Distributed-Order Non-Local Optimal Control
AArticle
Distributed-Order Non-Local Optimal Control
Faïçal Ndaïrou †,‡ and Delfim F. M. Torres ‡ * Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics,University of Aveiro, 3810-193 Aveiro, Portugal; [email protected] (F.N.); delfi[email protected] (D.F.M.T.) * Correspondence: delfi[email protected]; Tel.: +351-234-370-668† This research is part of first author’s Ph.D. project, which is carried out at the University of Aveiro under theDoctoral Program in Applied Mathematics of Universities of Minho, Aveiro, and Porto (MAP-PDMA).‡ These authors contributed equally to this work.Submitted: Sept 9, 2020; Revised: Sept 30, Oct 14 and 16, 2020; Accepted to Axioms: Oct 22, 2020
Abstract:
Distributed-order fractional non-local operators have been introduced and studied by Caputoat the end of the 20th century. They generalize fractional order derivatives/integrals in the sense thatsuch operators are defined by a weighted integral of different orders of differentiation over a certainrange. The subject of distributed-order non-local derivatives is currently under strong development dueto its applications in modeling some complex real world phenomena. Fractional optimal control theorydeals with the optimization of a performance index functional subject to a fractional control system. Oneof the most important results in classical and fractional optimal control is the Pontryagin MaximumPrinciple, which gives a necessary optimality condition that every solution to the optimization problemmust verify. In our work, we extend the fractional optimal control theory by considering dynamicalsystems constraints depending on distributed-order fractional derivatives. Precisely, we prove a weakversion of Pontryagin’s maximum principle and a sufficient optimality condition under appropriateconvexity assumptions.
Keywords: distributed-order fractional calculus; basic optimal control problem; Pontryagin extremals.
MSC:
1. Introduction
Distributed-order fractional operators were introduced and studied by Caputo at the end of theprevious century [1,2]. They can be seen as a kind of generalization of fractional order derivatives/integralsin the sense that these operators are defined by a weighted integral of different orders of differentiationover a certain range. This subject gained more interest at the beginning of the current century, byresearchers from different mathematical disciplines, through attempts to solve differential equations withdistributed-order derivatives [3–6]. Moreover, at the same time, in the domain of applied mathematics,those distributed-order fractional operators have started to be used, in a satisfactory way, to describe somecomplex phenomena modeling real world problem: see, for instance, works in viscoelasticity [7,8] and indiffusion [9]. Today, the study of distributed-order systems with fractional derivatives is a hot subject: see,e.g., [10–12] and references therein.Fractional optimal control deals with optimization problems involving fractional differential equationsas well as a performance index functional. One of the most important results is the Pontryagin MaximumPrinciple, which gives a first-order necessary optimality condition that every solution to the dynamicoptimization problem must verify. By applying such result, it is possible to find and identify candidate
This is a preprint of a paper whose final and definite form is published Open Access in
Axioms a r X i v : . [ m a t h . O C ] O c t of 13 solutions to the optimal control problem. For the state of the art on fractional optimal control we refer thereaders to [13–15] and references therein. Recently, distributed-order fractional problems of the calculus ofvariations were introduced and investigated in [16]. Here, our main aim is to extend the distributed-orderfractional Euler–Lagrange equation of [16] to the Pontryagin setting (see Remark 2).Regarding optimal control for problems with distributed-order fractional operators, the results arerare and reduce to the following two papers: [17] and [18]. Both works develop numerical methods while,in contrast, here we are interested in analytical results (not in numerical approaches). Moreover, ourresults are new and bring new insights. Indeed, in [17] the problem is considered with Riemann–Liouvilledistributed derivatives, while in our case we consider optimal control problems with Caputo distributedderivatives. It should be also noted an inconsistence in [17]: when one defines the control system witha Riemann–Liouville derivative, then in the adjoint system it should appear a Caputo derivative; whenone considers optimal control problems with a control system with Caputo derivatives, then the adjointequation should involve a Riemann–Liouville operator; as a consequence of integration by parts (cf.Lemma 1). This inconsistence has been corrected in [18], where optimal control problems with Caputodistributed derivatives (like we do here) are considered. Unfortunately, there is still an inconsistence inthe necessary optimality conditions of both [17] and [18]: the transversality conditions are written thereexactly as in the classical case, with the multiplier vanishing at the end of the interval, while the correctcondition, as we prove in our Theorem 1, should involve a distributed integral operator: see condition (3).The text is organized as follows. We begin by recalling definitions and necessary results of theliterature in Section 2 of preliminaries. Our original results are then given in Section 3. More precisely,we consider fractional optimal control problems where the dynamical system constraints depend ondistributed-order fractional derivatives. We prove a weak version of Pontryagin’s maximum principle forthe considered distributed-order fractional problems (see Theorem 1) and investigate a Mangasarian-typesufficient optimality condition (see Theorem 2). An example, illustrating the usefulness of the obtainedresults, is given (see Examples 1 and 2). We end with Section 4 of conclusions, mentioning also somepossibilities of future research.
