Weak stabilization for fractional differential inclusions with infinite delays

A control system governed by semilinear fractional differential inclusions with infinite delays is studied. We prove the stabilizability in a weak sense for mentioned system under the assumption that its linear part is strongly stabilizable.

Scientific Journal − No27/2018 35 WEAK STABILIZATION FOR FRACTIONAL DIFFERENTIAL INCLUSIONS WITH INFINITE DELAYS Tran Dinh Ke1, Tran Van Tuan2 Hanoi National University of Education Hanoi Pedagogical University Abstract: Abstract A control system governed by semilinear fractional differential inclusions with infinite delays is studied We prove the stabilizability in a weak sense for mentioned system under the assumption that its linear part is strongly stabilizable Keywords: Keywords fractional differential inclusion; infinite delay; weak stabilization; measure of noncompactness Email: ketd@hnue.edu.vn Received 25 July 2018 Accepted for publication 15 December 2018 INTRODUCTION Let X and U be Banach spaces Consider the following control system D0α x ( t ) ∈ Ax ( t ) + Bu ( t ) + F ( t , xt ) , t > 0, (1.1) x ( t ) = ϕ ( t ) , t ≤ 0, (1.2) where D0α , α ∈ ( 0,1) is the fractional derivative in the Caputo sense, A generates a C0 -semigroup on X , B : U → X is a bounded linear operator, ϕ ∈ B with B being an admissible phase space that will be specified later, F : [ 0, ∞ ) × B → P ( X ) is a multivalued function, and xt stands for the history of the state function defined by xt ( s ) = x ( t + s ) for s ∈ ( −∞,0] Fractional differential equations has been proved to be an effective tool for modeling various physical phenomena such as flows in porous media and oscillations (see [11, 19, 22]) A complete reference to basic notions and facts on fractional differential equations can be found in [15, 18, 23] The study of fractional differential inclusions is motivated by a number of problems Let us consult the one coming from the theory of differential Ha Noi Metroplolitan University 36 equations with discontinuous right-hand term (see [8]) Precisely, the multivalued function F in (1.1) is a regularization of a discontinuous single-valued function f, for instance F ( t , y ) is closed convex hull of the set Ξ( y) = {z : ∃yn → y, z = lim f (t , yn )}, n →∞ where the limit is understood in a specific context By this way, the obtained differential inclusion is considered as an approximate model for the previous differential equation with a note that F is more regular than f, that is, F is a u.s.c multivalued map The solvability of (1.1)-(1.2) has been investigated by many researchers under numerous assumptions on the semigroup etAand the nonlinearity F (see, e.g [4, 16, 17, 25]) Besides, the question of controllability for (1.1)-(1.2) has been taken into account in several works In this direction, we refer the reader to [20, 21, 25] In this work, we are interested in the question of stabilizability Since the uniqueness for (1.1)-(1.2) is unavailable, the Lyapunov theory fails to apply, even in the case α = Then we adopt the concept of weak stabilization as follows Definition 1.1 System (1.1)-(1.2) is said to be weak stabilizable by feedback if there exists a linear operator D : U → X such that with u ( t ) = Dx ( t ) , the solution set S (ϕ ) ≠ ∅ and satisfies (1) Stability: for anyϵ>0, there exists δ > such that for all x ∈ S (ϕ ) , supt >0 xt B in the strong topology of L ( X ) This family is called compact if S (t ) is a compact operator for each t > 2.1 Stabilization for linear fractional control system Assume that the linear operator A with domain D(A) is the generator of a C0 semigroup {etA }t ≥0 on X Denote by σ ( A) s ( A) the spectrum of A, the spectral bound of A, ω ( A) the growth bound of A, ωess ( A) the essential growth bound of A Specifically, s( A) = sup{Re(λ ) : λ ∈ σ ( A)}; ω ( A) = lim t →∞ ln ‖ etA ‖L ( X ) t ; ωess ( A) = ln ‖ etA ‖ess , here ‖ ⋅ ‖ess denote the essential norm of a linear operator on X (see the t →∞ t monograph of Engel and Nagel [7]) lim We gather here some results which provide properties of the above notions Their proofs can be found in [7] Lemma 2.1 Let A be the generator of a C0 -semigroup on a Banach