Controllability of Nonlinear Neutral Stochastic Differential Inclusions with Infinite Delay

Abstract

The paper is concerned with the controllability of nonlinear neutral stochastic differential inclusions with infinite delay in a Hilbert space. Sufficient conditions for the controllability are obtained by using a fixed-point theorem for condensing maps due to O'Regan.

1. Introduction

Let 𝐾 be a separable Hilbert space, and let (Ω,𝔉,𝔉_𝑡,𝑃) be a complete probability space furnished with a complete family of right continuous increasing 𝜎 algebras {𝔉_𝑡} satisfying 𝔉_𝑡⊂𝔉 for 𝑡≥0. Suppose 𝑤(𝑡) is a given 𝐾-valued, 𝔉_𝑡 adapted Brownian motion with a finite trace nuclear covariance operator 𝑄≥0. We are interested in the controllability of nonlinear stochastic differential inclusions with infinite delay 𝑑𝑥(𝑡)−𝑓𝑡,𝑥_𝑡∈𝐴𝑥(𝑡)𝑑𝑡+(𝐵𝑢)(𝑡)𝑑𝑡+𝐺𝑡,𝑥_𝑡(𝑡)[]𝑥𝑑𝑤,𝑡∈𝐽=0,𝑏,𝑡>0,₀=𝜙∈𝔅_ℎ,𝑡∈𝐽₀=(−∞,0], (1) where 𝜙 is 𝔉₀ measurable and 𝐴 is the infinitesimal generator of an analytic semigroup of bounded linear operator 𝑇(𝑡), 𝑡≥0, the state 𝑥(⋅) takes values in Hilbert space 𝐻 with the norm |⋅|, 𝐺∶𝐽×𝔅_ℎ→2^{𝐿(𝐾,𝐻)} is a bounded closed, convex-valued multivalued map, 𝑓∶𝐽×𝔅_ℎ→𝐻 is continuous, and the histories 𝑥_𝑡∶(−∞,0]→𝐻, 𝑥_𝑡(𝜃)=𝑥(𝑡+𝜃), and 𝜃≤0 belong to the space 𝔅_ℎ. For 𝜎₁,𝜎₂∈𝐿(𝐾,𝐻), define ⟨𝜎₁,𝜎₂⟩=tr(𝜎₁𝑄𝜎^∗₂), where 𝜎^∗₂ is the adjoint of the operator 𝜎₂, 𝑄∈𝐿⁺_𝑛(𝐾). 𝐿(𝐾,𝐻) furnished with the scalar product ⟨⋅,⋅⟩ is a pre-Hilbert space. The completion of 𝐿(𝐾,𝐻) with respect to the topology induced by the norm ‖⋅‖₂, where ‖𝜎‖²₂=⟨𝜎,𝜎⟩₂, is a Hilbert space.

At first, we present the abstract phase space 𝔅_ℎ. Assume that ℎ∶(−∞,0]→(0,+∞) is continuous function with 𝑙=∫⁰_−∞ℎ(𝑠)𝑑𝑠<+∞. Define 𝔅_ℎ=𝜑∶(−∞,0]⟶𝐻∶forany𝑎>0,𝐸|𝜑(𝜃)|^21/2[]isaboundedandmeasurablefunctionon−𝑎,0and_0−∞ℎ(𝑠)sup_{𝑠≤𝜃≤0}𝐸|𝜑(𝜃)|^21/2𝑑𝑠<+∞. (2) If 𝔅_ℎ is endowed with the norm 𝜙_𝔅ℎ=⁰_−∞ℎ(𝑠)sup_{𝑠≤𝜃≤0}𝐸|𝜙(𝜃)|^21/2𝑑𝑠,∀𝜙∈𝔅_ℎ, (3) then(𝔅_ℎ,‖⋅‖_𝔅ℎ) is a Banach space.

Controllability is one of the fundamental concepts in mathematical control theory and plays an important role in control systems. The problem of controllability is to show the existence of a control function, which steers the solution of the system from its initial state to a final state, where the initial and final states may vary over the entire space. Controllability generally means that it is possible to steer a dynamical control system from an arbitrary initial state to an arbitrary final state using the set of admissible controls. A standard approach is to transform the controllability problem into a fixed-point problem for an appropriate operator in a functional space. The problem of controllability for functional differential systems has been extensively studied in many papers [1–10]. For example, Sakthivel et al. [5] considered a class of fractional neutral control systems governed by abstract nonlinear fractional neutral differential equations and established a new set of sufficient conditions for the controllability of nonlinear fractional systems by using a fixed-point analysis approach. Using fixed-point technique, fractional calculations, stochastic analysis technique, and methods adopted directly from deterministic control problems, Sakthivel et al. [7] gave a new set of sufficient conditions for approximate controllability of fractional stochastic differential equations. In [8], Sakthivel and Ren investigated the complete controllability property of a nonlinear stochastic control system with jumps in a separable Hilbert space. In [9], Sakthivel et al. studied the approximate controllability for nonlinear dispersion system under the assumption that the corresponding linear control system is approximately controllable.

Benchohra et al. [11] discussed the controllability for one-order, second-order functional differential and integrodifferential inclusions in Banach space with finite delay. Park et al. [12] discussed the controllability for second-order neutral functional differential inclusions in Banach space with the help of some fixed-point theorems. In [13, 14], Liu investigated the controllability of neutral functional differential and integrodifferential inclusions with infinite delay.