2. Preliminaries
In this section, we recall necessary results and fix notations. We assume the reader to be familiar withthe standard Riemann–Liouville and Caputo fractional calculi [19,20].Let α be a real number in [
0, 1 ] and let ψ be a non-negative continuous function defined on [
0, 1 ] suchthat (cid:90) ψ ( α ) d α > ψ will act as a distribution of the order of differentiation. Definition 1 (See [1]) . The left and right-sided Riemann–Liouville distributed-order fractional derivatives of afunction x : [ a , b ] → R are defined, respectively, by D ψ ( · ) a + x ( t ) = (cid:90) ψ ( α ) · D α a + x ( t ) d α and D ψ ( · ) b − x ( t ) = (cid:90) ψ ( α ) · D α b − x ( t ) d α , where D α a + and D α b − are, respectively, the left and right-sided Riemann–Liouville fractional derivatives of order α . of 13 Definition 2 (See [1]) . The left and right-sided Caputo distributed-order fractional derivatives of a functionx : [ a , b ] → R are defined, respectively, by C D ψ ( · ) a + x ( t ) = (cid:90) ψ ( α ) · C D α a + x ( t ) d α and C D ψ ( · ) b − x ( t ) = (cid:90) ψ ( α ) · C D α b − x ( t ) d α , where C D α a + and C D α b − are, respectively, the left and right-sided Caputo fractional derivatives of order α . As noted in [16], there is a relation between the Riemann–Liouville and the Caputo distributed-orderfractional derivatives: C D ψ ( · ) a + x ( t ) = D ψ ( · ) a + x ( t ) − x ( a ) (cid:90) ψ ( α ) Γ ( − α ) ( t − a ) − α d α and C D ψ ( · ) b − x ( t ) = D ψ ( · ) b − x ( t ) − x ( b ) (cid:90) ψ ( α ) Γ ( − α ) ( b − t ) − α d α .Along the text, we use the notation I − ψ ( · ) b − x ( t ) = (cid:90) ψ ( α ) · I − α b − x ( t ) d α ,where I − α b − represents the right Riemann–Liouville fractional integral of order 1 − α .The next result has an essential role in the proofs of our main results, that is, in the proofs of Theorems 1and 2. Lemma 1 (Integration by parts formula [16]) . Let x be a continuous function and y a continuously differentiablefunction. Then, (cid:90) ba x ( t ) · C D ψ ( · ) a + y ( t ) dt = (cid:104) y ( t ) · I − ψ ( · ) b − x ( t ) (cid:105) ba + (cid:90) ba y ( t ) · D ψ ( · ) b − x ( t ) dt .Next, we recall the standard notion of concave function, which will be used in Section 3.3. Definition 3 (See [21]) . A function h : R n → R is concave ifh ( βθ + ( − β ) θ ) ≥ β h ( θ ) + ( − β ) h ( θ ) for all β ∈ [
0, 1 ] and for all θ , θ in R n . Lemma 2 (See [21]) . Let h : R n → R be a continuously differentiable function. Then h is a concave function if andonly if it satisfies the so called gradient inequality:h ( θ ) − h ( θ ) ≥ ∇ h ( θ )( θ − θ ) for all θ , θ ∈ R n . Finally, we recall a fractional version of Gronwall’s inequality, which will be useful to prove continuityof solutions in Section 3.1. of 13
Lemma 3 (See [22]) . Let α be a positive real number and let a ( · ) , b ( · ) , and u ( · ) be non-negative continuousfunctions on [ T ] with b ( · ) monotonic increasing on [ T ) . Ifu ( t ) ≤ a ( t ) + b ( t ) (cid:90) t ( t − s ) α − u ( s ) ds , then u ( t ) ≤ a ( t ) + (cid:90) t (cid:34) ∞ ∑ n = ( b ( t ) Γ ( α )) n Γ ( n α ) ( t − s ) n α − u ( s ) (cid:35) dsfor all t ∈ [ T ) .
