A necessary condition of Pontryagin type for fuzzy fractional optimal control problems

We prove necessary optimality conditions of Pontryagin type for a class of fuzzy fractional optimal control problems with the fuzzy fractional derivative described in the Caputo sense. The new results are illustrated by computing the extremals of three fuzzy optimal control systems, which improve recent results of Najariyan and Farahi.


1.
Introduction. Optimal control problems are usually solved with the help of the famous Pontryagin Maximum Principle (PMP), which provides a generalization of the classical Euler-Lagrange and Weierstrass necessary optimality conditions of the calculus of variations and is one of the central results of the mathematics of the XX century [30,32]. On the other hand, fractional (noninteger order) derivatives play an increasing role in mathematics, physics and engineering [17,19,23,31]. The two subjects have recently been put together and a theory of the calculus of variations and optimal control that deals with more general systems containing noninteger order derivatives is now available: see the books [2,21,29]. In particular, the fractional Hamiltonian perspective is a very active subject, being investigated in a series of publications: see, e.g., [4,7,13,24,25,33,39,40].
Uncertainty is inherent to most real world systems and fuzziness is a kind of uncertainty very common in real word problems [16]. In recent years, the notion of fuzzy set has been widely spread to various research areas, such as linear programming, optimization, differential equations and even fractional differential equations [34]. Thus, the study of a fuzzy optimal control theory forms a suitable setting for the mathematical modelling of real world problems in which uncertainties or vagueness pervade [14]. In the past few decades, the interest in the field of fuzzy optimal control has increased and fuzzy optimal control problems have attracted a great deal of attention. A large number of existing schemes of fuzzy optimal control for In [12], Farhadinia applies the fuzzy variational approach of [11] to fuzzy optimal control problems and derives necessary optimality conditions for fuzzy optimal control problems that depend on the Buckley and Feuring derivative (a Hukuhara derivative) [6]. In [8,10], the generalized Hukuhara derivative is used for a fuzzynumber-valued function, leading to solutions with decreasing length on their supports. Salahshour et al. [34] and Mazandarani and Kamyad [22] proposed, respectively, the concepts of Riemann-Liouville and Caputo fuzzy fractional differentiability, based on the Hukuhara difference, which strongly generalizes fuzzy differentiability. In [1,35], the generalized Hukuhara fractional Riemann-Liouville and Caputo concepts for fuzzy-valued functions are further investigated. For a Hukuhara approach valid on arbitrary nonempty closed sets of the real numbers (time scales) see [9]. In [8], Fard and Salehi investigate fuzzy fractional Euler-Lagrange equations for fuzzy fractional variational problems defined via generalized fuzzy fractional Caputo type derivatives. In [37], Soolaki et al. present necessary optimality conditions of Euler-Lagrange type for variational problems with natural boundary conditions and problems with holonomic constraints, where the fuzzy fractional derivative is described in a combined sense. Here, using the PMP and a novel form of the Hamiltonian approach, we achieve fuzzy solutions (state and control) by solving an appropriate system of differential equations. The proposed method is not limited to just optimal fuzzy linear time-invariant controlled systems, which were previously studied in [26,27] for integer-order problems. Since the Buckley and Feuring concept of differentiability [6] or even the Hukuhara notion of differentiability are not able to guarantee that the obtained solutions are fuzzy functions, in the present work we focus on the generalized Hukuhara differentiation. If the order of the derivatives appearing in the formulation of our problems approach integer values, then one obtains via our results the extremals of fuzzy optimal control problems investigated in [12,26,27].
The paper is organized as follows. Section 2 introduces necessary notations on fuzzy numbers and differentiability and integrability of fuzzy mappings. The notion of Caputo generalized Hukuhara fuzzy fractional derivative is recalled in Section 3. In Section 4 we establish our main result, Theorem 4.1, that provides Pontryagin conditions for fuzzy fractional optimal control problems. In Section 5 we consider three problems, illustrating the proposed method. In particular, it is shown that the candidates to minimizers given in [26,Example 4.2] and [28,Example 3] are not solutions to the considered problems. We end with Section 6 of conclusions and future work.
The converse is also true: ifã(t) = sup{r|a r ≤ t ≤ a r } is a fuzzy number with parametrization given by [ã] r = [a r , a r ], then functions a r and a r satisfy conditions (i)-(v).
Ifz =x ⊖ gHỹ exists as a fuzzy number, then its level cuts [z r , z r ] are obtained by z r = min{x r − y r , x r − y r }, z r = max{x r − y r , x r − y r } for all r ∈ [0, 1].
Definition 2.4 (See [18]). Let t ∈ (a, b) and h be such that t + h ∈ (a, b). The generalized Hukuhara derivative of a fuzzy-valued functionx : If D gHx (t) ∈ R f satisfying (1) exists, then we say thatx is generalized Hukuhara differentiable (gH-differentiable for short) at t. Also, we say thatx is , and thatx is If the fuzzy functionf (t) is continuous in the metric D, then its definite integral Definition 2.5 (See [11]). Letã,b ∈ R f . We writeã b , if a r ≤ b r and a r ≤ b r for all r ∈ [0, 1]. We also writeã ≺b, ifã b and there exists an r ′ ∈ [0, 1] so that We say thatã,b ∈ R f are comparable if eitherã b orã b ; and noncomparable otherwise.
3. The fuzzy fractional calculus. The Riemann-Liouville fractional derivative has one disadvantage when modelling real world phenomena: the fractional derivative of a constant is not zero. To eliminate this problem, one often considers fractional derivatives in the sense of Caputo. For this reason, in our work we restrict ourselves to problems defined by generalized Hukuhara fractional Caputo derivatives. Analogous results are, however, easily obtained for generalized Hukuhara fractional derivatives in the Riemann-Liouville sense.
The fuzzy gH-fractional Caputo derivative of a fuzzy valued function was introduced in [1]. Following ] be a fuzzy valued function and α > 0. Then the Riemann-Liouville fractional integral of order α is defined by where Γ(α) is the Gamma function and x > a.
] be a fuzzy valued function. The fuzzy (left) Riemann-Liouville integral off (x), based on its r-level representation, can be expressed as follows: Following [1,18], we now recall the definition of Caputo-type fuzzy fractional derivative under the gH-difference. The definition is similar to the concept of Caputo derivative in the crisp case [31] and gives a direct extension of gH-differentiability to the fractional context [5].
Remark 1. We use the notation gH−C a D β itx when the fuzzy-valued functionx is The definitions for the right fuzzy fractional operators of order α, are completely analogous.