History about the research of stochastic differential equations is over fifty years; many important results on the stochastic differential equations have been reported in the literature; see [15, 16]. Recently, stochastic partial differential equations (SPDE) arise in the mathematical modeling of various fields in physics and engineering science cited by Sobczyk [16]. About neutral stochastic differential equations with delay, Mahmudov [17] discussed the existence and uniqueness of neutral stochastic differential equations by using Picard technique. For the controllability of stochastic functional differential inclusions, Balasubramaniam and Ntouyas [18] discussed the controllability of semilinear stochastic delay evolution equations in Hilbert spaces. Balasubramaniam and Dauer [19] obtained the controllability result of stochastic differential inclusions with infinite delay in abstract space. Since many systems arising from realistic models heavily depend on histories (i.e., there is the effect of infinite delay on state equations) [20], there is a real need to discuss partial functional differential systems with infinite delay. So in the present paper, we will concentrate on the case with infinite delay and establish sufficient conditions for the controllability of systems (1) by relying on a fixed-point theorem for condensing maps due to O’Regan [21].

The structure of this paper is as follows. In Section 2 we briefly present some basic notations and preliminaries. The controllability result of system (1) is investigated by means of fixed-point theorem and operator theory in Section 3. Conclusion is given in Section 4.

2. Preliminaries

Let (𝐸,‖⋅‖) be a Banach space. A multivalued map 𝔍∶𝐸→2^𝐸 is convex (closed)-valued, if 𝔍(𝑥) is convex (closed) for all 𝑥∈𝐸. 𝔍 is bounded on bounded set if 𝔍(𝐵)=∪_𝑥∈𝐵𝔍(𝑥) is bounded in 𝐸 for any bounded set 𝐵 of 𝐸; that is, sup_𝑥∈𝐵sup𝑦∈𝔍(𝑥)<∞. (4)

𝔍 is called upper semicontinuous (Usc) on 𝐸 if for each 𝑥_∗∈𝐸, the set 𝔍(𝑥_∗) is nonempty, closed subset of 𝐸 and if for each open set 𝐵 of 𝐸 containing 𝔍(𝑥_∗), there exists an open neighborhood 𝑉 of 𝑥_∗ such that 𝔍(𝑉)⊆𝐵.

𝔍 is said to be completely continuous if 𝔍(𝐵) is relatively compact, for every bounded subset 𝐵⊆𝐸.

If the multivalued map 𝔍 is completely continuous with nonempty compact values, then 𝔍 is Usc if and only if 𝔍 has a closed graph (i.e., 𝑥_𝑛=𝑥_∗, 𝑦_𝑛=𝑦_∗, 𝑦_𝑛∈𝔍𝑥_𝑛 imply 𝑦_∗∈𝔍𝑥_∗).

Let BCC(𝐸) denote the sets of all the sets of all nonempty, bounded, closed, and convex subsets of 𝐸. For more details on multivalued maps see the books of Deimling [22] and Hu and Papageorgiou [23].

An upper semicontinuous map 𝐻∶𝐸→𝐸 is said to be condensing if for any subset 𝐵⊆𝐸 with 𝛼(𝐵)≠0, we have 𝛼(𝐻(𝐵))<𝛼(𝐵), where 𝛼 denotes the Kuratowski measure of noncompactness. It is easy to see that 𝑠 completely continuous multivalued map is a condensing map.

If 𝑇 is a uniformly bounded and analytic semigroup with infinitesimal generator 𝐴 such that 0∈𝜌(𝐴) then it is possible to define the fractional power (−𝐴)^𝛼, for 0<𝛼≤1, as a closed linear operator on its domain 𝐷((−𝐴)^𝛼). Furthermore, the subspace 𝐷((−𝐴)^𝛼) is dense in 𝑋 and the expression ‖𝑥‖_𝛼∶=(−𝐴)^𝛼𝑥,𝑥∈𝐷(−𝐴)^𝛼, (5) defines a norm on 𝐷((−𝐴)^𝛼). Hereafter we represent by 𝑋_𝛼 the space 𝐷((−𝐴)^𝛼) endowed with the norm ‖⋅‖_𝛼. Then the following properties are well known [24].

Lemma 1 (see [24]). Suppose that the preceding conditions are satisfied.(a)Let 0<𝛼≤1. Then 𝑋_𝛼 is a Banach space.(b)If 0<𝛾<𝛼≤1, then 𝑋_𝛼↪𝑋_𝛾 and the imbedding is compact whenever the resolvent operator of 𝐴 is compact.(c)For every 𝑎>0, there exists a positive constant 𝑐_𝛼 such that (−𝐴)^𝛼𝑇(𝑡≤𝑐)^𝛼𝑡^𝛼,0<𝑡≤𝑎. (6) The key tool in our approach is following fixed-point theorem [21].

Theorem 2 (see [21]). Let 𝑋 be a Hilbert space, and let 𝑈 be an open, convex subset of 𝑋 and 𝑢₀∈𝑈. If (a)𝐹∶𝑈→𝒫_𝑐𝑑(𝑋) has closed graph,(b)𝐹∶𝑈→𝒫_𝑐𝑑(𝑋) is condensing operator and 𝐹(𝑈) is bounded set of 𝑋,then either(i)𝐹 has fixed-point in 𝑈; or(ii)there exist 𝑢∈𝜕𝑈 and 𝜆∈(0,1) such that 𝑢∈𝜆𝐹(𝑢)+(1−𝜆){𝑢₀}.

3. Main Result

In the following, we will apply Theorem 2 to study the existence of solutions for system (1).

Definition 3. A function 𝑥∶(−∞,𝑏]→𝐻 is called a mild solution of system (1) if the following holds: let 𝜙 be 𝔉₀ measurable 𝐻-valued stochastic processes 𝑥₀=𝜙∈𝔅_ℎ on (−∞,0] and the integral equation 𝑥(𝑡)=𝑇(𝑡)𝜙(0)−𝑓0,𝜙+𝑓𝑡,𝑥_𝑡+^𝑡₀𝐴𝑇(𝑡−𝑠)𝑓𝑠,𝑥_𝑠𝑑𝑠+^𝑡₀𝑇(𝑠)+𝑡−𝑠)(𝐵𝑢)(𝑑𝑠_𝑡0𝑇()𝑔(𝑠)(𝑠)𝑥𝑡−𝑠𝑑𝑤,fora.e.𝑡∈𝐽,₀=𝜙∈𝔅_ℎ,𝑡∈𝐽₀ (7) is satisfied, where 𝑔∈𝑆_𝐺,𝑥=𝑔∈𝐿²(𝐽,𝐻)∶𝑔(𝑡)∈𝐺𝑡,𝑥_𝑡,fora.e.𝑡∈𝐽. (8)

