Minimax Solutions of Hamilton--Jacobi Equations with Fractional Coinvariant Derivatives

We consider a Cauchy problem for a Hamilton--Jacobi equation with coinvariant derivatives of an order $\alpha \in (0, 1)$. Such problems arise naturally in optimal control problems for dynamical systems which evolution is described by ordinary differential equations with the Caputo fractional derivatives of the order $\alpha$. We propose a notion of a generalized in the minimax sense solution of the considered problem. We prove that a minimax solution exists, is unique, and is consistent with a classical solution of this problem. In particular, we give a special attention to the proof of a comparison principle, which requires construction of a suitable Lyapunov--Krasovskii functional.


Introduction
Nowadays, the theory of differential equations with fractional-order derivatives (see, e.g., [8,22,41,43,46]) is an actively developing branch of mathematics, which attracts the interest of many researchers.In particular, attention is paid to optimal control problems for dynamical systems which evolution is described by differential equations with the Caputo fractional derivatives.Such problems appear in various fields of knowledge including, e.g., chemistry [10], biology [50], electrical engineering [19], and medicine [21].Main directions of research here are related to necessary optimality conditions (see, e.g., [4,30] and the references therein) and numerical methods for constructing optimal controls (see, e.g., [29,45,51] and the references therein).In addition, note that several problems for linear systems are considered and studied in detail in, e.g., [2,13,18,20,27,40].The reader is also referred to [5] for an overview of works on various control problems for fractional-order systems.
In [12], the dynamic programming principle was extended to a Bolza-type optimal control problem for a dynamical system described by a fractional differential equation with the Caputo derivative of an order α ∈ (0, 1).In particular, it was shown that the value of this problem should be introduced as a functional in a suitable space of paths.Further, the problem was associated to a Hamilton-Jacobi equation with coinvariant (ci-) derivatives of the order α.Note that these derivatives can be considered as an extension of the notion of ci-derivatives (of the first order) proposed and developed in, e.g., [23,37].It was proved that if the value functional is smooth enough (namely, if it is ci-smooth of the order α), then it satisfies the Hamilton-Jacobi equation and the natural boundary condition, and, therefore, the value functional can be treated as a solution of this Cauchy problem in the classical sense.However, by analogy with the case of optimal control problems for dynamical systems described by ordinary differential equations (i.e., when α = 1), the value functional usually does not possess the required smoothness properties, which leads to the need to introduce and study generalized solutions of the obtained Cauchy problem.
In the paper, we consider a Cauchy problem for a Hamilton-Jacobi equation with ci-derivatives of an order α ∈ (0, 1) and propose a notion of a minimax solution of this problem.The technique of minimax solutions originates in the positional differential games theory (see, e.g., [24,26]) and can be seen as the development of the unification constructions of differential games [25].Minimax solutions of Hamilton-Jacobi equations with first-order partial derivatives were proposed and comprehensively studied in [48] (see also [49]).Further, this technique was extended to Hamilton-Jacobi equations with first-order ci-derivatives, which arise in optimization problems for dynamical systems described by functional differential equations of a retarded type [37] (see also [17,[31][32][33][34]36], and [3] for an infinite dimensional case) and of a neutral type [38,39,42].Note that the minimax approach was also applied to investigate generalized solutions of systems of equations arising in mean field games [1].