3. Main Results
The basic problem of optimal control we consider in this work, denoted by (BP), consists to find apiecewise continuous control u ∈ PC and the corresponding piecewise smooth state trajectory x ∈ PC ,solution of the distributed-order non-local variational problem J [ x ( · ) , u ( · )] = (cid:90) ba L ( t , x ( t ) , u ( t )) dt −→ max, C D ψ ( · ) a + x ( t ) = f ( t , x ( t ) , u ( t )) , t ∈ [ a , b ] , x ( · ) ∈ PC , u ( · ) ∈ PC , x ( a ) = x a , (BP)where functions L and f , both defined on [ a , b ] × R × R , are assumed to be continuously differentiable inall their three arguments: L ∈ C , f ∈ C . Our main contribution is to prove necessary (Section 3.2) andsufficient (Section 3.3) optimality conditions. Before we can prove necessary optimality conditions to problem (BP), we need to establish continuityand differentiability results on the state solutions for any control perturbation (Lemmas 4 and 5), whichare then used in Section 3.2. The proof of Lemma 4 makes use of the following mean value theorem forintegration, that can be found in any textbook of calculus (see, e.g., Lemma 1 of [23]): if F : [
0, 1 ] → R is acontinuous function and ψ is an integrable function that does not change sign on the interval, then thereexists a number ¯ α such that (cid:90) ψ ( α ) F ( α ) d α = F ( ¯ α ) (cid:90) ψ ( α ) d α . Lemma 4 (Continuity of solutions) . Let u (cid:101) be a control perturbation around the optimal control u ∗ , that is, for allt ∈ [ a , b ] , u (cid:101) ( t ) = u ∗ ( t ) + (cid:101) h ( t ) , where h ( · ) ∈ PC is a variation and (cid:101) ∈ R . Denote by x (cid:101) its corresponding statetrajectory, solution of C D ψ ( · ) a + x (cid:101) ( t ) = f ( t , x (cid:101) ( t ) , u (cid:101) ( t )) , x (cid:101) ( a ) = x a . Then, we have that x (cid:101) converges to the optimal state trajectory x ∗ when (cid:101) tends to zero. Proof.
Starting from definition, we have, for all t ∈ [ a , b ] , that (cid:12)(cid:12)(cid:12) C D ψ ( · ) a + x (cid:101) ( t ) − C D ψ ( · ) a + x ∗ ( t ) (cid:12)(cid:12)(cid:12) = | f ( t , x (cid:101) ( t ) , u (cid:101) ( t )) − f ( t , x ∗ ( t ) , u ∗ ( t )) | . of 13 Then, by linearity, (cid:12)(cid:12)(cid:12) C D ψ ( · ) a + x (cid:101) ( t ) − C D ψ ( · ) a + x ∗ ( t ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) C D ψ ( · ) a + ( x (cid:101) ( t ) − x ∗ ( t )) (cid:12)(cid:12)(cid:12) = | f ( t , x (cid:101) ( t ) , u (cid:101) ( t )) − f ( t , x ∗ ( t ) , u ∗ ( t )) | and it follows, by definition of the distributed operator, that (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) ψ ( α ) C D α a + ( x (cid:101) ( t ) − x ∗ ( t )) d α (cid:12)(cid:12)(cid:12)(cid:12) = | f ( t , x (cid:101) ( t ) , u (cid:101) ( t )) − f ( t , x ∗ ( t ) , u ∗ ( t )) | .Now, using the mean value theorem for integration, and denoting m : = (cid:82) ψ ( α ) d α , we obtain that thereexists an ¯ α such that (cid:12)(cid:12)(cid:12) C D ¯ α a + ( x (cid:101) ( t ) − x ∗ ( t )) (cid:12)(cid:12)(cid:12) ≤ | f ( t , x (cid:101) ( t ) , u (cid:101) ( t )) − f ( t , x ∗ ( t ) , u ∗ ( t )) | m .Clearly, one has C D ¯ α a + ( x (cid:101) ( t ) − x ∗ ( t )) ≤ (cid:12)(cid:12)(cid:12) C D ¯ α a + ( x (cid:101) ( t ) − x ∗ ( t )) (cid:12)(cid:12)(cid:12) ≤ | f ( t , x (cid:101) ( t ) , u (cid:101) ( t )) − f ( t , x ∗ ( t ) , u ∗ ( t )) | m ,which leads to x (cid:101) ( t ) − x ∗ ( t ) ≤ I ¯ α a + (cid:20) | f ( t , x (cid:101) ( t ) , u (cid:101) ( t )) − f ( t , x ∗ ( t ) , u ∗ ( t )) | m (cid:21) .Moreover, because f is Lipschitz-continuous, we have (cid:12)(cid:12)(cid:12) f ( t , x (cid:101) , u (cid:101) ) − f ( t , x ∗ , u ∗ ) (cid:12)(cid:12)(cid:12) ≤ K (cid:12)(cid:12)(cid:12) x (cid:101) − x ∗ (cid:12)(cid:12)(cid:12) + K (cid:12)(cid:12) u (cid:101) − u ∗ (cid:12)(cid:12)(cid:12) .By setting K = max { K , K } , it follows that (cid:12)(cid:12)(cid:12) x (cid:101) ( t ) − x ∗ ( t ) (cid:12)(cid:12)(cid:12) ≤ Km I ¯ α a + (cid:16)(cid:12)(cid:12)(cid:12) x (cid:101) ( t ) − x ∗ ( t ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) (cid:101) h ( t ) (cid:12)(cid:12)(cid:12)(cid:17) = Km (cid:104) | (cid:101) | I ¯ α a + (cid:16)(cid:12)(cid:12)(cid:12) h ( t ) (cid:12)(cid:12)(cid:12)(cid:17) + I ¯ α a + (cid:16)(cid:12)(cid:12)(cid:12) x (cid:101) ( t ) − x ∗ ( t ) (cid:12)(cid:12)(cid:12)(cid:17)(cid:105) = Km (cid:20) | (cid:101) | I ¯ α a + (cid:16)(cid:12)(cid:12)(cid:12) h ( t ) (cid:12)(cid:12)(cid:12)(cid:17) + Γ ( ¯ α ) (cid:90) ta ( t − s ) ¯ α − (cid:12)(cid:12)(cid:12) x (cid:101) ( s ) − x ∗ ( s ) (cid:12)(cid:12)(cid:12) ds (cid:21) for all t ∈ [ a , b ] . Now, by applying Lemma 3 (the fractional Gronwall inequality), it follows that (cid:12)(cid:12)(cid:12) x (cid:101) ( t ) − x ∗ ( t ) (cid:12)(cid:12)(cid:12) ≤ Km (cid:34) | (cid:101) | I ¯ α a + (cid:16)(cid:12)(cid:12)(cid:12) h ( t ) (cid:12)(cid:12)(cid:12)(cid:17) + | (cid:101) | (cid:90) ta (cid:32) ∞ ∑ i = Γ ( i ¯ α ) ( t − s ) i ¯ α − I ¯ α a + (cid:16)(cid:12)(cid:12)(cid:12) h ( s ) (cid:12)(cid:12)(cid:12)(cid:17)(cid:33) ds (cid:35) = | (cid:101) | Km (cid:34) I ¯ α a + (cid:16)(cid:12)(cid:12)(cid:12) h ( t ) (cid:12)(cid:12)(cid:12)(cid:17) + (cid:90) ta (cid:32) ∞ ∑ i = Γ ( i ¯ α + ) ( t − s ) i ¯ α I ¯ α a + (cid:16)(cid:12)(cid:12)(cid:12) h ( s ) (cid:12)(cid:12)(cid:12)(cid:17)(cid:33) ds (cid:35) ≤ | (cid:101) | Km (cid:34) I ¯ α a + (cid:16)(cid:12)(cid:12)(cid:12) h ( t ) (cid:12)(cid:12)(cid:12)(cid:17) + (cid:90) ta (cid:32) ∞ ∑ i = δ i ¯ α Γ ( i ¯ α + ) I ¯ α a + (cid:16)(cid:12)(cid:12)(cid:12) h ( s ) (cid:12)(cid:12)(cid:12)(cid:17)(cid:33) ds (cid:35) .The series in the last inequality is a Mittag–Leffler function and thus convergent. Hence, by taking thelimit when (cid:101) tends to zero, we obtain the desired result: x (cid:101) → x ∗ for all t ∈ [ a , b ] . of 13 Lemma 5 (Differentiability of the perturbed trajectory) . There exists a function η defined on [ a , b ] such thatx (cid:101) ( t ) = x ∗ ( t ) + (cid:101)η ( t ) + o ( (cid:101) ) . Proof.
Since f ∈ C , we have that f ( t , x (cid:101) , u (cid:101) ) = f ( t , x ∗ , u ∗ ) + ( x (cid:101) − x ∗ ) ∂ f ( t , x ∗ , u ∗ ) ∂ x + ( u (cid:101) − u ∗ ) ∂ f ( t , x ∗ , u ∗ ) ∂ u + o ( | x (cid:101) − x ∗ | , | u (cid:101) − u ∗ | ) .Observe that u (cid:101) − u ∗ = (cid:101) h ( t ) and u (cid:101) → u ∗ when (cid:101) → x (cid:101) → x ∗ when (cid:101) → (cid:101) only, that is, the residue is o ( (cid:101) ) . Therefore, we have C D ψ ( · ) a + x (cid:101) ( t ) = C D ψ ( · ) a + x ∗ ( t ) + ( x (cid:101) − x ∗ ) ∂ f ( t , x ∗ , u ∗ ) ∂ x + (cid:101) h ( t ) ∂ f ( t , x ∗ , u ∗ ) ∂ u + o ( (cid:101) ) ,which leads to lim (cid:101) → C D ψ ( · ) a + ( x (cid:101) − x ∗ ) (cid:101) − ( x (cid:101) − x ∗ ) (cid:101) ∂ f ( t , x ∗ , u ∗ ) ∂ x − h ( t ) ∂ f ( t , x ∗ , u ∗ ) ∂ u = C D ψ ( · ) a + (cid:16) lim (cid:101) → x (cid:101) − x ∗ (cid:101) (cid:17) = lim (cid:101) → x (cid:101) − x ∗ (cid:101) ∂ f ( t , x ∗ , u ∗ ) ∂ x + h ( t ) ∂ f ( t , x ∗ , u ∗ ) ∂ u .We want to prove the existence of the limit lim (cid:101) → x (cid:101) − x ∗ (cid:101) = : η , that is, to prove that x (cid:101) ( t ) = x ∗ ( t ) + (cid:101)η ( t ) + o ( (cid:101) ) . This is indeed the case, since η is solution of the distributed order fractional differential equation C D ψ ( · ) a + η ( t ) = ∂ f ( t , x ∗ , u ∗ ) ∂ x η ( t ) + ∂ f ( t , x ∗ , u ∗ ) ∂ u h ( t ) , η ( a ) = The following result is a necessary condition of Pontryagin type [24] for the basic distributed-ordernon-local optimal control problem (BP).