Definition 4. System (1) is said to be controllable on the interval 𝐽 if for every continuous initial random process 𝜙∈𝔅_ℎ, 𝑥₁∈𝐻, there exists a control 𝑢∈𝐿²(𝐽,𝑈) such that the mild solution 𝑥(𝑡) of (1) satisfies 𝑥(𝑏)=𝑥₁.
To investigate the controllability of system (1), we use the following hypotheses.(𝐻₁)𝐴 is the infinitesimal generator of an analytic semigroup of bounded linear operators 𝑇(𝑡) in 𝐻, 0∈𝜌(𝐴), for 𝑡>0; there exist constants 𝑀₁ such that |𝑇(𝑡)|≤𝑀₁.(𝐻₂) The linear operator 𝑊∶𝐿²(𝐽,𝑈)→𝐻 defined by 𝑊𝑢=^𝑏₀𝑇(𝑏−𝑠)𝐵𝑢(𝑠)𝑑𝑠 (9) has an induced inverse operator 𝑊⁻¹ which takes values in 𝐿²(𝐽,𝑈)/ker𝑊 and there exist positive constants 𝑀₃,𝑀₄ such that |𝐵|≤𝑀₃ and |𝑊⁻¹|≤𝑀₄.(𝐻₃) There exist constants 0≤𝑐₁<1,1/2<𝛽<1, and 𝑐₂≥0 such that 𝑙²𝑐₁‖(−𝐴)^−𝛽‖²<1/5 and 𝐸(−𝐴)^𝛽𝑓(𝑡,𝑢)²≤𝑐₁‖𝑢‖²_𝔅ℎ+𝑐₂,𝑡∈𝐽,𝑢∈𝔅_ℎ, (10) where 𝑙=∫⁰_−∞ℎ(𝑠)𝑑𝑠<+∞. (𝐻₄)𝐺∶𝐽×𝔅_ℎ→BCC(𝐻); (𝑡,𝜙)→𝐺(𝑡,𝜙) is measurable with respect to 𝑡 for each 𝜙∈𝔅_ℎ, Usc with respect to 𝜙 for each 𝑡∈𝐽, and for each fixed 𝜙∈𝔅_ℎ, the set 𝑆_𝐺,𝑥=𝑔∈𝐿²(𝐽,𝐻)∶𝑔(𝑡)∈𝐺𝑡,𝑥_𝑡,fora.e.𝑡∈𝐽 (11) is nonempty. (𝐻₅)𝐸‖𝐺(𝑡,𝜙)‖²₂=sup{𝐸‖𝑣‖²₂∶𝑣∈𝐺(𝑡,𝜙)}≤𝑝(𝑡)𝜓(‖𝜙‖_𝔅ℎ)², 𝑡∈𝐽, 𝑝∈𝐿²(𝐽,𝑅⁺), 𝜙∈𝔅_ℎ, and 𝜓∶[0,+∞)→(0,+∞) is a continuous nondecreasing function. There exists a positive constant 𝑟 such that 𝑟𝐹^∗+𝑀𝑟+𝑘₃𝜓3𝑙²𝑟+3𝑞^∗∫^𝑏₀𝑝(𝑠)𝑑𝑠>1, (12) where 𝐹=5𝑀²₁(−𝐴)^−𝛽2𝑐₁𝜙²_𝔅ℎ+𝑐₂+5(−𝐴)^−𝛽2𝑐₂+5𝑏^2𝛽𝑐₂𝑐²_1−𝛽2𝛽−1+25𝑏²𝑀²₁𝑀²₃𝑀²₄×𝐸𝑥₁²+2𝑀²₁𝐸𝜙(0)²+2𝑀²₁(−𝐴)^−𝛽2×𝑐₁𝜙²_𝔅ℎ+𝑐₂+(−𝐴)^−𝛽2𝑐₂+𝑏^2𝛽−1𝑐₂𝑐²_1−𝛽,𝑞2𝛽−1_∗=𝜙²_𝔅ℎ+𝑙²𝑀²₁𝐸𝜙(0)²,𝐹^∗=𝐹+𝑘₁𝑞^∗+𝑘₂𝑏^2𝛽−1𝑞2𝛽−1^∗,𝑀=𝑘₁𝑙²+𝑘₂𝑏^2𝛽−1𝑙²,𝑘2𝛽−1₁=15(−𝐴)^−𝛽2𝑐₁+75𝑏²𝑀²₁𝑀²₃𝑀²₄(−𝐴)^−𝛽2𝑐₁,𝑘₂=15𝑏𝑐₁𝑐²_1−𝛽+75𝑏²𝑀²₁𝑀²₃𝑀²₄,𝑘₃=5𝑀²₁+25𝑏²𝑀₁𝑀₂𝑀₃𝑀₄². (13)(𝐻₆) The operator 𝐹 with values (𝐹(𝑥))(𝑡)=𝑓(𝑡,𝑥_𝑡), 𝑡∈𝐽, 𝐹 is completely continuous in 𝐶(𝐽,𝐻) and for any bounded set 𝑉⊆𝐶(𝐽,𝐻), the set {𝑡→𝑓(𝑡,𝑥_𝑡)∶𝑥∈𝑉} is equicontinuous in 𝐶(𝐽,𝐻). (𝐻₇) There exist constants 1/2<𝛽<1 and 𝑞>0 such that (−𝐴)^𝛽𝑓 is continuous and for 𝑡∈𝐽, 𝜙₁,  𝜙₂∈𝔅_ℎ such that |(−𝐴)^𝛽𝑓(𝑡,𝜙₁)−(−𝐴)^𝛽𝑓(𝑡,𝜙₂)|≤𝑞‖𝜙₁−𝜙₂‖_𝔅ℎ,  0<𝑄=2‖(−𝐴)^−𝛽‖²𝑞²+2𝑐²_1−𝛽(1/((2𝛽−1)))𝑏^2𝛽−1𝑞²𝑙²<1.