space X Then (i ) ω ( A) = max{s ( A), ωess ( A)}; (ii) If the semigroup {etA } is norm - continuous, then s ( A) = ω ( A); (iii) ωess ( A) = ωess ( A + K ), where K ∈ L( X ) is a compact operator; (iv) For every ω > ωess ( A) the set σ ( A) ∩ {λ ∈ ℂ : Re(λ ) ≥ ω} is bounded Ha Noi Metroplolitan University 38 Recall that for a fixed ω > ωess ( A) ( ω is called a spectral separator) then σ ( A) can be decomposed as σ ( A) = σ u ( A) ∪ σ s ( A) (see, for example, [7, Theorem V.3.7]), where σ u ( A) = σ ( A) ∩ {λ ∈ ℂ : Re(λ ) ≥ ω}, σ s ( A) = σ ( A) ∩ {λ ∈ ℂ : Re(λ ) < ω} One can see that σ u ( A) is bounded and separated from σ s ( A) by a closed curve ΓC Following Kato [14, Theorem 6.17], we have a corresponding decomposition of the state space X and of the opeator A as follows X = X u ⊕ X s , A = Au ⊕ As , where X u = PX , X s = ( I − P ) X , P is the projection operator P= (λ I − A) −1 d λ , ∫ Γ 2π i C Au = A |X u , As = A |X s and In fact, X u and X s are two A -invariant subspaces, that is AX u ⊂ X u , AX s ⊂ X s Furthermore, σ ( Au ) = σ u ( A), σ ( As ) = σ s ( A) and Au is a bounded operator on X u We denote by L1 (0, T ; X ) the space of functions on [0,T] taking values in X , which are integrable in the sense of Bochner Definition 2.1 The fractional integral of order α > of a function f ∈ L1 (0, T ; X ) is defined by I 0α f (t ) = t (t − s)α −1 f ( s)ds, ∫ Γ(α ) where Γ is the Gamma function, provided the integral converges Definition 2.2 For a function f ∈ C1 ([0, T ]; X ) , the Caputo fractional derivative of order α ∈ (0,1) is defined by D0α f (t ) = t (t − s ) −α f ′ ( s )ds ∫ Γ(1 − α ) We now consider the Cauchy problem D0α x (t ) = Ax (t ) + f (t ), t > 0, α ∈ (0,1], (2.1) x(0) = x0 (2.2) Scientific Journal − No27/2018 39 Definition 2.3 A function x ∈ C ([0, T ]; X ) is called an integral solution of (2.1)-(2.2) on interval [0,T] iff t x(t ) = Sα (t ) x0 + ∫ (t − s)α −1 Pα (t − s ) f ( s) ds, ∀t ∈ [0, T ], (2.3) where ∞ Sα (t ) = ∫ φα (θ )et α θA ∞ dθ , (2.4) α (2.5) Pα (t ) = α ∫ θφα (θ )et θA φα (θ ) = ∞ ∑ (−1) απ n −1 dθ , θ n −1 n =1 Γ (nα + 1) sin nπα , θ ∈ (0, ∞) n! (2.6) The formulation of (2.3) and the operators Sα (⋅), Pα (⋅) can be found in [26] We refer to Sα (⋅) and Pα (⋅) as the fractional resolvent operators generated by A Let f(t) = Bu(t) in (2.1)-(2.2), then we are concerned with the linear control system D0α x (t ) = Ax (t ) + Bu (t ), t > 0, (2.7) x(0) = x0 (2.8) For the sake of brevity, we use the notation ( A, B)α to indicate the last system and make ( A, B ) stand for ( A, B)1 Recall that ( A, B ) is said to be exponentially stabilizable by feedback if one can find a linear operator D : U → X such that the operator A = A + BD generates an exponentially stable semigroup {S (t )}t ≥0 , i.e ‖ S (t ) ‖L ( X ) ≤ Me− at , for some M ≥ 1, a > In what follows, a bounded linear operator D : U → X is called a feedback operator if A = A + BD generates a C0 -semigroup We now adopt the following definition Definition 2.4 We say that system ( A, B)α is stabilizable by feedback if there exists a feedback operator D : U → X such that the integral solution of the Cauchy problem D0α x(t ) = Ax(t ), t > 0, x(0) = x0 , exists and ‖ x(t ) ‖→ as t → +∞ , here A = A + BD Ha Noi Metroplolitan University 40 It will be shown in the sequel that, if ( A, B ) is stabilizable, so is ( A, B)α To this end, we establish some important estimates Let < α < 1, δ > −1 be given constants We make use of function Ψα ,δ :[0, ∞) → [0, ∞) defined by ∞ Ψ α ,δ ( s ) = ∫ e − sθ θ δ φα (θ ) dθ , s ≥ 0, (2.9) where φα is defined by (2.6) The following important estimate was proved in [1] Lemma 2.2 ([1]) There exists a constant Dδ such that ≤ Ψα ,δ ( s) ≤ Dδ , ∀s > s1+α (2.10) Now for a given operator A , let S (⋅) and {Sα (⋅), Pα (⋅)} be the C0 -semigroup, the fractional resolvent operators generated by A , respectively Then one has ∞ Sα (t ) = ∫ φα (θ )S (t αθ ) dθ , (2.11) ∞ Pα (t ) = α ∫ θφα (θ ) S (t α θ )dθ (2.12) Using Lemma 2.2, we have the following result Lemma 2.3 Let S (⋅) be an exponentially stable semigroup, ‖ S (t ) ‖L ( X ) ≤ Me − at , ∀t ≥ 0, (2.13) for some M ≥ 1, a > Then for any fixed t ≥ 0, Sα (t ) and Pα (t ) are linear bounded