Theorem 1 (Pontryagin Maximum Principle for (BP)) . If ( x ∗ ( · ) , u ∗ ( · )) is an optimal pair for (BP) , then thereexists λ ∈ PC , called the adjoint function variable, such that the following conditions hold for all t in the interval [ a , b ] : • the optimality condition ∂ L ∂ u ( t , x ∗ ( t ) , u ∗ ( t )) + λ ( t ) ∂ f ∂ u ( t , x ∗ ( t ) , u ∗ ( t )) =
0; (1)• the adjoint equation D ψ ( · ) b − λ ( t ) = ∂ L ∂ x ( t , x ∗ ( t ) , u ∗ ( t )) + λ ( t ) ∂ f ∂ x ( t , x ∗ ( t ) , u ∗ ( t )) ; (2) of 13 • the transversality condition I − ψ ( · ) b − λ ( b ) =
0. (3)
Proof.
Let ( x ∗ ( · ) , u ∗ ( · )) be solution to problem (BP), h ( · ) ∈ PC be a variation, and (cid:101) a real constant. Define u (cid:101) ( t ) = u ∗ ( t ) + (cid:101) h ( t ) , so that u (cid:101) ∈ PC . Let x (cid:101) be the state corresponding to the control u ∗ , that is, the statesolution of C D ψ ( · ) a + x (cid:101) ( t ) = f ( t , x (cid:101) ( t ) , u (cid:101) ( t )) , x (cid:101) ( a ) = x a . (4)Note that u (cid:101) ( t ) → u ∗ ( t ) for all t ∈ [ a , b ] whenever (cid:101) →
0. Furthermore, ∂ u (cid:101) ( t ) ∂(cid:101) (cid:12)(cid:12)(cid:12) (cid:101) = = h ( t ) . (5)Something similar is also true for x (cid:101) : because f ∈ C , it follows from Lemma 4 that, for each fixed t , x (cid:101) ( t ) → x ∗ ( t ) as (cid:101) →
0. Moreover, by Lemma 5, the derivative ∂ x (cid:101) ( t ) ∂(cid:101) (cid:12)(cid:12)(cid:12) (cid:101) = exists for each t . The objectivefunctional at ( x (cid:101) , u (cid:101) ) is J [ x (cid:101) , u (cid:101) ] = (cid:90) ba L ( t , x (cid:101) ( t ) , u (cid:101) ( t )) dt .Next, we introduce the adjoint function λ . Let λ ( · ) be in PC , to be determined. By the integration byparts formula (see Lemma 1), (cid:90) ba λ ( t ) · C D ψ ( · ) a + x (cid:101) ( t ) dt = (cid:104) x (cid:101) ( t ) · I − ψ ( · ) b − λ ( t ) (cid:105) ba + (cid:90) ba x (cid:101) ( t ) · D ψ ( · ) b − λ ( t ) dt ,and one has (cid:90) ba λ ( t ) · C D ψ ( · ) a + x (cid:101) ( t ) dt − (cid:90) ba x (cid:101) ( t ) · D ψ ( · ) b − λ ( t ) dt − x (cid:101) ( b ) · I − ψ ( · ) b − λ ( b ) + x (cid:101) ( a ) · I − ψ ( · ) b − λ ( a ) = J [ x (cid:101) , u (cid:101) ] gives φ ( (cid:101) ) = J [ x (cid:101) , u (cid:101) ] = (cid:90) ba (cid:104) L ( t , x (cid:101) ( t ) , u (cid:101) ( t )) + λ ( t ) · C D ψ ( · ) a + x (cid:101) ( t ) − x (cid:101) ( t ) · D ψ ( · ) b − λ ( t ) (cid:105) dt − x (cid:101) ( b ) · I − ψ ( · ) b − λ ( b ) + x (cid:101) ( a ) · I − ψ ( · ) b − λ ( a ) ,which by (4) is equivalent to φ ( (cid:101) ) = J [ x (cid:101) , u (cid:101) ] = (cid:90) ba (cid:104) L ( t , x (cid:101) ( t ) , u (cid:101) ( t )) + λ ( t ) · f ( t , x (cid:101) ( t ) , u (cid:101) ( t )) − x (cid:101) ( t ) · D ψ ( · ) b − λ ( t ) (cid:105) dt − x (cid:101) ( b ) · I − ψ ( · ) b − λ ( b ) + x a · I − ψ ( · ) b − λ ( a ) . of 13 Since the process ( x ∗ , u ∗ ) = ( x , u ) is assumed to be a maximizer of problem (BP), the derivative of φ ( (cid:101) ) with respect to (cid:101) must vanish at (cid:101) =