4.
Optimality of fuzzy fractional optimal control problems. The fuzzy fractional optimal control problem in the sense of Caputo is introduced, without loss of generality, in Lagrange form: wherex : [a, b] → R n F satisfies appropriate boundary conditions, u r (t) and u r (t) are piecewise continuous, and β ∈ (0, 1). The LagrangianL : are assumed to be functions of class C F 1 with respect to all their arguments and We say that an admissible fuzzy curve (x * ,ũ * ) is solution of problem (2), if for all admissible curve (x,ũ) of problem (2), It follows, from the definition of partial ordering given in Definition 2.5, that the inequalityJ(x * ,ũ * ) J (x,ũ) holds if and only if for all r ∈ [0, 1], where the r-level set of fuzzy curvesx * ,ũ * ,x andũ are respectively.
and is a particular case of our problem (2): one just need to chooseφ(x,ũ, t) =ũ.  (2)). Let controlũ * have the lower and upper bounds u * r and u * r , andx * be the corresponding state with lower and upper bounds x * r and x * r . If (x * ,ũ * ) is solution to (2), then there exist costate functions p 1 and p 2 such that the quadruple (x * r , x * r , u * r , u * r ) satisfies • the Hamiltonian adjoint system ∂H ∂x * r , • and the stationary conditions ∂H ∂u r = 0, ∂H ∂u r = 0, where the partial derivatives are evaluated at with the Hamiltonian H defined as follows: ifx * is [(1)-gH]-differentiable, then Proof. Consider a variation u r = u * r +δu r and a variation u r = u * r +δu r of u * r and u * r , respectively, with corresponding state (x * r + δx r , x * r + δx r ). The consequent change∆(J ) inJ is Using the gH-difference, one gets where [x + δx,ũ + δũ] r = (x * r + δx r , x * r + δx r , u * r + δu r , u * r + δu r , t), [x,ũ] r = (x * r , x * r , u * r , u * r , t).
Without loss of generality, we consider L r x * r + δx r , x * r + δx r , u * r + δu r , u * r + δu r , t dt − b a L r (x * r , x * r , u * r , u * r , t) dt L r x * r + δx r , x * r + δx r , u * r + δu r , u * r + δu r , t dt − b a L r (x * r , x * r , u * r , u * r , t)dt.
If we evaluate the derivatives in the integrand along the optimal trajectory, then we arrive at is an optimal solution for the crisp functions J r and J r . Let δJ r and δJ r denote the first variation. If u * r and u * r are optimal, from the classical theory of optimal control, it is necessary that the first variation δJ r and δJ r are zero. Thus, on optimal trajectories, one has for all variations. Now, we simply need to introduce two Lagrange multipliers p 1 (t) and p 2 (t). Ifx is [(1)-gH]-differentiable, then we consider the integrals and Ifx is [(2)-gH]-differentiable, then we consider the integrals Let us assume [(1)-gH]-differentiablity ofx * . The proof for the other case is completely similar, so it is here omitted. We begin by computing the variation δφ r of functional (5): Becausex(a) andx(b) are specified, we have δx r (a) = δx r (b) = δx r (a) = δx r (b) = 0. Using fractional integration by parts [21], equation (7) is equivalent to ∂ϕ r ∂x r δx r + ∂ϕ r ∂u r δu r + ∂ϕ r ∂u r δu r p 1 dt since φ r = 0 for all u r and u r and δφ r = 0. Therefore, the condition δJ r = 0 can now be replaced by δJ r + δφ r = 0. With substitutions of δJ r and δφ r , we have If we use the Hamiltonian function as in (3), then by summing Eqs. (8) and (9) we arrive at The intended necessary conditions follow.
A pair (x * ,ũ * ) satisfying Theorem 4.1 is said to be an extremal for problem (2).