Following the general methodology, we define a minimax solution of the considered Cauchy problem in terms of a pair of non-local stability properties of this solution with respect to so-called characteristic differential inclusions, which in this case become fractional differential inclusions with the Caputo derivatives of the order α.We prove that a minimax solution exists, is unique, and is consistent with a classical solution of the problem.In particular, we establish a comparison principle.In general, the proofs of these results are carried out by the schemes of the proofs of the corresponding statements for Hamilton-Jacobi equations with partial derivatives [48] and with first-order ci-derivatives [37] (see also [31] and [3]).They are based on properties [15] of the sets of solutions of the characteristic differential inclusions.However, in order to prove the comparison principle, it is required to construct a suitable Lyapunov-Krasovskii functional with a number of prescribed properties (in this connection, see, e.g., Sect.15 of [37] and also Sect. 5 of [33], Sect.4.1 of [17]).Observe that the functionals proposed earlier in [17,31,36] for the case of first-order ci-derivatives cannot be applied in the fractional setting due to formal reasons, and, furthermore, it turns out that dealing with direct fractional counterparts of these functionals also does not lead to a satisfactory result (see Sect. 7.3 for discussion).To overcome this difficulty, we develop a technique that explicitly takes into account features of fractional-order integrals and derivatives and build the required functional via a finite sum (the number of terms depends on α) of some integral functionals with weakly singular kernels.In addition, note that each of these functionals can be treated as a modification of the quadratic functional used in, e.g., [11] (see also the references therein).Thus, the construction of the Lyapunov-Krasovskii functional substantially differs from the previous studies and can be considered as the main contribution of the paper.
The paper is organized as follows.In Section 2, we recall definitions of Riemann-Liouville integrals and Caputo derivatives of a fractional order, describe some of their properties, and introduce special functional spaces.Auxiliary facts from the theory of differential inclusions with the Caputo fractional derivatives are presented in Section 3. In Section 4, we discuss a notion of ci-derivatives of a fractional order.In Section 5, a Cauchy problem for a Hamilton-Jacobi equation with ci-derivatives of a fractional order is considered, and a definition of a minimax solution of this problem is given.Consistency of minimax and classical solutions of the problem is studied in Section 6.A comparison principle is established in Section 7. Existence and uniqueness of a minimax solution are proved in Section 8. Concluding remarks are given in Section 9.
Finally, for every α ∈ (0, 1], we introduce the following two subsets of G n : (2.9) Remark 2.1.Instead of the metric dist from (2.7), the set G n can be endowed with the metric (see, e.g., [3]) where a ∧ b min{a, b} for all a, b ∈ R. It can be shown that the metrics dist and dist 0 are not strongly equivalent, but they induce the same topology on G n (see, e.g., [47], Chap.2).In particular, we obtain that any functional ϕ : G n → R is continuous with respect to the metric dist if and only if it is continuous with respect to the metric dist 0 .The reader is referred to Section 5.1 of [17] for details.

Differential inclusions with fractional derivatives
This section deals with ordinary and functional differential inclusions with the Caputo fractional derivatives of an order α ∈ (0, 1) and provides auxiliary results concerning properties of the sets of solutions of such differential inclusions.The presented notions and statements are useful in order to give a definition of a minimax solution in Section 5.2 below and constitute a basis for the proofs of the main results of the paper.

Ordinary differential inclusions with fractional derivatives
Suppose that a set-valued function [0, T ] × R n (t, x) → F (t, x) ⊂ R n × R satisfies the following conditions: (F.1)For every t ∈ [0, T ] and x ∈ R n , the set F (t, x) is nonempty, convex, and compact in R n × R. (F.2) The set-valued function F is upper semicontinuous (in the Hausdorff sense).It means that, for every (t, x) ∈ [0, T ] × R n and ε > 0, there exists δ > 0 such that, for any (t , x ) ∈ [0, T ] × R n , the inequality Here and below, for ε > 0 and F ⊂ R n × R, the symbol [F ] ε stands for the ε-neighbourhood of F , given by (F.3)There exists c F > 0 such that, for any (t, x) Given (t 0 , w 0 (•)) ∈ G α n and z 0 ∈ R, consider the Cauchy problem for the differential inclusion where (x(t), z(t)) ∈ R n × R and t ∈ [t 0 , T ], under the initial condition By a solution of problem (3.1), (3.2), we mean a pair of functions ( ) denote the set of such solutions.Proposition 3.1.For any (t 0 , w 0 (•)) ∈ G α n and z 0 ∈ R, the set XZ α 0 (t 0 , w 0 (•), z 0 ) is nonempty and compact in Then, for every t ∈ [t 0 , T ] and (x (•), z (•)) ∈ XZ α 0 (t , x t (•), z(t )), the inclusion (x (•), z (•)) ∈ XZ α 0 (t 0 , w 0 (•), z 0 ) holds, where the function z (•) is defined by z (t) z(t) for t ∈ [0, t ] and z (t) z (t) for t ∈ (t , T ]. In the case when there is no additional variable z(t), similar statements are proved in [15] by adapting the proofs of the corresponding results for ordinary and functional differential inclusions with first-order derivatives (see, e.g., [9] and also [28,37]).The proofs of Propositions 3.1, 3.2, and 3.3 can be carried out by the same scheme with only minor technical changes, and, therefore, they are omitted.