0, that is,0 = φ (cid:48) ( ) = dd (cid:101) J [ x (cid:101) , u (cid:101) ] | (cid:101) = = (cid:90) ba (cid:20) ∂ L ∂ x ∂ x (cid:101) ( t ) ∂(cid:101) (cid:12)(cid:12)(cid:12) (cid:101) = + ∂ L ∂ u ∂ u (cid:101) ( t ) ∂(cid:101) (cid:12)(cid:12)(cid:12) (cid:101) = + λ ( t ) (cid:18) ∂ f ∂ x ∂ x (cid:101) ( t ) ∂(cid:101) (cid:12)(cid:12)(cid:12) (cid:101) = + ∂ f ∂ u ∂ u (cid:101) ( t ) ∂(cid:101) (cid:12)(cid:12)(cid:12) (cid:101) = (cid:19) − D ψ ( · ) b − λ ( t ) ∂ x (cid:101) ( t ) ∂(cid:101) (cid:12)(cid:12)(cid:12) (cid:101) = (cid:21) dt − ∂ x (cid:101) ( b ) ∂(cid:101) (cid:12)(cid:12)(cid:12) (cid:101) = I − ψ ( · ) b − λ ( b ) ,where the partial derivatives of L and f , with respect to x and u , are evaluated at ( t , x ∗ ( t ) , u ∗ ( t )) .Rearranging the term and using (5), we obtain that (cid:90) ba (cid:20)(cid:16) ∂ L ∂ x + λ ( t ) ∂ f ∂ x − D ψ ( · ) b − λ ( t ) (cid:17) ∂ x (cid:101) ( t ) ∂(cid:101) (cid:12)(cid:12)(cid:12) (cid:101) = + (cid:16) ∂ L ∂ u + λ ( t ) ∂ f ∂ u (cid:17) h ( t ) (cid:21) dt − ∂ x (cid:101) ( b ) ∂(cid:101) (cid:12)(cid:12)(cid:12) (cid:101) = I − ψ ( · ) b − λ ( b ) = H ( t , x , u , λ ) = L ( t , x , u ) + λ f ( t , x , u ) , it follows that (cid:90) ba (cid:20)(cid:16) ∂ H ∂ x − D ψ ( · ) b − λ ( t ) (cid:17) ∂ x (cid:101) ( t ) ∂(cid:101) (cid:12)(cid:12)(cid:12) (cid:101) = + ∂ H ∂ u h ( t ) (cid:21) dt − ∂ x (cid:101) ( b ) ∂(cid:101) (cid:12)(cid:12)(cid:12) (cid:101) = I − ψ ( · ) b − λ ( b ) = H are evaluated at ( t , x ∗ ( t ) , u ∗ ( t ) , λ ( t )) . Now, choosing D ψ ( · ) b − λ ( t ) = ∂ H ∂ x ( t , x ∗ ( t ) , u ∗ ( t ) , λ ( t )) , with I − ψ ( · ) b − λ ( b ) = (cid:90) ba ∂ H ∂ u ( t , x ∗ ( t ) , u ∗ ( t ) , λ ( t )) h ( t ) = ∂ H ∂ u ( t , x ∗ ( t ) , u ∗ ( t ) , λ ( t )) = Remark 1.
If we change the basic optimal control problem (BP) by changing the boundary condition given on thestate variable at initial time, x ( a ) = x a , to a terminal condition, then the optimality condition and the adjointequation of the Pontryagin Maximum Principle (Theorem 1) remain exactly the same. Changes appear only on thetransversality condition: • a boundary condition at final/terminal time, that is, fixing the value x ( b ) = x b with x ( a ) remaining free,leads to I − ψ ( · ) a − λ ( a ) = in case when no boundary conditions is given (i.e., both x ( a ) and x ( b ) are free), then we have I − ψ ( · ) b − λ ( b ) = and I − ψ ( · ) a − λ ( a ) = of 13 Remark 2.