Lemma 5 (Lasota and Opial [25]). Let 𝐼 be a compact real interval and let 𝐸 be a Banach space. Let 𝐺 be a multivalued map satisfying (𝐻₄) and let Γ be a linear continuous mapping from 𝐿²(𝐼,𝐸) to 𝐶(𝐼,𝐸). Then the operator Γ∘𝑆_𝐺∶𝐶(𝐼,𝐸)⟶BCC(𝐶(𝐼,𝐸)),𝑥⟶Γ∘𝑆_𝐺(𝑥)=Γ𝑆_𝐺,𝑥 (14) is a closed graph operator in 𝐶(𝐼,𝐸)×𝐶(𝐼,𝐸).

Lemma 6 (see [26]). Let 𝑣(⋅),𝑟(⋅)∶[0,𝑇]→[0,∞) be continuous functions. If 𝑟(⋅) is nondecreasing and there are constants 𝜃>0, 0<𝜏<1 such that 𝑣(𝑡)≤𝑟(𝑡)+𝜃^𝑡₀𝑣(𝑠)(𝑡−𝑠)^1−𝜏𝑑𝑠,𝑡∈[0,𝑇], (15) then 𝑣(𝑡)≤𝑒^{𝜃𝑛Γ(𝜏)𝑛𝑡𝑛𝜏/Γ(𝑛𝜏)𝑛−1}_𝑗=0𝜃𝑇^𝜏𝜏^𝑗𝑟(𝑡), (16) for every 𝑡∈[0,𝑇] and every 𝑛∈𝑁 such that 𝑛𝜏>1, and Γ(⋅) is the Gamma function.

Now we consider the space 𝔅_𝑏; let 𝔅_𝑏 be the space of all 𝐻-valued stochastic processes 𝑥(𝑡), −∞<𝑡≤𝑏, such that 𝑥₀=𝜙∈𝔅_ℎ, let ‖⋅‖ be a seminorm in 𝔅_𝑏 defined by ‖𝑥‖_𝑏=𝑥_0𝔅ℎ+sup_{𝑠∈[0,𝑏]}𝐸|𝑥(𝑠)|^21/2,𝑥∈𝔅_𝑏. (17)

Lemma 7 (see [27]). Suppose 𝑥∈𝔅_𝑏; then for 𝑡∈𝐽, 𝑥_𝑡∈𝔅_ℎ. Moreover 𝑙𝐸^1/2|𝑥(𝑡)|²≤𝑥_𝑡𝔅ℎ≤𝑙sup_{[]𝑠∈0,𝑡}𝐸|𝑥(𝑠)|^21/2+𝑥_0𝔅ℎ, (18) where 𝑙=∫⁰_−∞ℎ(𝑠)𝑑𝑠<+∞.

Now, consider the multivalued map 𝔏∶𝔅_𝑏→2^𝔅𝑏 defined by 𝔏𝑥 the set of 𝜌∈𝔅_𝑏 such that 𝜌(𝑡)=𝜙(𝑡),(],𝑇(𝑡𝜙(0)𝑡∈−∞,0)−𝑓0,𝜙+𝑓𝑡,𝑥_𝑡+∫^𝑡₀𝐴𝑇(𝑡−𝑠)𝑓𝑠,𝑥_𝑠𝑑𝑠+∫^𝑡₀𝑇(𝑡−𝑠)𝑔(𝑠)𝑑𝑤(𝑠)+∫^𝑡₀𝑇𝑡−𝜂𝐵𝑊⁻¹×𝑥₁−𝑇(𝑏)𝜙(0)−𝑓0,𝜙−𝑓𝑏,𝑥_𝑏−∫^𝑏₀𝐴𝑇(𝑏−𝑠)𝑓𝑠,𝑥_𝑠𝑑𝑠−∫^𝑏₀𝑇(𝑏−𝑠)𝑔(𝑠)𝑑𝑤(𝑠)𝜂𝑑𝜂,𝑡∈𝐽, (19) where 𝑔∈𝑆_𝐺,𝑥.