Finally, note that, since the right-hand side of differential inclusion (3.1) does not depend on z(t), then, for any

Functional differential inclusions with fractional derivatives
Let us also give an analogue of Proposition 3.1 for the case when the right-hand side of differential inclusion (3.1) depends not only on a single value x(t) of an unknown solution, but on values x(τ ) for all τ ∈ [0, t] or, in other words, on the function x t (•) given by (2.6).
A functional ϕ : G α n → R is said to be ci-smooth of the order α if it is continuous, ci-differentiable of the order α at every point (t, w(•)) ∈ G α• n , and the functionals n ⊂ G n is endowed with the metric dist from (2.7) (see also Rem. 2.1).Note that, if α = 1, then the notion of ci-differentiability of the order α (i.e., of the first order) agrees with the notion of ci-differentiability from, e.g., [23,37].
The following proposition expresses one of the key properties of ci-smooth of the order α functionals (see, e.g., Lem.2.1 of [37], Lem.9.2 of [12], and also Prop. 1 of [17]).Proposition 4.1.Let a functional ϕ : G α n → R be ci-smooth of the order α.Then, for every function x(•) ∈ AC α ([0, T ], R n ), the function ω(t) ϕ(t, x t (•)), t ∈ [0, T ], is continuous on [0, T ] and is Lipschitz continuous on [0, ϑ] for any fixed ϑ ∈ [0, T ).Moreover, the equality below holds: 3) seem rather unusual.Nevertheless (see [12]), we need formula (4.3) with precisely the Caputo fractional derivative ( C D α x)(t) and not with the first-order derivative ẋ(t) in order to derive the Hamilton-Jacobi equation associated with an optimal control problem for a dynamical system described by a differential equation with the Caputo fractional derivative of the order α.Moreover, to the best of our knowledge, this approach to the development of the theory of Hamilton-Jacobi equations corresponding to such fractional-order dynamical systems is the only one that has been proposed so far.
In order to illustrate the notion of ci-differentiability of the order α in the fractional setting α ∈ (0, 1), let us present three examples.
Example 4.3.Take a functional ψ : G 1 n → R and, for every (t, Consider the functional ϕ : G α n → R given by Then, it can be verified directly that, in view of (2.5), the functional ϕ is ci-differentiable of the order α at a point (t, w(•)) ∈ G α• n provided that the functional ψ is ci-differentiable of the first order at the point (t, h(• | t, w(•))).Moreover, in this case, we have is continuous, we obtain that ci-smoothness of the first order of the functional ψ implies ci-smoothness of the order α of the functional ϕ.In other words, on the basis of every ci-smooth of the first order functional ψ, we can define the ci-smooth of the order α functional ϕ according to (4.5).Observe that the class of ci-smooth of the first order functionals is rather wide and wellstudied.Some examples of such functionals and formulas for calculating the corresponding ci-derivatives can be found in, e.g., Section 2 of [23] and Section 2 of [37].
Remark 4.4.In general, every functional ϕ : G α n → R can be represented in the form (see, e.g., [14], Sect.4) Here, the function h(• | t, w(•)) is defined by (4.4), and where we denote As in Example 4.3, there is a connection between ci-differentiability of the order α of the functional ϕ and ci-differentiability of the first order of the functional ψ (with respect to (t, h(•))).However, let us observe that the value l(τ | t, h 0 , h(•)) depends explicitly on the values of the derivatives ḣ(ξ) for a.e.ξ ∈ [0, τ ], which leads to difficulties with continuity properties of the mapping In particular, this prevents us from dealing directly with the functional ψ instead of the original functional ϕ.
