Existence of optimal solutions to Lagrange problem for a fractional nonlinear control system with Riemann-Liouville derivative

. In the paper, a nonlinear control system containing the Riemann-Liouville derivative of order α ∈ (0 , 1) with a nonlinear integral performance index is studied. We discuss the existence of optimal solutions to such problem under some convexity assumption. Our study relies on the implicit function theorem for multivalued mappings.


EXISTENCE OF OPTIMAL SOLUTIONS TO
The proof is completed.
In [39, Theorem 1] the following fractional Gronwall lemma has been proved.

DARIUSZ IDCZAK AND RAFA L KAMOCKI
Proof. Let us note that functions a and g satisfy assumptions of Lemma 2.3. Consequently, condition (8) is satisfied. It is easy to check that Thus, using condition (8) we obtain The proof is completed.
From the above lemma, we immediately obtain for all t ∈ [0, T ).

Remark 1.
From the proofs of lemmas 2.3 and 2.4 it follows that the interval [0, T ) can be replaced with an arbitrary interval [a, b) ⊂ R. In such a case, inequality (9) takes the form for all t ∈ [a, b).
3.1. Homogoneous problem. Let us consider the optimal control problem (1)-(4) with x 0 = 0. i.e. where In [22], the following theorem on the existence and uniqueness of a solution x u ∈ AC α,p a+ to system (11)-(12), corresponding to a fixed control u ∈ U M , has been proved.
a.e. and all x ∈ R n ; a.e. and all u ∈ M , then, for any fixed u ∈ U M , there exists a unique solution x u ∈ AC α,p a+ to (11)-(12). Remark 2. The solution x u to system (11)-(12), obtained in [22], is given by In the proof of the main result (Theorem 3.5), we shall use the following lemmas.
for t ∈ [a, b] a.e. and all u ∈ M , with λ > − 1 p , then there exists a function for any u ∈ U M .

DARIUSZ IDCZAK AND RAFA L KAMOCKI
Consequently, we obtain the inequality a.e., u ∈ U M . Finally, using inequality (10) from Remark 1, we obtain p and M is bounded, then assumption (16) can be replaced with condition (3 f ). Proof is the function from Lemma 3.2, N is the Lipschitz constant from assumption (2 f ) and h is the function from condition (17). Then, we obtain

3.2.
Existence of optimal solutions. Definition 3.4. We say that a pair (x * , u * ) ∈ AC α,p a+ ×U M is an optimal solution to problem (11)- (14), if x * is a solution to system (11)-(12), corresponding to control u * and J(x * , u * ) J(x, u) for all pairs (x, u) ∈ AC α,p a+ × U M satisfying (11)- (12). Remark 4. In view of the uniqueness of the solution to control system (11)-(12), we can replace the functional J with a functionalĴ : U M → R given bŷ  In an elementary way, one can prove: Proposition 2. Let W be a metric space, Z -a compact metric space and Y -a metric space with a metric ρ.
uniformly with respect to z ∈ Z, i.e.
Now, we shall prove the main result of this paper, namely, a theorem on the existence of optimal solutions to problem (11)- (14). We have Theorem 3.5. Let α ∈ (0, 1) and 1 p < ∞. Moreover, let us assume that (a) the set M is compact, , |x| κ(t) and u ∈ M . Then problem (11) − (14) possesses an optimal solution (x * , u * ) ∈ AC α,p a+ × U M . Proof. Let us consider an extended problem of the form: It is clear that in order to find the optimal solution (x * , u * ) = (x u * , u * ) to problem (11)- (14), it suffices to find a control u * such that the extended responsex * = (x 0 * , x * ) satisfies the following equalitŷ J(u * ) = (I 1−α From assumption (f) and condition (17) it follows that −∞ < s < ∞.
Let(u l ) l∈N ∈ U M be a minimizing sequence of functionalĴ, i.e.
Step 2. Now, we shall prove that Indeed, first, let us observe that from the weak convergence (27) and complete continuity of the operator I α a+ : Consequently, there exists a subsequence, still denoted by (x l ) l∈N , convergent to x * (t) a.e. on [a, b]. In other words, where S 1 ⊂ [a, b] and µ(S 1 ) = b − a (µ denotes one-dimensional Lebesque measure). Let us define the set From Proposition 1 it follows that S is measurable. Let us suppose that µ(S) > 0, so also µ(S ∩ S 1 ) > 0. Let us fix t ∈ S ∩ S 1 . Assumptions (a) and (e) guarantee convexity and compactness of the setf (t, x * (t), M ). Consequently, there exist a constant γ(t) ∈ R and a vector b(t) ∈ R 1+n \{0} such that b(t)f (t, x * (t), u) < γ(t) < b(t)(D α a+x * )(t) for all u ∈ M. From the convergence (29) and Proposition 2 it follows that there exists l 0 ∈ N such that b(t)f (t, x l (t), u) < γ(t) for l l 0 and u ∈ M . In particular, for l l 0 and consequently, Thus, Since the last inequality is strong, therefore lim sup l→∞ qf (t, x l (t), u l (t)) < q(D α a+x * )(t) for some q ∈ Q, where Q is the set of points q = (q 1 , . . . , q n+1 ) such that q i , i = 1, . . . , n + 1, are rational numbers. It means that From the fact that µ(S ∩ S 1 ) > 0 it follows that there exists q 0 ∈ Q such that Since the function t → χ A (t)q 0 is essentially bounded, therefore the above strong inequality contradicts the weak convergence of the sequence (D α a+xl ) l∈N to D α a+x * in L 1 ([a, b], R 1+n ). It means that inclusion (28) is satisfied. Using the implicit function theorem for multivalued mappings ([26, Sec. II, Theorem 3.12]), we assert that there exists a measurable function u * : [a, b] → M such that (D α a+x * )(t) =f (t,x * (t), u * (t)), t ∈ [a, b] a.e. Thus, Since M is bounded, it is clear that u * ∈ U M . To finish the proof, we shall show that u * is a minimizer of functionalĴ. Indeed, from conditions (25) and (26) it follows that ([a, b], R) and (I 1−α a+ x 0 * )(a) = 0. Consequently, from equality (30), we obtain In particular, The proof is completed.
It is easy to show that if a pair x * (·), u * (·) ∈ AC α,p a + ×U M is an optimal solution to problem (11)- (14) with functions f and f 0 of the form then the pair is an optimal solution to problem (31)-(34). Using Theorem 3.5, we shall prove the following (C) the function g is such that for t ∈ [a, b] a.e., |y| κ(t) and all u ∈ M , then problem ( Indeed, since 1 p < 1 1−α , therefore the interval − 1 p , α − 1 is nonempty. Moreover, since g is lipschtzian in y, therefore a.e. and all u ∈ M . If t − a 1, then (t − a) α−1 1 and, consequently, If 0 < t − a < 1, then (t − a) α−1 < (t − a) λ for λ < α − 1 and, consequently,
It is easy to check that all assumptions of Theorem 3.7 are satisfied. In particular, the setṼ from assumption (F ) is convex, although both functions g and g 0 are not convex with respect to variable u. Consequently, there exists a pair (y * (t), u * (t)) which is an optimal solution to investigated problem. Using the maximum principle (cf. [22,Theorem 9]) we assert that there exists λ(·) ∈ I for t ∈ [0, 2] a.e. From [22,Theorem 11] it follows that a solution of problem (43) -(44) is given by Consequently, condition (45) is equivalent to the following one A k t α(k+1)−1 Γ(α(k + 1)) . Consequently,