We will show that the operator 𝔏 has fixed-points, which are then a solution of system (1). For 𝜙∈𝔅_ℎ, we define 𝜙 by 𝜙(𝑡)=𝜙(𝑡)𝑇(𝑡)𝜙(0),−∞<𝑡≤0,,0≤𝑡≤𝑏, (20) then 𝜙∈𝔅_𝑏. Set 𝑥(𝑡)=𝑦(𝑡)+𝜙(𝑡),−∞<𝑡≤𝑏. (21) It is clear to see that 𝑥 satisfies (7) if and only if 𝑦 satisfies 𝑦₀=0 and 𝑦(𝑡)=−𝑇(𝑡)𝑓0,𝜙+𝑓𝑡,𝑦_𝑡+𝜙_𝑡+^𝑡₀𝐴𝑇(𝑡−𝑠)𝑓𝑠,𝑦_𝑠+𝜙_𝑠+𝑑𝑠_𝑡0𝑇(𝑡−𝑠)𝑔(𝑠)𝑑𝑤(𝑠)+^𝑡₀𝑇𝑡−𝜂𝐵𝑊⁻¹×𝑥₁−𝑇(𝑏)𝜙(0)−𝑓0,𝜙−𝑓𝑏,𝑦_𝑏+𝜙_𝑏−^𝑏₀𝐴𝑇(𝑏−𝑠)𝑓𝑠,𝑦_𝑠+𝜙_𝑠−𝑑𝑠_𝑏0𝑇(𝑏−𝑠)𝑔(𝑠)𝑑𝑤(𝑠)𝜂𝑑𝜂,𝑡∈𝐽. (22) Let 𝔅⁰_𝑏={𝑦∈𝔅_𝑏∶𝑦₀=0∈𝔅_ℎ}. For any 𝑦∈𝔅⁰_𝑏, 𝑦_𝑏=𝑦_0𝔅ℎ+sup_{[]𝑠∈0,𝑏}𝐸^1/2𝑦(𝑠)²=sup_{[]𝑠∈0,𝑏}𝐸^1/2𝑦(𝑠)². (23) Thus (𝔅⁰_𝑏,‖⋅‖_𝑏) is a Banach space. Set 𝔅_𝑞={𝑦∈𝔅⁰_𝑏∶‖𝑦‖_𝑏≤𝑞} for some 𝑞≥0; then 𝔅_𝑞⊆𝔅⁰_𝑏 is uniformly bounded, for any 𝑦∈𝔅_𝑞; from Lemma 7, we have 𝑦_𝑡+𝜙_𝑡𝔅ℎ≤𝑦_𝑡𝔅ℎ+𝜙_𝑡𝔅ℎ≤𝑙sup_{0≤𝑠≤𝑡}𝐸^1/2𝑦(𝑠)²+𝑦_0𝔅ℎ+𝑙sup_{0≤𝑠≤𝑡}𝐸^1/2𝜙(𝑠)²+𝜙_0𝔅ℎ≤𝑙𝑞+𝜙_𝔅ℎ+𝑙sup_{0≤𝑠≤𝑡}|𝑇(𝑠)|𝐸^1/2𝜙(0)²≤𝑙𝑞+𝑀₁𝐸^1/2𝜙(0)²+𝜙_𝔅ℎ=𝑞^′. (24) Define the multivalued map 𝔏₁∶𝔅⁰_𝑏→2^𝔅0𝑏 defined by 𝔏₁𝑦 the set of 𝜌∈𝔅⁰_𝑏 such that 𝜌(𝑡)=(],(𝑡)𝑓0,𝑡∈−∞,0−𝑇0,𝜙+𝑓𝑡,𝑦_𝑡+𝜙_𝑡+∫^𝑡₀𝐴𝑇(𝑡−𝑠)𝑓𝑠,𝑦_𝑠+𝜙_𝑠𝑑𝑠+∫^𝑡₀𝑇(𝑡−𝑠)𝑔(𝑠)𝑑𝑤(𝑠)+∫^𝑡₀𝑇𝑡−𝜂𝐵𝑊⁻¹×𝑥₁−𝑇(𝑏)𝜙(0)−𝑓0,𝜙−𝑓𝑏,𝑦_𝑏+𝜙_𝑏−∫^𝑏₀𝐴𝑇(𝑏−𝑠)𝑓𝑠,𝑦_𝑠+𝜙_𝑠𝑑𝑠−∫^𝑏₀𝑇(𝑏−𝑠)𝑔(𝑠)𝑑𝑤(𝑠)𝜂𝑑𝜂,𝑡∈𝐽. (25)

Theorem 8. Assume that hypotheses (𝐻₁)−−(𝐻₇) hold; then system (1) is controllable on 𝐽.