Example 4.5.Let f : [0, T ] × R n → R n be a continuously differentiable function for which there exists c f > 0 such that, for any t ∈ [0, T ] and For every point (t 0 , w 0 (•)) ∈ G α n , consider the Cauchy problem for the differential equation where x(t) ∈ R n and t ∈ [t 0 , T ], under the initial condition A solution of problem (4.6), (4.7) is defined as a function x(•) ∈ X α (t 0 , w 0 (•)) satisfying differential equation (4.6) for a.e.t ∈ [t 0 , T ].According to, e.g., Proposition 2 of [14], such a solution exists and is unique, and we denote it by x(• | t 0 , w 0 (•)).Now, given a continuously differentiable function σ : R n → R, consider the functional Then, similarly to Theorem 3.1 of [16], it can be proved that the functional ϕ is ci-differentiable of the order α at every point (t, w(•)) ∈ G α• n , and, moreover, formulas for calculating the ci-derivatives of the order α of ϕ can be obtained.In particular, this example illustrates the fact that the notion of ci-differentiability of the order α can be a useful tool when dealing with functionals defined in terms of solutions of differential equations with the Caputo fractional derivatives of the order α.
The next example shows that even simplest functionals can be not ci-differentiable of the order α.
Example 4.6.Take ∈ R n \ {0} and consider the functional This functional ϕ is ci-smooth of the first order, and its ci-derivatives of the first order are given by Nevertheless, let us prove that the functional ϕ (more precisely, its restriction to G α n ) is not ci-differentiable of the order α at every point (t, w(•)) ∈ G α• n .Arguing by contradiction, assume that ϕ is ci-differentiable of the order α at some point (t 0 , w 0 At the same time, due to the assumption made, there exists Thus, we obtain Dividing this equality by (t − t 0 ) α and, after that, passing to the limit as t → t + 0 , we derive , f − g = 0, which contradicts the choice of f and g.

Hamilton-Jacobi equation with fractional coinvariant derivatives
In this section, we consider a Cauchy problem for a Hamilton-Jacobi equation with fractional ci-derivatives of an order α ∈ (0, 1) and propose a definition of a minimax solution of this problem.

Hamilton-Jacobi equation
Consider the Cauchy problem for the Hamilton-Jacobi equation with ci-derivatives of the order α and the boundary condition In this problem, ϕ : G α n → R is an unknown functional, and the given mappings (H.3)For every R ≥ 0, there exists λ H > 0 such that, for any t ∈ [0, T ] and x, x , s ∈ R n , if x ≤ R and x ≤ R, then (σ) The functional σ is continuous.
Cauchy problem (5.1), (5.2) arises [12] when studying infinitesimal properties of the value functional in Bolza-type optimal control problems for dynamical systems described by fractional differential equations with the Caputo derivatives of the order α.In this connection, assumptions (H.1)-(H.3)and (σ) seem quite natural since they are fulfilled in a sufficiently wide range of such problems.If the value functional is ci-smooth of the order α, then, according to Theorem 10.1 of [12], it satisfies Hamilton-Jacobi equation (5.1) and boundary condition (5.2), and, therefore, it can be considered as a solution of problem (5.1), (5.2) in the classical sense.In particular, this allows us to efficiently construct optimal control strategies ( [12], Cor.11.4).However, by analogy with the case of optimal control problems for dynamical systems described by ordinary differential equations (i.e., when α = 1), the value functional usually does not possess such smoothness properties, which leads to the need to introduce and study generalized solutions of problem (5.1), (5.2).Let us present an example.