operators, moreover, ‖ Sα (t ) ‖L ( X ) ≤ M 1, D0 α  at  ,  (2.14) and  D , 12α  Γ(1 + α ) a t ‖ Pα ‖L( X ) ≤ α M   ,  (2.15) for all t > , where D0 , D1 are constants defined in Lemma 2.2 The following result give a sufficient condition for system (2.7)-(2.8) to be stabilizable by feedback Scientific Journal − No27/2018 41 Theorem 2.4 Suppose that ( A1) The C0 − semigroup {etA }t ≥0 is norm − continuous, ( A2) ωess ( A) < 0, ( A3) ( Au , PB ) is exponentially stabilizable by feedback , where P is a projection with respect to a spectral separator ω ∈ (ωess ( A), 0) and Au = A |PX Then the control system (2.7)-(2.8) is stabilizable by feedback Proof Since the semigroup {etA }t ≥0 is norm-continuous, it implies from Lemma 2.1(i) that s ( A) = ω ( A) We notice that under the hypothesis (A1), the semigroup {Ss (t )}t ≥0 generated by As is also norm-continuous In addition, for a spectral separator ω such that ωess ( A) < ω < , we get that s( As ) = ω( As ) < thanks to (A2) Let Du be the feedback operator that makes ( Au , PB) stabilizable, and put D = ( Du ,0) ∈L( X , U ) Then using the arguments similar to those in the proof [5, Theorem 3.32] (see also [24, Theorem 6.1]), we infer that the operator A = A + BD generates a C0 semigroup {S (t )}t ≥0 which is exponentially stable, i.e there exists positive constants M ≥ 1, a > such that ‖ S (t ) ‖L ( X ) ≤ Me − at , t ≥ (2.19) By the feedback u ( t ) = Dx ( t ) , the integral solution of (2.7)-(2.8) is given by x(t ) = Sα (t ) x0 Then ‖ x(t ) ‖ ≤ ‖ Sα (t ) ‖L ( X ) ‖ x0 ‖ = O (t −α ) as t → +∞, thanks to Lemma 2.3 The proof is complete 2.2 Phase space In this and the next subsection, we will utilize the axiomatic definition of the phase space B introduced by Hale and Kato in [10] The space B is a linear space of functions mapping ( −∞, 0] into X endowed with a seminorm | ⋅ |B and satisfying the following fundamental axioms If a function y: (−∞,T +σ] → X is such that y |[σ ,T +σ ]∈ C ([σ , T + σ ]; X ) and yσ ∈ B , then Ha Noi Metroplolitan University 42 (B1) yt ∈ B for t ∈ [σ , T + σ ]; (B2) the function t ֏ yt is continuous in t ∈ [σ , T + σ ]; (B3) | yt |B ≤ K (t − σ ) sup ‖ y ‖X + M (t − σ ) | yσ |B where K , M :[0, ∞) → [0, ∞), σ ≤ s ≤t K is continuous, M is locally bounded, and they are independent of y In this work, we need an additional assumption on B : (B4) there exists ̻ > such that ‖ φ (0) ‖X ≤ ̻ | φ |B , for all φ ∈ B We now show an example of phase spaces For more examples, we refer to the book by Hino, Mukarami and Naito [12] Let Cγ = {φ ∈ C ((−∞,0]; X ) : lim eγθ φ (θ ) exists in X }, θ →−∞ (2.20) where γ is a positive number This phase space satisfies (B1)-(B3) with K (t ) = 1, M (t ) = e−γ t , (2.21) and it is a Banach space with the norm | φ |B = sup eγθ ‖ φ (θ ) ‖ θ ≤0 2.3 Measure of compactness and condensing multivalued maps In this subsection, we recall some notations and basic results related to multivalued analysis and measure of noncompactness in Banach spaces Let E be a Banach space Denote P ( E ) = {Y ⊂ E : Y ≠ ∅}, Pb ( E ) = {Y ∈ P ( E ) : Y is bounded}, K ( E ) = {Y ∈ P ( E ) : Y is compact}, Kv( E ) = {Y ∈ K ( E ) : Y is convex} We will use the following definition of measure of noncompactness (see, e.g [13]) Definition 2.5 A function β : Pb ( E ) → ℝ + is called a measure of noncompactness (MNC) on E if β (coΩ) = β (Ω) for every Ω∈ Pb ( E ), where coΩ is the closure of convex hull of Ω An MNC β is said to be: Scientific Journal − No27/2018 43 (i) monotone if for each Ω0 , Ω1 ∈ Pb ( E ) such that Ω0 ⊆ Ω1 , we have β (Ω0 ) ≤ β (Ω1 ); (ii) nonsingular if β ({a} ∪ Ω) = β (Ω) for any a ∈ E , Ω ∈ Pb ( E ); (iii) invariant with respect to the union with a compact set, if β ( K ∪ Ω) = β (Ω) for every relatively compact set K ∈ E and Ω ∈ Pb ( E ); (iv) algebraically semi - additive if β (Ω0 + Ω1 ) ≤ β (Ω0 ) + β (Ω1 ) for any Ω0 , Ω1 ∈ Pb ( E ); (v) regular if β (Ω) = is equivalent to the relative compactness of Ω An important example of MNC satisfying all properties stated in Definition 2.5 is the Hausdorff MNC χ (⋅) , which is defined as follows χ (Ω ) = inf{ε > : Ω has a finite ε − net} Now we define some useful MNCs which will be employed in the sequel For given L > and D ⊂ E = C ([0, T ]; X ) , put ωT ( D ) = sup e − Lt χ ( D (t )), where D (t ) := { x(t ) : x ∈ D}, (2.22) t∈[0,T ] modT ( D ) = lim sup max δ → x∈D t , s∈[0,T ],|t − s| : χ ( L(Ω)) ≤ C χ (Ω) or all bounded set Ω ⊂ E} We are now in a position to recall some basic estimates based on MNCs Proposition 2.5 ([13]) If {ω n } ⊂ L1 (0, T ; X ) such that ‖ ωn (t ) ‖X ≤ ν (t ), for a.e t ∈[0, T ], for some ν ∈ L1 (0, T ; ℝ) , then we have χ ({∫ ω (s)ds}) ≤ 2∫ χ ({ω (s)})ds , for t ∈ [0, T ] t t n n (2.24) Ha Noi Metroplolitan University 44 Proposition 2.6 ([1]) Let D ⊂ L1 (0, T ; X ) be such that (i) ‖ ζ (t ) ‖≤ ν (t ), for all ζ ∈ D and for a.e t ∈ [0, T ], (ii) χ ( D (t )) ≤ q (t ) for a.e t ∈ [0, T ], where ν , q ∈ L1 (0, T ; ℝ ) Then χ here t t 0 ( ∫ D(s)ds ) ≤ 4∫ q(s)ds, t t 0 ∫ D(s)ds = {∫ ζ (s) ds : ζ ∈ D} We also need some notions and facts from set-valued analysis Let Y be a metric space Definition 2.6 ([13]) A multivalued map (multimap) F : Y → P ( E ) is said to be: (i) upper semicontinuous (u.s.c) if F −1 (V ) := { y ∈ Y : F ( y ) ∩ V ≠ ∅} is a closed subset of Y for every closed set V ⊂ E ; (ii) weakly upper semicontinuous (weakly u.s.c) if F −1 (V ) is closed subset of Y for all weakly closed set V ⊂ E ; (iii) closed if its graph Γ F := {( y, z ) : z ∈ F ( y )} is closed subset of Y × E; (iv) compact if F (Y ) is relatively compact in E; (v) quasicompact if its restriction to any compact subset A ⊂ Y is compact Lemma 2.7 ([13, Theorem 1.1.12]) Let G : Y → P ( E ) be a closed quasicompact multimap with compact values Then G is u.s.c Lemma 2.8 ([3, Proposition 2]) Let X be a Banach space and Ω be a nonempty subset of another Banach space Assume that G : Ω → P ( X ) is a multimap with weakly compact, convex values Then G is weakly u.s.c iff {xn } ⊂ Ω with the strong convergence xn → x0 ∈ Ω and yn ∈G ( xn ) implies the weak convergence yn ⇀ y0 ∈ G ( x0 ) , up to a subsequence We end this subsection by recalling a fixed point principle for condensing multimaps Definition 2.7 A multimap F : Z ⊆ E → P ( E ) is said to be condensing with respect to an MNC β ( β −condensing) if for any bounded set Ω ⊂ Z, the relation β (Ω ) ≤ β ( F (Ω )) implies the relatively compactness of Ω Scientific Journal − No27/2018 45 Let β be a monotone nonsingular MNC in E We have the following fixed point principle Theorem 2.9 ([13, Corollary 3.3.1]) Let M be a bounded convex closed subset of E and let F : M → Kv(M) be a closed and β-condensing multimap Then Fix( F ) := { x ∈ M : x ∈ F ( x )} is nonempty Consequently, we have the following theorem Theorem 2.10 Let M be a compact convex subset of E and F : M → Kv (M) a closed multimap with convex values Then Fix( F ) ≠ ∅ Consider the operator Qα : Lp (0, T ; X ) → C ([0, T ]; X ) defined by t Qα ( f )(t ) = ∫ (t − s )α −1Pα (t − s ) f ( s) ds, where p > α (2.25) Given T > and ϕ ∈ B , we denote by Cϕ the set of all continuous functions y : [0, T ] → X such that y (0) = ϕ (0) Then Cϕ is a closed subspace of C ([0, T ]; X ) with the norm ‖ y ‖C = sup ‖ y(t ) ‖ t∈[0,T ] For ϕ ∈ B and y ∈ C ([0, T ]; X ) , we define the function y[ϕ ] : ( −∞ , T ] → X as follows:  y (t ) for t ∈ [0, T ], y[ϕ ](t ) =  ϕ (t ) for t < For x ∈ Cϕ , we denote PFp ( x ) = { f ∈ Lp (0, T ; X ) : f (t ) ∈ F (t , x[ϕ ]t )} By using the same arguments as in [9], we have that PFp is well-defined In accordance with formula (2.3), we derive the definition of integral solution for (3.1)-(3.2) in the following way Definition 2.8 A function x : ( −∞ , T ] → X is called an integral solution of problem (3.1)-(3.2) on ( −∞ , T ] iff x (t ) = ϕ (t ) for t ≤0, and there exists a function f ∈ PFp ( x |[0,T ] ) such that t x(t ) = Sα (t )ϕ (0) + ∫ (t − s )α −1Pα (t − s ) f ( s ) ds, t > 0, Ha Noi Metroplolitan University 46 where Sα (⋅) and Pα (⋅) are defined by (2.11) and (2.12), respectively For given ϕ ∈ B , we define the solution operator Σ : Cϕ → P (Cϕ ) as follows Σ( y )(t ) = Sα (t )ϕ (0) + {∫ (t − s )α −1Pα (t − s ) f ( s ) ds : f ∈ PFp ( y )}, t (2.26) or equivalently, Σ( y ) = Sα (⋅)ϕ (0) + Qα PFp ( y ), where Qα defined by (2.25) Since F has convex values, so does PFp This implies that Σ has convex values as well WEAK STABILIZATION In this section, we assume that there exists a