Proof. We divide the proof into several steps.
Step 1. We show that there exists an open set 𝑈 and there does not exist 𝑦∈𝜕𝑈, for some 𝜆∈(0,1), such that 𝑦∈𝜆𝔏₁(𝑦).
Let 𝜆∈(0,1) and let 𝑦∈𝜆𝔏₁𝑦. Then there exists 𝑔∈𝑆_𝐺,𝑦 such that 𝑦(𝑡)=−𝜆𝑇(𝑡)𝑓0,𝜙+𝜆𝑓𝑡,𝑦_𝑡+𝜙_𝑡+𝜆^𝑡₀𝐴𝑇(𝑡−𝑠)𝑓𝑠,𝑦_𝑠+𝜙_𝑠𝑑𝑠+𝜆^𝑡₀𝑇(𝑡−𝑠)𝑔(𝑠)𝑑𝑤(𝑠)+𝜆^𝑡₀𝑇𝑡−𝜂𝐵𝑊⁻¹×𝑥₁−𝑇(𝑏)𝜙(0)−𝑓0,𝜙−𝑓𝑏,𝑦_𝑏+𝜙_𝑏−^𝑏₀𝐴𝑇(𝑏−𝑠)𝑓𝑠,𝑦_𝑠+𝜙_𝑠−𝑑𝑠_𝑏0𝑇()𝑔(𝑠)(𝑠𝜂𝐸𝑦(𝑡𝑏−𝑠𝑑𝑤)𝑑𝜂,𝑡∈𝐽.)²≤5𝑀²₁𝐸𝑓0,𝜙²+5𝐸𝑓𝑡,𝑦_𝑡+𝜙_𝑡²+5𝑏^𝑡₀𝑐_1−𝛽(𝑡−𝑠)^1−𝛽2𝑐₁𝑦_𝑠+𝜙_𝑠²_𝔅ℎ+𝑐₂𝑑𝑠+5𝑀²₁^𝑡₀𝑝(𝑠)𝜓𝑦_𝑠+𝜙_𝑠²_𝔅ℎ𝑑𝑠+25𝑏²𝑀²₁𝑀²₃𝑀²₄×𝐸𝑥₁²+2𝑀²₁𝐸𝜙(0)²+2𝑀²₁(−𝐴)^−𝛽2𝑐₁𝜙²_𝔅ℎ+𝑐₂+(−𝐴)^−𝛽2𝑐₁𝑦_𝑡+𝜙_𝑡²_𝔅ℎ+𝑐₂+^𝑏₀𝑐_1−𝛽(𝑏−𝑠)^1−𝛽2𝑐₁𝑦_𝑠+𝜙_𝑠²_𝔅ℎ+𝑐₂𝑑𝑠+𝑀²₁^𝑏₀𝑝(𝑠)𝜓𝑦_𝑠+𝜙_𝑠²_𝔅ℎ𝑑𝑠≤5𝑀²₁(−𝐴)^−𝛽2𝑐₁𝜙²_𝔅ℎ+𝑐₂+5(−𝐴)^−𝛽2𝑐₁𝑦_𝑡+𝜙_𝑡²_𝔅ℎ+𝑐₂+5𝑏^2𝛽𝑐₂𝑐²_1−𝛽2𝛽−1+5𝑏𝑐₁𝑐²_1−𝛽^𝑡₀𝑦_𝑠+𝜙_𝑠²_𝔅ℎ(𝑡−𝑠)^2−2𝛽𝑑𝑠+5𝑀²₁^𝑡₀𝑝(𝑠)𝜓𝑦_𝑠+𝜙_𝑠²_𝔅ℎ𝑑𝑠+25𝑏²𝑀²₁𝑀²₃𝑀²₄×𝐸𝑥₁²+2𝑀²₁𝐸𝜙(0)²+2𝑀²₁(−𝐴)^−𝛽2𝑐₁𝜙²_𝔅ℎ+𝑐₂+(−𝐴)^−𝛽2𝑐₁𝑦_𝑡+𝜙_𝑡²_𝔅ℎ+𝑐₂+𝑏^2𝛽−1𝑐₂𝑐²_1−𝛽2𝛽−1+𝑐₁𝑐²_1−𝛽^𝑏₀𝑦_𝑠+𝜙_𝑠²_𝔅ℎ(𝑏−𝑠)^2−2𝛽𝑑𝑠+𝑀²₁^𝑏₀𝑝(𝑠)𝜓𝑦_𝑠+𝜙_𝑠²_𝔅ℎ𝑑𝑠. (26)
Since 𝑦_𝑡+𝜙_𝑡𝔅ℎ≤𝑙sup_{0≤𝑠≤𝑡}𝐸^1/2𝑦(𝑠)²+𝜙_𝔅ℎ+𝑙𝑀₁𝐸^1/2𝜙(0)²,𝑦_𝑏=sup_{[]𝑡∈0,𝑏}𝐸𝑦(𝑡)^21/2≤5𝑀²₁(−𝐴)^−𝛽2𝑐₁𝜙²_𝔅ℎ+𝑐₂+5(−𝐴)^−𝛽2𝑐₂+5𝑏^2𝛽𝑐₂𝑐²_1−𝛽2𝛽−1+25𝑏²𝑀²₁𝑀²₃𝑀²₄×𝐸𝑥₁²+2𝑀²₁𝐸𝜙(0)²+2𝑀²₁(−𝐴)^−𝛽2𝑐₁𝜙²_𝔅ℎ+𝑐₂+(−𝐴)^−𝛽2𝑐₂+𝑏^2𝛽−1𝑐₂𝑐²_1−𝛽+)2𝛽−115(−𝐴^−𝛽2𝑐₁+75𝑏²𝑀²₁𝑀²₃𝑀²₄(−𝐴)^−𝛽2𝑐₁×𝑙²sup_{0≤𝑠≤𝑏}𝐸𝑦(𝑠)²+𝜙²_𝔅ℎ+𝑙²𝑀²₁𝐸𝜙(0)²+15𝑏𝑐₁𝑐²_1−𝛽+75𝑏²𝑀²₁𝑀²₃𝑀²₄×^𝑏₀𝑙²sup_{0≤𝑠≤𝑏}𝐸𝑦(𝑠)²+𝜙²_𝔅ℎ+𝑙²𝑀²₁𝐸𝜙(0)²(𝑏−𝑠)^2−2𝛽+5𝑀²₁+25𝑏²𝑀₁𝑀₂𝑀₃𝑀₄²×^𝑏₀𝑝(𝑠)𝜓3𝑙²sup_{0≤𝑠≤𝑏}𝐸𝑦(𝑠)²+3𝜙²_𝔅ℎ+3𝑙²𝑀²₁𝐸𝜙(0)²𝑑𝑠^1/2=𝐹+𝑘₁𝑙²sup_{0≤𝑠≤𝑏}𝐸𝑦(𝑠)²+𝑞^∗+𝑘₂^𝑏₀𝑙²sup_{0≤𝑠≤𝑏}𝐸𝑦(𝑠)²+𝑞^∗(𝑏−𝑠)^2−2𝛽𝑑𝑠+𝑘₃^𝑏₀𝑝(𝑠)𝜓3𝑙²sup_{0≤𝑠≤𝑏}𝐸𝑦(𝑠)²+3𝑞^∗𝑑𝑠^1/2, (27) where 𝐹,𝑘₁,𝑘₂,𝑘₃, and 𝑞^∗ are defined in 𝐻₅.