Example 5.1.Suppose that n = 1 and, following Section 12 of [12], consider the optimal control problem for the dynamical system described by the differential equation with the Caputo fractional derivative of the order α , and t ∈ [0, T ], and for the cost functional J = |x(T )| to be minimized.It can be verified directly (or, e.g., based on [13], Thm.1), that the value functional ρ : G α 1 → R in this problem is given by where the auxiliary functional ρ * : Note that (see, e.g., [12], Sect.12) the functional ρ * is ci-smooth of the order α, and However, let us choose a point (t 0 , w α and prove that the value functional ρ is not ci-differentiable of the order α at this point.Suppose that ρ * (t 0 , w 0 (•)) = (T − t 0 ) α for definiteness.For every f ∈ R, considering the function Then, for a fixed f > −1 and all t ∈ [t 0 , T ], we obtain ρ * (t, x and, hence, On the other hand, take g ≤ −1.Put t * T − (T − t 0 )(1/g + 1) 1/α and observe that t * ∈ (t 0 , T ].Then, for any t ∈ [t 0 , t * ], we have 0 ≤ ρ * (t, x [g] t (•)) ≤ (T − t) α , and, consequently, ρ(t, x [g] t (•)) = 0. Thus, if we assume that ρ is ci-differentiable of the order α at the point (t 0 , w 0 (•)), we get (5.3) Dividing (5.3) by t − t 0 and passing to the limit as t → t + 0 , we derive Since this equality must hold for all f > −1 and g ≤ −1, we come to a contradiction and complete the proof.
Below, we give a definition of a minimax solution of problem (5.1), (5.2), which is a modification of the corresponding definitions in the case of Hamilton-Jacobi equations with partial derivatives (see, e.g., [48], Sect.6.2) and with first-order ci-derivatives (see, e.g., Sect.6 of [37], [31], and also [3]).We prove that the minimax solution exists, is unique, and is consistent with a classical solution of problem (5.1), (5.2).In particular, we establish a comparison principle.

Minimax solution
Note that (see, e.g., Sect.6.2 of [48]) the set E(t, x, s) is nonempty, convex, and compact in R n × R for every t ∈ [0, T ] and x, s ∈ R n , the set-valued function E is continuous (in the Hausdorff sense) due to assumption (H.1), and the inequality below holds: In addition, it follows from (H.2) that E(t, x, s) ∩ E(t, x, s ) = ∅ for any t ∈ [0, T ] and x, s, s ∈ R n .Given (t 0 , w 0 (•)) ∈ G α n , z 0 ∈ R, and s ∈ R n , consider the Cauchy problem for the differential inclusion where (x(t), z(t)) ∈ R n × R and t ∈ [t 0 , T ], under the initial condition Here, s is treated as a constant parameter.Let CH(t 0 , w 0 (•), z 0 , s) be the set of solutions (x(•), z(•)) of problem (5.5), (5.6).According to Proposition 3.1 and the described above properties of the function E, the set CH(t 0 , w 0 (•), z 0 , s) is nonempty and compact in C([0, T ], R n × R).Following the conventional terminology, differential inclusion (5.5) is called a characteristic differential inclusion, and any element of the set CH(t 0 , w 0 (•), z 0 , s) is called a (generalized) characteristic of equation (5.1).We say that a functional ϕ : G α n → R is an upper solution of problem (5.1), (5.2) if it is lower semicontinuous, satisfies the boundary condition and possesses the following property: Respectively, a lower solution of this problem is an upper semicontinuous functional ϕ : and the statement below holds: In (ϕ + ) and (ϕ − ), as usual, the function x t (•) is the restriction of the function x(•) to the interval [0, t] (see (2.6)).
A functional ϕ : G α n → R is called a minimax solution of problem (5.1), (5.2) if it is an upper solution as well as a lower solution of this problem.

Consistency
This section deals with issues of consistency of a minimax solution of problem (5.1), (5.2) with a solution of this problem in the classical sense.
By a classical solution of problem (5.1), (5.2), we mean a ci-smooth of the order α functional ϕ : G α n → R that satisfies Hamilton-Jacobi equation (5.1) and boundary condition (5.2).