feedback operator D : U → X , that is A = A + BD generates a C0 -semigroup, denoted by {S (t )}t ≥0 We consider the solvability of (1.1)-(1.2) with u (t ) = Dx (t ), t ≥ , which can be written as D0α x(t ) ∈ Ax(t ) + F (t , xt ), t > 0, (3.1) x(t ) = ϕ (t ), t ≤ (3.2) It is easily seen that if y is a fixed point of Σ then y[ϕ ] is an integral solution of (3.1)(3.2) on ( −∞ , T ] To investigate the weak stabilizability for the problem (3.1)-(3.2), we first prove the weak attractivity Consider the solution operator Σ on the following space: BCϕ0 = { y ∈ C ([0, +∞); X ) : y (0) = ϕ (0) and lim ‖ y (t ) ‖= 0} t →+∞ One can easily verify that BCϕ0 is a closed subspace of BC = { y ∈ C ([0, +∞); X ) :‖ y ‖BC < +∞}, with the norm ‖ y ‖ BC = sup ‖ y (t ) ‖ t ≥0 We are now in a position to define a regular MNC on BCϕ0 and show the condensivity of the solution operator on this space Let π T , T > 0, be the truncate function on BC, that is, for z ∈ BC π T ( z ) ∈ C ([0, T ]; X ) is the restriction of z to the interval [0, T] Let χ ∞ ( D ) = sup ωT (π T ( D)) + sup modT (π T ( D)), T >0 T >0 Scientific Journal − No27/2018 47 then χ ∞ is an MNC on BC Unfortunately, this MNC is not regular (see [1]) We need another MNC to combine with χ ∞ Put dT ( D ) = sup sup ‖ x (t ) ‖ , x∈D t ≥T d ∞ ( D ) = lim dT ( D ), (3.3) χ * ( D ) = χ ∞ ( D ) + d ∞ ( D ) (3.4) T →∞ The following result was proved in [1] Proposition 3.1 ([1]) The MNC χ * defined by (3.4) is regular We deal with the weak attractivity for (3.1)-(3.2) as follows We first find a closed convex bounded subset of BCϕ0 , which is invariant under Σ , in the following form BRγ ( ρ ) = { y ∈ BCϕ0 :‖ y ‖BC ≤ R and t γ ‖ y (t ) ‖≤ ρ , ∀t ≥ 0}, where < γ < min(α ,1 − α ) , R and ρ are positive numbers Then we show that Σ : BRγ ( ρ ) → P ( BRγ ( ρ )) is a closed multimap with compact and convex values, and is χ∗- condensing To this end, we assume that: (A*) The C0 -semigroup generated by A is norm-continuousand (A,B) is stabilizable by feedback (B*) The space B satisfies (B1)-(B4) with K ∈ BC (ℝ+ ) and M being such that t γ M (t ) = O(1) as t → +∞ , here BC (ℝ+ ) denotes the space of continuous bounded functions on ℝ + (F*) The multimap F : ℝ + × B → Kv( X ) satisfies (1) for a.e t ∈ ℝ + the multimap F (t , ⋅) : B → Kv ( X ) is u.s.c; (2) for every fixed v ∈B the multimap F (⋅, v) : ℝ + → Kv( X ) admits a strongly measurable selection; (3) there exists nonnegative functions m ∈ L∞ ( ℝ + ) with p > for every v ∈B , one has ‖ F (t , v) ‖ ≤ m(t ) | v |B , for a.e t ∈ ℝ + , here ‖ F (t , v) ‖ = sup{‖ ξ ‖: ξ ∈ F (t , v)} ; α such that Ha Noi Metroplolitan University 48 (4) if the semigroup generated by A is non-compact then for any bounded set Ω ⊂ B , we have χ ( F (t , Ω)) ≤ k (t ) sup χ (Ω(θ )) θ ≤0 for a.e t ∈ ℝ + , where k ∈ L∞ ( ℝ + ) is a nonnegative function By the assumption (A*), we can assume that there exists a feedback operator D such that A = A + BD generates the C0 -semigroup S (⋅) satisfying ‖ S (t ) ‖L ( X ) ≤ Me − at , ∀t ≥ 0, here M ≥ 1, a > By (B*) we have K ∞ = sup K (t ) < +∞, M ∞ = sup M (t ) < +∞, t ≥0 γ t ≥0 γ M ∞ = sup t M (t ) < +∞ t ≥0 By (F*) we get m∞ = ess supt∈ℝ + m(t ) < +∞, k∞ = ess supt∈ℝ + k (t ) < +∞ Lemma 3.2 Let (A*), (B*) and (F*) hold Then Σ( BC Rγ ( ρ )) ⊂ BC Rγ ( ρ ) for some R , ρ > 0, provided that t Θ∞ = sup ∫ (t − s )α −1 ‖ Pα (t − s) ‖L ( X ) m( s) M ( s ) ds < +∞, t ≥0 t Λ ∞ = sup ∫ (t − s)α −1 ‖ Pα (t − s ) ‖L ( X ) m( s ) K ( s ) ds < 1, t ≥0 Ξ ∞ = sup ∫ t ≥0 δt ‖ Pα (t − s) ‖L ( X ) m( s )ds < +∞, t ≥0 δ 2γ δt K∞ for some δ ∈ (0,1) Proof We proceed the proof in two steps Step We show that there exists R > such that the ball B R = {x ∈ BCϕ0 : ‖ x ‖ BC ≤ R} (3.6) (3.7) t Υ ∞ = sup ∫ (t − s)α −1 ‖ Pα (t − s ) ‖L ( X ) m( s )ds < (3.5) , (3.8) Scientific Journal − No27/2018 49 is invariant under Σ, i.e Σ(B R ) ⊂ B R Assume to the contrary that for each n ∈ℕ there exists xn ∈ BCϕ0 with ‖ xn ‖BC ≤ n and zn ∈Σ( xn ) with ‖ zn ‖BC > n From the formulation of Σ, we can find f n ∈ PFp ( xn ) such that t zn (t ) = Sα (t )ϕ (0) + ∫ (t − s )α −1Pα (t − s ) f n ( s ) ds, ∀t > 0 It follows that t ‖ zn (t ) ‖ ≤ ‖ Sα (t ) ‖ L ( X ) ‖ ϕ (0) ‖ + ∫0 (t − s )α −1 ‖ Pα (t − s ) ‖ L ( X ) m( s ) | xn [ϕ ]s | B ds (3.9) One observes that | xn [ϕ ]s |B ≤ K ( s ) sup r∈[0, s ] ‖ xn ( s ) ‖ + M ( s ) | ϕ |B ≤ nK ( s ) + M ( s ) | ϕ |B (3.10) Then combining (3.9) with (3.10), we get ‖ zn (t ) ‖ ≤ ‖ Sα (t ) ‖L ( X ) ‖ ϕ (0) ‖ t + ∫ (t − s)α −1 ‖ Pα (t − s ) ‖L ( X ) m( s ) ( nK ( s ) + M ( s ) | ϕ |B ) ds t ≤ M ‖ ϕ (0) ‖ + n ∫ (t − s)α −1 ‖ Pα (t − s) ‖L ( X ) m( s ) K ( s ) ds + | ϕ |B Θ(t ), (3.11) (3.12) where t Θ(t ) = ∫ (t − s )α −1 ‖ Pα (t − s ) ‖L ( X ) m( s ) M ( s ) ds Hence (3.12) implies ‖ z n‖ BC 1< n ≤ Λ∞ + M ‖ ϕ (0)‖ +Θ ∞ | ϕ |B n (3.13) Passing to the limit as n → ∞ in the last inequality, we get a contradiction with (3.6) Step Now we show the existence of a number ρ > such that BRγ ( ρ ) = B R ∩ {x ∈ BCϕ0 : t γ ‖ x(t )‖ ≤ ρ , ∀t ≥ 0}, is invariant under Σ Assume the opposite: there exists yn ∈ BRγ (n) and zn ∈ Σ( yn ) such that sup t γ ‖ zn (t )‖ > n Take f n ∈ PFp ( yn ) such that t ≥0 t zn (t ) = Sα (t )ϕ (0) + ∫ (t − s )α −1Pα (t − s ) f n ( s ) ds, Ha Noi Metroplolitan University 50 then ‖ zn (t ) ‖ ≤ ‖ Sα (t ) ‖ L ( X ) ‖ ϕ (0) ‖ + (∫ δt + t ∫δ ) (t − s) α −1 t ‖ Pα (t − s ) ‖ L ( X ) m( s) | ys | B ds = I1 (t ) + I (t ) + I (t ), where I1 (t ) =‖ Sα (t )‖ L ( X ‖ ) ϕ (0)‖ , δt I (t ) = ∫ (t − s)α −1‖ Pα (t − s)‖ L ( X ) m( s ) | ys | B ds, t I (t ) = ∫ (t − s )α −1‖ Pα (t − s )‖ L ( X ) m( s ) | ys | B ds δt Using Lemma 2.2, we have γ −α t γ I1 (t ) = t γ ‖ Sα (t )‖ L ( X ‖ ‖ ϕ (0)‖ → as t → +∞ ) ϕ (0)‖ ≤ Mt So sup t γ I1 (t ) < +∞, thanks to the fact that t γ I1 (t ) → as t → t ≥0 Now it follows from (B3) with σ = that | ys |B ≤ K ( s ) sup r∈[0, s ] ‖ y ( r ) ‖ + M ( s ) | ϕ |B (3.14) ≤ RK ∞ + M ∞ | ϕ |B So regarding I ( t ) one has t t γ I (t ) ≤ Ct γ ∫ (t − s )α −1 ‖ Pα (t − s ) ‖L ( X ) m( s ) ds δt ≤ Ct [(1 − δ )t ]α −1 ∫ γ t δt ‖ Pα (t − s ) ‖L ( X ) m( s )ds ≤ C (1 − δ )α −1 t γ −(1−α ) Ξ ∞ , ∀t > 0, where C = RK ∞ + M ∞ | ϕ |B Then t γ I (t ) → as t → +∞ On the other hand, we see that t γ I (t ) ≤ CMm∞ γ t CMm∞ t ∫ (t − s )α −1ds = (1 − δ )α t γ +α δ t Γ(α ) Γ(α + 1) → as t → Thus sup t γ I (t ) < +∞ t ≥0 To deal with I3 (t ) , we employ (B3) with σ = δ s to get (3.15) Scientific Journal − No27/2018 51 | ys |B ≤ K ( s − δ s) sup r∈[δ s , s ] ≤ K∞ sup r∈[δ s , s ] ‖ y (r ) ‖ + M (s − δ s) | yδ s |B ‖ y (r ) ‖ +CM (s(1 − δ )), thanks to (3.14) So s γ | y s |B ≤ K ∞ s γ (δ s ) − γ sup r γ ‖ y ( r ) ‖ +C (1 − δ ) −γ [(1 − δ ) s ]γ M ((1 − δ ) s ) r∈[δ s , s ] ≤ K ∞δ − γ n + C (1 − δ ) − γ M ∞γ Plugging this estimate into t α I (t ) , we get t t γ I (t ) = t γ ∫ (t − s )α −1s −γ δt ‖ Pα (t − s ) ‖L ( X ) m( s ) s γ | ys |B ds t ≤ δ −γ ∫ (t − s )α −1 ‖ Pα (t − s ) ‖L ( X ) m ( s )[ K ∞δ −γ n + C (1 − δ ) −γ M ∞γ ]ds δt ≤ ( K ∞δ −2γ n + Cδ −γ (1 − δ ) −γ M ∞γ ) Υ ∞ (3.16) Putting (3.13), (3.15) and (3.16) together, we obtain 1< sup t γ n t ≥0 ‖ z (t ) ‖ ≤  sup t γ I1 (t ) + sup t γ I (t ) + sup t γ I (t )  n t ≥0 t ≥0 t ≥0  D ≤ K ∞δ −2γ Υ ∞ + , n (3.17) where D = sup t γ I1 (t ) + sup t γ I (t ) + Cδ −γ (1 − δ ) − γ M ∞γ Υ ∞ < +∞ t ≥0 t ≥0 Passing to the limit in (3.17) as n → ∞ , we get a contradiction with (3.8) The proof is complete Now let us state that one can choose L in the definition of ωT in (2.22) such that t ℓ := 4sup ∫ (t − s )α −1‖ Pα (t − s )‖ L ( X ) e − L (t − s ) k ( s ) ds < t >0 This is possible since we have t α ∫ (t − s) ‖ Pα (t − s)‖ −1 L( X ) Mk∞ t (t − s )α −1e − L ( t − s ) ds ∫ Γ(α ) Mk∞ t α −1 − Lr = r e dr Γ(α ) ∫0 Mk ≤ α ∞ → as L → +∞ L e − L (t − s ) k ( s ) ds ≤ We will prove the χ * -condensing property of Σ in the following lemma Ha Noi Metroplolitan University 52 Lemma 3.3 Let (A*) (B*) and (F*) hold If the conditions (3.5)-(3.8) are satisfied then Σ is χ * -condensing on BRγ ( ρ ) Proof We need to show that for any subset Ω ⊂ BRγ ( ρ ) , it holds that χ * (Σ(Ω)) ≤ ℓ ⋅ χ * (Ω), where ℓ is given in (3.18) We first claim that d∞ (Σ(Ω)) = Let z ∈Σ(Ω), then z ∈ BRγ ( ρ ) By the formulation of BRγ ( ρ ) one has t γ ‖ z (t )‖ ≤ ρ , ∀t ≥ For T > we have ‖ z(t ) ‖ ≤ ρT −γ , ∀t ≥ T Since z ∈ Σ (Ω) is arbitrary, we get d T ( Σ (Ω )) = sup sup‖ z (t )‖ ≤ ρ T − γ z∈Σ ( Ω ) t ≥T Letting T → +∞ , we obtain d∞ (Σ(Ω)) = In the sequel, we show that χ ∞ (Σ(Ω)) ≤ ℓ ⋅ χ ∞ (Ω) Let T > be arbitrary but fixed