Consequently 𝑦²_𝑏≤𝐹+𝑘₁𝑞^∗+𝑘₁𝑙²𝑦²_𝑏+𝑘₂𝑏^2𝛽−1𝑙2𝛽−1²𝑦²_𝑏+𝑞^∗+𝑘₃𝜓3𝑙²𝑦²_𝑏+3𝑞^∗𝑏₀𝑝(𝑠)𝑑𝑠=𝐹^∗+𝑀𝑦²_𝑏+𝑘₃𝜓3𝑙²𝑦²_𝑏+3𝑞^∗𝑏₀𝑝(𝑠)𝑑𝑠. (28)
So 𝑦²_𝑏𝐹^∗+𝑀𝑦²_𝑏+𝑘₃𝜓3𝑙²𝑦²_𝑏+3𝑞^∗∫^𝑏₀𝑝(𝑠)𝑑𝑠≤1. (29) Then by 𝐻₅ there exists 𝑟 such that ‖𝑦‖²_𝑏≠𝑟; that is, ‖𝑦‖_𝑏≠𝑟^1/2. Define the set: 𝑈={𝑧∈ℬ⁰_𝑏∶‖𝑧‖_𝑏<𝑟^1/2}. From the choice of 𝑈, we know that there does not exist 𝑦∈𝜕𝑈, for some 𝜆∈(0,1), such that 𝑦∈𝜆𝔏₁(𝑦).
Step 2.  𝔏₁𝑦 is convex for each 𝑦∈𝔅⁰_𝑏.
In fact, if 𝜌₁, 𝜌₂ belong to 𝔏₁𝑦, then there exist 𝑔₁, 𝑔₂∈𝑆_𝐺,𝑦 such that for each 𝑡∈𝐽 we have 𝜌_𝑖(𝑡)=−𝑇(𝑡)𝑓0,𝜙+𝑓𝑡,𝑦_𝑡+𝜙_𝑡+^𝑡₀𝐴𝑇(𝑡−𝑠)𝑓𝑠,𝑦_𝑠+𝜙_𝑠+𝑑𝑠_𝑡0𝑇(𝑡−𝑠)𝑔_𝑖(𝑠)(𝑠)+𝑑𝑤_𝑡0𝑇𝑡−𝜂𝐵𝑊⁻¹×𝑥₁−𝑇(𝑏)𝜙(0)−𝑓0,𝜙−𝑓𝑏,𝑦_𝑏+𝜙_𝑏−^𝑏₀𝐴𝑇(𝑏−𝑠)𝑓𝑠,𝑦_𝑠+𝜙_𝑠−𝑑𝑠_𝑏0𝑇(𝑏−𝑠)𝑔_𝑖(𝑠)𝑑𝑤(𝑠)𝜂𝑑𝜂,𝑡∈𝐽. (30) Let 𝜆∈[0,1]; we have 𝜆𝜌₁+(1−𝜆)𝜌₂(𝑡)=−𝑇(𝑡)𝑓0,𝜙+𝑓𝑡,𝑦_𝑡+𝜙_𝑡+^𝑡₀𝐴𝑇(𝑡−𝑠)𝑓𝑠,𝑦_𝑠+𝜙_𝑠+𝑑𝑠_𝑡0𝑇(𝑡−𝑠)𝜆𝑔₁(𝑠)+(1−𝜆)𝑔₂(𝑠)𝑑𝑤(𝑠)+^𝑡₀𝑇𝑡−𝜂𝐵𝑊⁻¹×𝑥₁−𝑇(𝑏)𝜙(0)−𝑓0,𝜙−𝑓𝑏,𝑦_𝑏+𝜙_𝑏−^𝑏₀𝐴𝑇(𝑏−𝑠)𝑓𝑠,𝑦_𝑠+𝜙_𝑠−𝑑𝑠_𝑏0𝑇(𝑏−𝑠)𝜆𝑔₁(𝑠)+(1−𝜆)𝑔₂(𝑠)𝑑𝑤(𝑠)𝜂𝑑𝜂. (31) Since 𝑆_𝐺,𝑦 is convex, we have 𝜆𝜌₁+(1−𝜆)𝜌₂∈𝔏₁𝑦.
Step 3. 𝔏₁ maps bounded set into bounded set in 𝔅⁰_𝑏. Indeed, it is enough to show that there exists a positive constant Λ such that for each 𝜌∈𝔏₁𝑦, 𝑦∈𝔅_𝑞={𝑦∈𝔅⁰_𝑏∶‖𝑦‖_𝑏≤𝑞} one has ‖𝜌‖_𝑏≤Λ. If 𝜌∈𝔏₁𝑦, then there exist 𝑔∈𝑆_𝐺,𝑦, such that for each 𝑡∈𝐽, 