The schemes of the proof of the statements below go back to the proofs of the corresponding results for Hamilton-Jacobi equations with partial derivatives (see, e.g., Sect.2.4 of [48]) and with first-order ci-derivatives (see, e.g., Sects.4 and 5 of [37], Prop.5.1 of [31], and also Sect.B.1 of [3]).Theorem 6.1.A classical solution of problem (5.1), (5.2) is a minimax solution of this problem.
We also establish the following result.
The proof of this proposition is based on the lemma below.
Lemma 6.3.If a functional ϕ : G α n → R is lower semicontinuous, then ϕ satisfies condition (ϕ + ) if and only if the following statement holds: Respectively, for an upper semicontinuous functional ϕ : G α n → R, condition (ϕ − ) is equivalent to the following: Proof.We prove only the first part of the lemma since the proof of the second one is essentially the same.It is clear that (ϕ + ) follows from (ϕ * + ), so it remains to verify the reverse implication.Let ϕ : G α n → R be a lower semicontinuous functional satisfying (ϕ + ), and let (t 0 , w 0 k, such that the following relations hold for every i ∈ 1, k: Then, by induction, based on Proposition 3.3 (see also the remark in the end of Sect.3.1), we can prove that, for every i ∈ 0, k, Thus, for the functions Due to compactness of CH(t 0 , w 0 (•), 0, s), we can assume that the sequence {(x [k] . Hence, passing to the limit as k → ∞ in inequality (6.2), in view of lower semicontinuity of ϕ, we derive So, the functional ϕ possesses property (ϕ * + ), and the lemma is proved.
In particular, from Proposition 6.2, we derive Theorem 6.4.If a minimax solution of problem (5.1), (5.2) is ci-smooth of the order α, then it is a classical solution of this problem.
Theorems 6.1 and 6.4 allow us to conclude that the introduced notion of a minimax solution of problem (5.1), (5.2) is consistent with the notion of a solution of this problem in the classical sense.Remark 6.5.The question of under what conditions problem (5.1), (5.2) admits a classical solution seems interesting and important, but is beyond the scope of the present paper.Nevertheless, let us note that, based on the results of [12] and [16], it can be verified that the Cauchy problem for the Hamilton-Jacobi equation and the boundary condition has a classical solution provided that the functions f and σ are as in Example 4.5.In addition, applying the results of [13] on the reduction of optimal control problems for linear fractional-order dynamical systems to optimal control problems for ordinary dynamical systems, some particular examples can be constructed when the Cauchy problem for the Hamilton-Jacobi equation and boundary condition (6.6) has a classical solution.In equation (6.7),A : [0, T ] → R n×n , f : [0, T ] × U → R n , and χ : [0, T ] × U → R are continuous functions, R n×n is the space of (n × n)-matrices endowed with the norm induced by the Euclidean norm

Comparison principle
The goal of this section is to prove the result below, which is often called a comparison principle.In the next section, it is used in the proof of existence and uniqueness of a minimax solution of problem (5.1), (5.2).Theorem 7.1.Let ϕ + and ϕ − be respectively an upper and a lower solutions of problem (5.1), (5.2).Then, the inequality below holds: In general, this theorem is proved by the same scheme as the corresponding statements for Hamilton-Jacobi equations with partial derivatives (see, e.g., [48], Thm.7.3) and with first-order ci-derivatives (see, e.g., [31], Lem.7.7).However, the key point of the proof, which concerns construction of a Lyapunov-Krasovskii functional with a number of prescribed properties (in this connection, see, e.g., Sect. 5 of [33] and Sect.4.1 of [17]), substantially differs from the previous studies owing to features of fractional-order integrals and derivatives (see Sect. 7.3 for discussion).

Lyapunov-Krasovskii functionals
The construction of the required Lyapunov-Krasovskii functional is carried out in four steps.
If t = 0, inequality (7.3) holds automatically.So, let t > 0. Note that and, if τ > 0, we derive Thus, inequality (7.3) is valid.2. Now, let us prove statement (V.1).Let (t 0 , r , are uniformly bounded and equicontinuous (see Sect. 2), there exists R > 0 such that r Fix k ∈ N. Assume that t 0 ≤ t k .Then, we have For the first term, by virtue of (2.8) and (7.4), we derive and for the second term, in view of (2.8) and ( 7.3), we get Thus, we obtain inequality (7.5).In the case t k < t 0 , this inequality can be proved in a similar way.