Illustrative examples.
In this section, we apply the necessary conditions of Pontryagin type given by Theorem 4.1 to three fuzzy optimal control problems.

5.1.
A non-autonomous fuzzy fractional optimal control problem. We begin with a non-autonomous fuzzy fractional optimal control problem.
Example 1. Consider the following problem: We assume that (2t − 1)x(t) ⊖ gH sin(t)ũ(t) exists and Using Theorem 4.1, we consider two cases to obtain the extremals of (10).
The extremals for (12) are obtained from (11) by considering β → 1: We solved (13) numerically, with Matlab's built-in solver bvp4c. Figure 1 shows the control and state extremals, where the solid lines in the center corresponds to The optimality conditions of Theorem 4.1, the initial conditions and the control system assert that Note that it is difficult to solve the above fractional equations to get the extremals. For 0 < β < 1, a numerical method should be used. When β goes to 1, problem (10) reduces to problem (12). The extremals for (12) are obtained from (14) and considering β → 1: Similarly as before, we solved (15) with Matlab's built-in solver bvp4c. Figure 2 shows the graphic of the control and state extremals, where the solid lines at the center correspond to r = 1, the dashed lines are the upper bounds, and the doted lines are the lower bounds for fuzzy control and state functions for r = 0. Comparing

5.2.
On two examples of Najariyan and Farahi. In the recent paper [26], Najariyan and Farahi characterize extremals for fuzzy linear time-invariant (autonomous) optimal control systems. Precisely, they investigate a method for solving the following time-invariant fuzzy optimal control problem: b aũ 2 (t)dt → min, Main result of [26] asserts that the fuzzy optimal control problem (16) is equivalent to the crisp complex optimal control system b a (u r (t)) 2 + i(u r (t)) 2 dt → min, where the elements of the matrices B and D are determined from those of A and C as follows: with e : a + bi → a + bi and g : a + bi → b + ai. The extremals for the crisp optimal control problem (17) are given by the classical PMP [32].
Example 2 (Example 4.2 of Najariyan and Farahi [26]). Consider the following problem: x 1 (0) =x 2 (0) = (1, 2, 3), In [26] the authors provide a figure (see [26, Figure 2]) with what they claim to be the fuzzy control and state extremals for problem (18). It turns out that the provided functions are not extremals for the optimal control problem (18). Indeed, in the crisp case, i.e., when the variablesx 1 (t),x 2 (t) andũ(t) and2 = (1, 2, 3) and 0 = (−0.5, 0, 0.5) are crisp quantities, the fuzzy optimal control problem (18) is transformed into the following crisp optimal control problem: x 1 (0) = x 2 (0) = 2, The extremals for (19) are easily obtained from the classical PMP [30,32]. Figure 3 shows the graphics of the control and state extremals for problem (19). Comparing these functions with the ones given in [26,Example 4.2], one may conclude that there is an inconsistency in [26,Example 4.2]. Let us use Theorem 4.1 to obtain the extremals for (18). Suppose thatx 1 is a [(1)-gH]-differentiable function andx 2 is a [(2)-gH]-differentiable function. The analysis of the other three cases are similar and are left to the reader. Our assumption leads to H = −((u r ) 2 + (u r ) 2 ) + p 1 (−2x r 2 + u r ) + p 2 (−2x r 2 + u r ) + p 3 (2x r 1 ) + p 4 (2x r 1 ). From the optimality conditions of Theorem 4.1, the initial conditions and the control system of problem (18), and considering β = 1, we obtain that By solving (20), the control and state extremals can be found straightforwardly. Figure 4 shows the graphics of the fuzzy control and state extremals, where the continuous lines in the center correspond to r = 1. We clearly see from Figures 3 and 4 that the fuzzy extremals of the time-invariant linear optimal control problem (18) are related with the extremals of the crisp optimal control problem (19), which is in agreement with the results of [26].  In [28], Najariyan and Farahi also propose a method to find extremals for linear non-autonomous fuzzy optimal control problems with fuzzy boundary conditions. Here we show that fuzzy minimizers for [28, Example 3] do not exist.
We have just discussed necessary optimality conditions. Much remains to be done and we end by mentioning some possible lines of research. The obtained fuzzy fractional optimality conditions are, in general, difficult to solve and it would be good to develop specific numerical methods to address the issue. To obtain second order necessary optimality conditions is presently a big challenge. Other open lines of research consist to prove sufficient optimality conditions and existence results. While here we have assumed that the optimal solution exists, and necessary optimality conditions have been obtained under such assumption, as Example 3 shows, this is not always the case. As future work, we intend to prove conditions assuring the existence of optimal solutions to fuzzy fractional optimal control problems.