If f ( t , x , u ) = u, that is, C D ψ ( · ) a + x ( t ) = u ( t ) , then our problem (BP) gives a basic problem of thecalculus of variations, in the distributed-order fractional sense of [16]. In this very particular case, we obtain fromour Theorem 1 the Euler–Lagrange equation of [16] (cf. Theorem 2 of [16]). Remark 3.
Our distributed-order fractional optimal control problem (BP) can be easily extended to the vectorsetting. Precisely, let x : = ( x , . . . , x n ) and u : = ( u , . . . , u n ) with ( n , m ) ∈ N such that m ≤ n, and functionsf : [ a , b ] × R n × R m → R n and L : [ a , b ] × R n × R m → R be continuously differentiable with respect to all itscomponents. If ( x ∗ , u ∗ ) is an optimal pair, then the following conditions hold for t ∈ [ a , b ] : • the optimality conditions ∂ L ∂ u i ( t , x ∗ ( t ) , u ∗ ( t )) + λ ( t ) · ∂ f ∂ u i ( t , x ∗ ( t ) , u ∗ ( t )) = i =
1, . . . , m ;• the adjoint equations D ψ ( · ) b − λ j ( t ) = ∂ L ∂ x j ( t , x ∗ ( t ) , u ∗ ( t )) + λ ( t ) · ∂ f ∂ x j ( t , x ∗ ( t ) , u ∗ ( t )) , j =
1, . . . , n ;• the transversality conditions I − ψ ( · ) b − λ j ( b ) = j =
1, . . . , n . (6) Definition 4.
The candidates to solutions of (BP) , obtained by the application of our Theorem 1, will be called(Pontryagin) extremals.
We now illustrate the usefulness of our Theorem 1 with an example.
Example 1.
The triple ( ˜ x , ˜ u , λ ) given by ˜ x ( t ) = t , ˜ u ( t ) = t ( t − ) ln t , and λ ( t ) = , for t ∈ [
0, 1 ] , is an extremalof the following distributed-order fractional optimal control problem:J [ x ( · ) , u ( · )] = (cid:90) − (cid:16) x ( t ) − t (cid:17) − (cid:18) u − t ( t − ) ln t (cid:19) −→ max, C D ψ ( · ) + x ( t ) = u ( t ) , t ∈ [
0, 1 ] , x ( ) =
0. (7)
Indeed, by defining the Hamiltonian function asH ( t , x , u , λ ) = − (cid:34) ( x − t ) + (cid:18) u − t ( t − ) ln t (cid:19) (cid:35) + λ u , (8) it follows: • from the optimality condition ∂ H ∂ u = , λ ( t ) = (cid:18) u − t ( t − ) ln t (cid:19) ; (9) • from the adjoint equation D ψ ( α ) + λ ( t ) = ∂ H ∂ x , D ψ ( α ) + λ ( t ) = − ( x − t ) ; (10)• from the transversality condition, I − ψ ( α ) b − λ ( b ) =
0. (11)
We easily see that (9) , (10) and (11) are satisfied forx ( t ) = t , u ( t ) = t ( t − ) ln t , λ ( t ) = We now prove a Mangasarian type theorem for the distributed-order fractional optimal controlproblem (BP).
Theorem 2.
Consider the basic distributed-order fractional optimal control problem (BP) . If ( x , u ) → L ( t , x , u ) and ( x , u ) → f ( t , x , u ) are concave and ( ˜ x , ˜ u , λ ) is a Pontryagin extremal with λ ( t ) ≥ , t ∈ [ a , b ] , thenJ [ ˜ x , ˜ u ] ≥ J [ x , u ] for any admissible pair ( x , u ) . Proof.