3. Further, let us prove (V.2).Fix r(•) ∈ AC γ ([0, T ], R) such that r(0) = 0 and consider the function v(t) V (t, r t (•)), t ∈ [0, T ].For every θ ≥ 0, based on the equality which can be verified by direct calculation, we derive and, consequently, according to (2.1) and (7.2), the function v(•) can be represented as follows: and v2 (t) = r(t) for any t ∈ (0, T ).Thus, it remains to investigate the properties of the function 4. To this end, let us introduce the auxiliary function and describe some of its properties.It follows from (7.6) that 0 In particular, we obtain M (θ) → 0 as θ → 0 + .Further, by virtue of the integration by parts formula, we derive Consequently, we have and, therefore, for any θ > 0 and θ > θ, (7.11)

Now, based on the representation
let us prove first that the function v 3 (•) satisfies the Lipschitz condition |v 3 (t ) − v 3 (t)| ≤ L|t − t| for every t, t ∈ [0, T ] with the constant Fix t, t ∈ [0, T ] such that t > t.If t = 0, then, taking into account that v 3 (0) = 0 and using (7.8), we derive Suppose that t > 0.Then, we have For the first term, according to (7.8), we get and, for the second term, by virtue of (7.11), we conclude Thus, we obtain the desired estimate.6.Since v 3 (•) ∈ Lip([0, T ], R), then the derivative v3 (t) exists for a.e.t ∈ [0, T ].In order to obtain an explicit formula for this derivative, let us calculate the right-hand side derivative v+ 3 (t) of v 3 (•) at every t ∈ (0, T ).For the first term in (7.12), owing to (7.13), we have and, therefore, Let us consider the second term in (7.12).For any τ ∈ [0, t), we get lim and, moreover, due to (7.11), Then, applying Lebesgue's dominated convergence theorem, we conclude Hence, we derive v+ 3 (t) lim As a result, in view of (7.9), we get dξ dτ for a.e.t ∈ [0, T ]. (7.14) 7. Summarizing the above, we obtain that v(•) ∈ Lip([0, T ], R) and and, therefore, This completes the proof of the lemma.
Before proving Lemma 7.4, we present an auxiliary proposition.
Proof.According to (2.1), for every t ∈ [0, T ], we have which proves the proposition.
3. In the general case, choose m ∈ N such that α ∈ [2 −m , 2 −(m−1) ).The cases m = 1 and m = 2 were considered above, so we can assume that m > 2. Put Note that β 1 = 0, and, due to the choice of m, Further, define numbers µ i > 0, i ∈ 1, m, by the following recurrent relations: and set Finally, take the corresponding functionals V * βi,µi , i ∈ 1, m, from (7.15) and put The proofs of properties (V * .1)and (V * .3)for the specified λ * and V * are carried out by the same scheme as in the two particular cases considered above, and, therefore, they are omitted.Let us prove (V * .2).

Discussion
As can be seen, properties (V.1)-(V.4) of the functionals V ε , p ε , and s ε , established in Lemma 7.7, allow us to prove Theorem 7.1 by following along the same lines as in the case of Hamilton-Jacobi equations with partial derivatives (see, e.g., [48], Thm.7.3) and with first-order ci-derivatives (see, e.g., [31], Lem.7.7).The fact that Cauchy problem (5.1), (5.2) involves fractional-order ci-derivatives is reflected in property (V.3), since relation (7.32) contains the term s ε (t, x t (•)), ( C D α x)(t) .Further, note that properties (V.1)-(V.4)are in turn a consequence of the definition of the functional V ε , which is close in form to the definitions of the functionals proposed earlier in [17,31,36] for the case of first-order ci-derivatives, and properties (V * .1)-(V* .3) of the auxiliary functional V * from Lemma 7.4.Thus, the construction of the functional V * is one of the main features of the presented proof of Theorem 7.1.The purpose of this section is to make some additional remarks concerning this construction.
