the nontrivial couple (p,p0) of Theorem 3, which is a Lagrange multiplier, is defined up to a multiplicative scalar. Defining as usual an extremal as a quadruple (q,p,p0,u) solution of the extremal equations (6), an extremal is said to be normal whenever p0 (cid:54)= 0 and abnormal whenever p0 = 0. In the normal case p0 (cid:54)= 0, it is usual to normalize the Lagrange multiplier so that p0 =−1. Remark 8. Theorem 3 encompasses the time scale version of the PMP derived in [16] when the control is permanent, that is, when T1 = T. Indeed, in that case, for every r ∈ [a,b∗)T∩RS, r ∈[a,b∗)T1∩RS1 andσ1(r)=σ(r)≤b∗. Thenthecondition(7)canbewrittenasthenonpositive gradient condition (cid:28) (cid:29) ∂H (r,q∗(r),p(σ(r)),p0,u∗(r)), y−u∗(r) ≤0, ∂u Rm for every y ∈DΩ (u∗(r)). Moreover, in the case of a free final time, under the assumptions made stab in the fourth item of Theorem 3, b∗ also belongs to the interior of T = T, and then in that case 1 we recover the classical condition maxH(b∗,q∗(b∗),p(b∗),p0,z)=0. z∈Ω 10