Since π T (Ω) is bounded in C([0, T]; X) we have PFp (Ω) is an integrably bounded in Lp (0, T ; X ) Using this fact, we get π T (Σ(Ω)) = Sα (⋅)ϕ (0) + Qα PFp (π T (Ω)) is equicontinuous in C([0, T]; X) Thus modT (π T (Σ(Ω)) ) = (3.19) For each t ∈ [0, T ] , by using Proposition 2.6, we have χ (Σ(Ω)(t )) ≤ χ ( ∫ (t − s) t α −1 Pα (t − s)PFp (Ω)( s) ds ) t ≤ 4∫ (t − s )α −1χ ( Pα (t − s )PFp (Ω)( s ) ) ds (3.20) If the semigroup S (⋅) is compact, then χ (Σ (Ω )(t )) = 0, thanks to the fact that χ ( Pα (t − s ) PFp (Ω )( s )) = for a.e s ∈ (0, t ) In the oppositecase, by using (F*)(4) we have χ (Pα (t − s )PFp (Ω)( s)) ≤ ‖ Pα (t − s ) ‖ L ( X ) χ (PFp (Ω)( s)) ≤ ‖ Pα (t − s ) ‖ L ( X ) k ( s ) sup χ ( Ω[ϕ ]( s + θ ) ) θ ≤0 ≤ ‖ Pα (t − s) ‖ L ( X ) k ( s ) sup χ (Ω(r )), r∈[0, s ] (3.21) Scientific Journal − No27/2018 53 thanks to the fact that Ω[ϕ ]( r ) = {ϕ ( r )}, ∀r ≤ , which is singleton Putting (3.21) in (3.20), we get t χ (Σ(Ω)(t )) ≤ ∫ (t − s)α −1 ‖ Pα (t − s) ‖L ( X ) k ( s) sup χ (Ω(r )) ds r∈[0, s ] This implies t e − Lt χ (Σ(Ω)(t )) ≤ 4∫ (t − s )α −1‖ Pα (t − s)‖ L ( X ) e − L (t − s ) k ( s) sup e − Lr χ (Ω(r )) ds ( r∈[0, s ] ) t ≤ 4∫ (t − s )α −1‖ Pα (t − s )‖ L ( X ) e − L (t − s ) k ( s ) ds ωT (π T (Ω) ) Consequently, ωT (π T (Σ(Ω))) ≤  sup ∫ (t − s)α −1 ‖ Pα (t − s ) ‖L ( X ) e− L ( t − s ) k ( s ) ds  ωT (π T (Ω) ) t  t >0  (3.22) Combining the inequalities (3.19) and (3.22) yields χ ∞ (Σ(Ω)) ≤ ℓ ⋅ χ ∞ (Ω), with t ℓ = ∫ (t − s )α −1 ‖ Pα (t − s ) ‖L ( X ) e − L ( t − s ) k ( s) ds < Therefore χ * (Σ(Ω)) = χ ∞ (Σ(Ω)) + d ∞ (Σ(Ω)) = χ ∞ (Σ(Ω)) ≤ ℓ ⋅ χ ∞ (Ω) ≤ ℓ ⋅ χ * (Ω) The proof is complete The main result of this section is now formulated as follows Theorem 3.4 Let the hypotheses of Lemma 3.3 hold Then problem (3.1)-(3.2) has at least one integral solution on ( −∞ , +∞ ) satisfying ‖ x(t ) ‖ = O(t −γ ) as t →∞ Proof By Lemma 3.2, one can consider the solution operator Σ : BRγ ( ρ ) → Kv ( BRγ ( ρ )) for some R , ρ > We get that Σ is closed In addition, Σ is χ * -condensing by Lemma 3.3 So the conclusion follows from Theorem 2.9 We are in a position to formulate the weak stabilizability of the system (1.1)(1.2) It suffices to show the weakly asymptotical stability of (3.1)-(3.2) under the hypotheses (A*), (B*) and (F*) By the definition of solution and (F*)(3), one has Ha Noi Metroplolitan University 54 t ‖ x(t ) ‖ ≤ ‖ Sα (t ) ‖ L ( X ) ‖ ϕ (0) ‖ + ∫0 (t − s)α −1 ‖ Pα (t − s ) ‖ L ( X ) m( s ) | xs |B ds t ≤ ‖ Sα (t ) ‖ L ( X ) ‖ ϕ (0) ‖ + ∫ (t − s)α −1 ‖ Pα (t − s ) ‖ L ( X ) m( s ) K ( s ) ‖ x ‖ BC ds t + ∫ (t − s )α −1 ‖ Pα (t − s) ‖ L ( X ) m( s) M ( s) | ϕ |B ds ≤ M ̻ | ϕ |B +Λ ∞ ‖ x ‖ BC +Θ ∞ | ϕ |B This implies (1 − Λ∞ )‖ x‖ BC ≤ ( M ̻ + Θ∞ ) | ϕ |B Now using (B3) with σ = gives | xt |B ≤ K ∞ sup s∈[0,t ] ‖ x( s ) ‖ + M ∞ | ϕ |B ≤ K ∞ ‖ x ‖ BC + M ∞ | ϕ |B  K  ≤  ∞ ( M ̻ + Θ ∞ ) + M ∞  | ϕ |B , ∀t ≥ 1 − Λ ∞  (3.23) Thus the first statement in Definition 1.1 follows In order to check the second one, we employ (B3) with σ = t to get that t | xt |B ≤ K ∞ sup ‖ x( s ) ‖ + M   | x t |B t 2 ≤ s ≤t t ≤ K∞   2 −γ −γ γ  t   t   t  sup s ‖ x ( s) ‖ +     M    | x t |B t        ≤ s ≤t γ −γ  t  ≤   K ∞ sup sγ t 2   ≤ s ≤t  ‖ x( s) ‖ + M ∞ | x t |B    γ If x is the solution obtained by Theorem 3.4, we see that sup s γ t ≤ s ≤t ‖ x ( s ) ‖ ≤ sup s γ ‖ x( s ) ‖< +∞ s ≥0 Hence it follows from (3.23)-(3.24) that | xt |B → as t → +∞ (3.24) Scientific Journal − No27/2018 55 REFERENCES N.T Anh, T.D Ke(2015), “Decay integral solutions for neutral fractional differential equations with infinite delays”, Math Methods Appl Sci., 38, 1601-1622 R.R Akhmerov, M.I Kamenskii, A.S Potapov, A.E Rodkina, B.N Sadovskii (1992), “Measures of Noncompactness and Condensing Operators”, Birkh-@ user, Boston-Basel-Berlin D Bothe(1998), “Multivalued perturbations of m-accretive differential inclusions”, Israel J Math,108, 109-138 4 A Cernea(1998), “On the existence of mild solutions for nonconvex fractional semilinear differential inclusions”, Electron J Qual Theory Differ Equ., No 64, 1-15 R.F Curtain, A Pritchard (1978), “Infinite Dimensional Linear Systems Theory”, Lecture Notes in Control and Information Sciences, Springer Verlag I.Ekeland, R.Temam (1999), “Convex