Because L is concave as a function of x and u , we have from Lemma 2 that L ( t , ˜ x ( t ) , ˜ u ( t )) − L ( t , x ( t ) , u ( t )) ≥ ∂ L ∂ x ( t , ˜ x ( t ) , ˜ u ( t )) · ( ˜ x ( t ) − x ( t )) + ∂ L ∂ u ( t , ˜ x ( t ) , ˜ u ( t )) · ( ˜ u ( t ) − u ( t )) for any control u and its associated trajectory x . This gives J [ ˜ x ( · ) , ˜ u ( · )] − J [ x ( · ) , u ( · )] = (cid:90) ba [ L ( t , ˜ x ( t ) , ˜ u ( t )) − L ( t , x ( t ) , u ( t ))] dt ≥ (cid:90) ba (cid:20) ∂ L ∂ x ( t , ˜ x ( t ) , ˜ u ( t )) · ( ˜ x ( t ) − x ( t )) + ∂ L ∂ u ( t , ˜ x ( t ) , ˜ u ( t )) · ( ˜ u ( t ) − u ( t )) (cid:21) dt = (cid:90) ba (cid:20) ∂ L ∂ x ( t , ˜ x ( t ) , ˜ u ( t )) · ( ˜ x ( t ) − x ( t )) − ∂ L ∂ u ( t , ˜ x ( t ) , ˜ u ( t )) · ( ˜ u ( t ) − u ( t )) (cid:21) dt . (12)From the adjoint equation (2), we have ∂ L ∂ x ( t , ˜ x ( t ) , ˜ u ( t )) = D ψ ( · ) b − λ ( t ) − λ ( t ) ∂ f ∂ x ( t , ˜ x ( t ) , ˜ u ( t )) .From the optimality condition (1), we know that ∂ L ∂ u ( t , ˜ x ( t ) , ˜ u ( t )) = − λ ( t ) ∂ f ∂ u ( t , ˜ x ( t ) , ˜ u ( t )) . It follows from (12) that J [ ˜ x ( · ) , ˜ u ( t )] − J [ x ( · ) , u ( · )] ≥ (cid:90) ba (cid:18) D ψ ( · ) b − λ ( t ) − λ ( t ) ∂ f ∂ x ( t , ˜ x ( t ) , ˜ u ( t )) (cid:19) · ( ˜ x ( t ) − x ( t )) − λ ( t ) ∂ f ∂ u ( t , ˜ x ( t ) , ˜ u ( t )) · ( ˜ u ( t ) − u ( t )) dt . (13)Using the integration by parts formula of Lemma 1, (cid:90) ba λ ( t ) · C D ψ ( · ) a + ( ˜ x ( t ) − x ( t )) dt = (cid:104) ( ˜ x ( t ) − x ( t )) · I − ψ ( · ) b − λ ( t ) (cid:105) ba + (cid:90) ba ( ˜ x ( t ) − x ( t )) · D ψ ( · ) b − λ ( t ) dt ,meaning that (cid:90) ba ( ˜ x ( t ) − x ( t )) · D ψ ( · ) b − λ ( t ) dt = (cid:90) ba λ ( t ) · C D ψ ( · ) a + ( ˜ x ( t ) − x ( t )) dt − (cid:104) ( ˜ x ( t ) − x ( t )) · I − ψ ( · ) b − λ ( t ) (cid:105) ba . (14)Substituting (14) into (13), we get J [ ˜ x ( · ) , ˜ u ( · )] − J [ x ( · ) , u ( · )] ≥ (cid:90) ba λ ( t ) [ f ( t , ˜ x ( t ) , ˜ u ( t )) − f ( t , x ( t ) , u ( t )) − ∂ f ∂ x ( t , ˜ x ( t ) , ˜ u ( t )) · ( ˜ x ( t ) − x ( t )) − ∂ f ∂ u ( t , ˜ x ( t ) , ˜ u ( t )) · ( ˜ u ( t ) − u ( t )) (cid:21) dt .Finally, taking into account that λ ( t ) ≥ f is concave in both x and u , we conclude that J [ ˜ x ( · ) , ˜ u ( · )] − J [ x ( · ) , u ( · )] ≥ Example 2.
The extremal ( ˜ x , ˜ u , λ ) given in Example 1 is a global minimizer for problem (7) . This is easily checkedfrom Theorem 2 since the Hamiltonian defined in (8) is a concave function with respect to both variables x and u and,furthermore, λ ( t ) ≡ . In Figure 1, we give the plots of the optimal solution to problem (7) .
4. Conclusion
In this paper we investigated fractional optimal control problems depending on distributed-orderfractional operators. We have proved a necessary optimality condition of Pontryagin’s type and aMangasarian-type sufficient optimality condition. The new results were illustrated with an example.As future work, it would be interesting to develop proper numerical approaches to solve problems ofoptimal control with distributed-order fractional derivatives. In this direction, the approaches found in[17] and [18] can be easily adapted.
Author Contributions:
The authors equally contributed to this paper, read and approved the final manuscript: Formalanalysis, Faïçal Ndaïrou and Delfim F. M. Torres; Investigation, Faïçal Ndaïrou and Delfim F. M. Torres; Writing –original draft, Faïçal Ndaïrou and Delfim F. M. Torres; Writing – review & editing, Faïçal Ndaïrou and Delfim F. M.Torres.
Funding:
This research was funded by the Portuguese Foundation for Science and Technology (FCT), grant numberUIDB/04106/2020 (CIDMA). Ndaïrou was also supported by FCT through the PhD fellowship PD/BD/150273/2019.
Acknowledgments:
The authors are grateful to two anonymous reviewers for several comments and suggestions thathave helped them to improve the manuscript.
Conflicts of Interest:
The authors declare no conflict of interest. * (t) = t u * (t) = t(t − 1)/lnt Figure 1.
The optimal control u ∗ and corresponding optimal state variable x ∗ , solution of problem (7). References
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