First of all, observe that an important constituent part of the Lyapunov-Krasovskii functionals used in [17,31,36]  However, in the case when α ∈ (0, 1), a function x(•) ∈ AC α ([0, T ], R n ) is not necessarily Lipschitz continuous (it is only Hölder continuous with exponent α, see (2.3)), and, furthermore, it can happen that, at all points t ∈ [0, T ], the function x(•) is not first-order differentiable (see, e.g., [44]).Consequently, we cannot expect the corresponding function v(t) V (t, x t (•)) = x(t) 2 , t ∈ [0, T ], to have any first-order differentiability properties.This fact does not allow us to directly use the functionals from [17,31,36] in the present paper.
A possible way to overcome this difficulty is to modify the functional V as follows (see Ex.Note also that the functional V 1 (more precisely, its restriction to G α n ) is ci-smooth of the order α.Nevertheless, since, for a given (t, w(•)) ∈ G n , the value V 1 (t, w(•)) depends on the function w(•) only via the value of the integral (I 1−α (w(•) − w(0)))(t), it can happen that V 1 (t, w(•)) = 0 while w(•) − w(0) [0,t] = 0.This circumstance implies that the proposed modification V 1 of the functional V is not entirely suitable for taking it as a basis for constructing the desired functional V * .
V * (t, w(•)) e −λ * t V * 0,µ1 (t, w(•)), (t, w(•)) ∈ G n , where the functional V * 0,µ1 is defined by (7.17).We see that the functional V * 0,µ1 is obtained from the functional V 2 by multiplying the integrand by the function e −µ1(t−τ ) α , τ ∈ [0, t].Moreover, observe that, due to (7.7), the following representation holds (recall that µ 1 = Γ(1 − α)λ): and, hence, the functional V * 0,µ1 can be treated as a regularization of the functional V 4 , which, in particular, overcomes the difficulties with properties (V * .1)and (V * .3).On the other hand, in order to get now property (V * .2),we need to take into account the new additional term in (7.47).To this end, owing to Proposition 7.6, it suffices to multiply the functional V * 0,µ1 by the function e −λ * t , t ∈ [0, T ], for the suitably chosen λ * .However, in the remaining case α ∈ (0, 1/2), it can be shown that there is no constant b > 0 such that 1 Γ(α) which does not allow us to obtain an estimate of type (7.24) and, consequently, to deal with the additional term in (7.47) similarly to above.For this reason, the functionals V * β,µ are introduced (see (7.16)), which are more complex modifications of the functional V 2 , and, finally, according to (7.27), the functional V * is constructed on the basis of the finite sum of the functionals V * βi,µi .All details can be found in the proof of Lemma 7.4.

Existence and uniqueness
The main result of the paper is the following theorem, which is valid under assumptions (H.1)-(H.3)and (σ) from Section 5.1.
Proof.Taking into account Theorem 7.1, in order to prove the statement, it is sufficient to show that there exist an upper solution ϕ • + : G α n → R and a lower solution ϕ • − : G α n → R of problem (5.1), (5.2) such that ϕ • + (t, w(•)) ≤ ϕ • − (t, w(•)) for every (t, w(•)) ∈ G α n .Construction of such functionals ϕ • + and ϕ • − repeats essentially the arguments given in Section 7 of [31] and follows the scheme from Theorem 8.2 of [48] (see also Sect. 5 of [3] and [42], Thm. 1).The basis of this construction are the properties of the set-valued function E from (5.4) and the sets of characteristics, which are provided by Propositions 3.1, 3.2, and 3.3.For the reader's convenience, we briefly outline the main steps of the proof below.
Let Φ + be the set of functionals ϕ : G α n → R that satisfy boundary condition (5.7) and possess property (ϕ + ).Respectively, by Φ − , we denote the set of functionals ϕ : G α n → R such that (5.8) and (ϕ − ) are valid.1.For a given s ∈ R n , consider the functionals ψ