analysis and variational problems”,Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, K.J Engel, R Nagel (2000), “One-parameter semigroups for linear evolution equations”, Graduate Texts in Mathematics, vol.194, Springer-Verlag, New York A F Filippov (1988), “Differential equations with discontinuous righthand sides”,Translated from the Russian Mathematics and its Applications (Soviet Series), Kluwer Academic Publishers Group, Dordrecht C Gori, V Obukhovskii, M Ragni, P Rubbioni(2002), “Existence and continuous dependence results for semilinear functional differential inclusions with infinite delay”, Nonlinear Anal 51, 765-782 10 J Hale, J Kato(1978), “Phase space for retarded equations with infinite dealy”, Funkcial Ekvac., 21,11-41 11 R Hilfer (edited, 2000), “Applications of fractional calculus in physics”,World Scientific Publishing Co., Inc., River Edge, NJ, ISBN: 981-02-3457-0 12 Y Hino, S Murukami, T Naito (1991), “Functional Differential Equations with Unbounded Delay”, in: Lecture Notes in Mathematics, vol 1473, Springer-Verlag, Berlin 13 M Kamenskii, V Obukhovskii, and P Zecca (2001), “Condensing multivalued maps and semilinear differential inclusions in Banac spaces”, de Gruyter Series in Nonlinear Analysis and Applications, vol 7, Walter de Gruyter, Berlin, New York 14 T Kato (1995), “Perturbation theory for linear operators”,Reprint of the 1980 edition Classics in Mathematics, Springer-Verlag, Berlin 15 A.A Kilbas, H.M Srivastava, and J.J Trujillo (2006), “Theory and Applications of Fractional Differential Equations”, North-Holland Mathematics Studies, Vol 204, Elsevier Science B.V., Amsterdam 16 X Liu, Z Liu(2013), “Existence results for fractional semilinear differential inclusions in Banach spaces”, J Appl Math Comput 42, 171-182 17 A Ouahab(2012), “Fractional semilinear differential inclusions”, Comput Math Appl 64, 3235-3252 56 Ha Noi Metroplolitan University 18 K.S Miller and B Ross (1993), “An Introduction to the Fractional Calculus and Fractional Differential Equations”, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York 19 J Sabatier, O P Agrawal and J A Tenreiro Machado (edited, 2007), “Advances in fractional calculus”,Theoretical developments and applications in physics and engineering Springer, Dordrecht 20 R Sakthivel, R Ganesh, S.M Anthoni(2013), “Approximate controllability of fractional nonlinear differential inclusions”, Appl Math Comput.,225, 708-717 21 R Sakthivel, Yong Ren, N.I Mahmudov, On the approximate controllability of semilinear fractional differential systems, Comput Math Appl 62, 1451-1459 22 D Shantanu(2011), “Functional fractional calculus”,Second edition Springer-Verlag, Berlin 23 I Podlubny (1999), “Fractional Differential Equations An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications”, Academic Press 24 R Triggiani, On the stabilizability problem in Banach space, J Math Anal Appl 52 (1975) 383-403 25 J Wang, Y Zhou(2011), “Existence and controllability results for fractional semilinear differential inclusions”, Nonlinear Anal.: Real World Applications, 12, 3642-3653 26 Y Zhou, F Jiao(2010), “Existence of mild solutions for fractional neutral evolution equations”, Comput Math Appl., 59, 1063-1077 SỰ ỔN ĐỊNH HÓA YẾU ĐỐI VỚI BAO HÀM THỨC VI PHÂN PHÂN THỨ VỚI TRỄ VƠ HẠN Tóm tắ tắt: Trong báo chúng tơi phân tích tính ổn định hóa nghiệm dừng hệ điều khiển xác định bao hàm thức vi phân phân thứ nửa tuyến tính có trễ vơ hạn Nếu phần tuyến tính hệ ổn định hóa hệ ổn định hóa theo nghĩa yếu Từ khóa: khóa Bao hàm thức vi phân, trễ vơ hạn, ổn định yếu, độ đo không compăc  ... dependence results for semilinear functional differential inclusions with infinite delay”, Nonlinear Anal 51, 765-782 10 J Hale, J Kato(1978), “Phase space for retarded equations with infinite dealy”,... No27/2018 55 REFERENCES N.T Anh, T.D Ke(2015), “Decay integral solutions for neutral fractional differential equations with infinite delays , Math Methods Appl Sci., 38, 1601-1622 R.R Akhmerov, M.I Kamenskii,... perturbations of m-accretive differential inclusions , Israel J Math,108, 109-138 4 A Cernea(1998), “On the existence of mild solutions for nonconvex fractional semilinear differential inclusions , Electron