𝜌(𝑡)=−𝑇(𝑡)𝑓0,𝜙+𝑓𝑡,𝑦_𝑡+𝜙_𝑡+^𝑡₀𝐴𝑇(𝑡−𝑠)𝑓𝑠,𝑦_𝑠+𝜙_𝑠+𝑑𝑠_𝑡0𝑇()𝑔(𝑠)(𝑠)+𝑡−𝑠𝑑𝑤_𝑡0𝑇𝑡−𝜂𝐵𝑊⁻¹×𝑥₁−𝑇(𝑏)𝜙(0)−𝑓0,𝜙−𝑓𝑏,𝑥_𝑏−^𝑏₀𝐴𝑇(𝑏−𝑠)𝑓𝑠,𝑦_𝑠+𝜙_𝑠−𝑑𝑠_𝑏0𝑇(𝑏−𝑠)𝑔(𝑠)𝑑𝑤(𝑠)𝜂𝑑𝜂. (32) So, from Lemma 1(c) for 𝑡∈𝐽𝐸𝜌(𝑡)²=𝐸−𝑇(𝑡)𝑓0,𝜙+𝑓𝑡,𝑦_𝑡+𝜙_𝑡+^𝑡₀𝐴𝑇(𝑡−𝑠)𝑓𝑠,𝑦_𝑠+𝜙_𝑠+𝑑𝑠_𝑡0𝑇()𝑔(𝑠)(𝑠)+𝑡−𝑠𝑑𝑤_𝑡0𝑇𝑡−𝜂𝐵𝑊⁻¹×𝑥₁−𝑇(𝑏)𝜙(0)−𝑓0,𝜙−𝑓𝑏,𝑥_𝑏−^𝑏₀𝐴𝑇(𝑏−𝑠)𝑓𝑠,𝑦−𝑠+𝜙_𝑠−𝑑𝑠_𝑏0𝑇(𝑏−𝑠)𝑔(𝑠)𝑑𝑤(𝑠)𝜂𝑑𝜂²≤5𝐸−𝑇(𝑡)𝑓0,𝜙²+5𝐸𝑓𝑡,𝑦_𝑡+𝜙_𝑡²+5𝐸^𝑡₀𝐴𝑇(𝑡−𝑠)𝑓𝑠,𝑦_𝑠+𝜙_𝑠𝑑𝑠²+5𝐸^𝑡₀𝑇(𝑡−𝑠)𝑔(𝑠)𝑑𝑤(𝑠)²₂+5𝐸^𝑡₀𝑇𝑡−𝜂𝐵𝑊⁻¹×𝑥₁−𝑇(𝑏)𝜙(0)−𝑓0,𝜙−𝑓𝑏,𝑥_𝑏−^𝑏₀𝐴𝑇(𝑏−𝑠)𝑓𝑠,𝑦_𝑠+𝜙_𝑠−𝑑𝑠_𝑏0𝑇(𝑏−𝑠)𝑔(𝑠)𝑑𝑤(𝑠)𝜂𝑑𝜂²≤5𝑀²₁(−𝐴)^−𝛽2𝐸(−𝐴)^𝛽𝑓0,𝜙²+5(−𝐴)^−𝛽2𝐸(−𝐴)^𝛽𝑓𝑡,𝑦_𝑡+𝜙_𝑡²+5𝐸^𝑡₀(−𝐴)^1−𝛽𝑇(𝑡−𝑠)(−𝐴)^𝛽𝑓𝑠,𝑦−𝑠+𝜙_𝑠𝑑𝑠²+5^𝑡₀𝐸𝑇(𝑡−𝑠)𝑔(𝑠)²₂𝑑𝑠+5𝑏^𝑡₀𝐸𝑇𝑡−𝜂𝐵𝑊⁻¹×𝑥₁−𝑇(𝑏)𝜙(0)−𝑓0,𝜙−𝑓𝑏,𝑦_𝑏+𝜙_𝑏−^𝑏₀𝐴𝑇(𝑏−𝑠)𝑓𝑠,𝑦_𝑠+𝜙_𝑠−𝑑𝑠_𝑏0𝑇(𝑏−𝑠)𝑔(𝑠)𝑑𝑤(𝑠)²𝜂𝑑𝜂≤5𝑀²₁(−𝐴)^−𝛽2𝑐₁𝜙²+𝑐₂+5(−𝐴)^−𝛽2𝑐₁𝑞^′2+𝑐₂+5^𝑡₀(−𝐴)^1−𝛽𝑇(𝑡−𝑠)²𝑑𝑠^𝑡₀𝐸(−𝐴)^𝛽𝑓𝑠,𝑦_𝑠+𝜙_𝑠²𝑑𝑠+5𝑀²₁sup_{𝑥∈[0,𝑞′]}𝜓𝑥^2𝑏₀𝑝(𝑠)𝑑𝑠+5𝑏𝑀²₁𝑀²₃𝑀²₄×^𝑡₀𝐸𝑥₁−𝑇(𝑏)𝜙(0)−𝑓0,𝜙−𝑓𝑏,𝑦_𝑏+𝜙_𝑏−^𝑏₀𝐴𝑇(𝑏−𝑠)𝑓𝑠,𝑦_𝑠+𝜙_𝑠−𝑑𝑠_𝑏0𝑇(𝑏−𝑠)𝑔(𝑠)𝑑𝑤(𝑠)²𝜂𝑑𝜂≤5𝑀²₁(−𝐴)^−𝛽2𝑐₁𝜙²+𝑐₂+5(−𝐴)^−𝛽2𝑐₁𝑞^′2+𝑐₂+5^𝑡₀𝑐_1−𝛽(𝑡−𝑠)^1−𝛽2𝑑𝑠^𝑡₀𝑐₁𝑦_𝑠+𝜙_𝑠²+𝑐₂𝑑𝑠+5𝑀²₁sup_{𝑥∈[0,𝑞′]}𝜓𝑥^2𝑏₀𝑝(𝑠)𝑑𝑠+5𝑏𝑀²₁𝑀²₃𝑀²₄×^𝑡₀5𝐸𝑥₁²+5𝑀²₁𝐸𝜙(0)−𝑓0,𝜙²+5𝐸𝑓𝑏,𝑦_𝑏+𝜙_𝑏²+5𝐸^𝑏₀𝐴𝑇(𝑏−𝑠)𝑓𝑠,𝑦_𝑠+𝜙_𝑠𝑑𝑠²+5𝐸^𝑏₀𝑇(𝑏−𝑠)𝑔(𝑠)𝑑𝑤(𝑠)²₂𝜂𝑑𝜂≤5𝑀²₁(−𝐴)^−𝛽2𝑐₁𝜙²+𝑐₂+5(−𝐴)^−𝛽2𝑐₁𝑞^′2+𝑐₂+5𝑐²_1−𝛽1𝑏2𝛽−1^2𝛽𝑐₁𝑞^′2+𝑐₂+5𝑀²₁sup_{𝑥∈[0,𝑞′]}𝜓𝑥^2𝑏₀𝑝(𝑠)𝑑𝑠+5𝑏𝑀²₁𝑀²₃𝑀²₄×^𝑡₀5𝑥₁²+5𝑀²₁2𝐸𝜙(0)²+2(−𝐴)^−𝛽2𝑐₁𝜙²+𝑐₂+5(−𝐴)^−𝛽2𝑐₁𝑦_𝑏+𝜙_𝑏²+𝑐₂+5^𝑏₀(−𝐴)^1−𝛽𝑇(𝑏−𝑠)²×𝑑𝑠_𝑏0𝐸(−𝐴)^𝛽𝑓𝑠,𝑦_𝑠+𝜙_𝑠𝑑𝑠²𝑑𝑠+5^𝑏₀𝐸𝑇(𝑏−𝑠)𝑔(𝑠)²₂𝜂𝑑𝜂≤5𝑀²₁(−𝐴)^−𝛽2𝑐₁𝜙²+𝑐₂+5(−𝐴)^−𝛽2𝑐₁𝑞^′2+𝑐₂𝑐+5^21−𝛽𝑏^2𝛽𝑐2𝛽−1¹𝑞^′2+𝑐₂+5𝑀²₁sup_{𝑥∈[0,𝑞′]}𝜓𝑥^2𝑏₀𝑝(𝑠)𝑑𝑠+5𝑏𝑀²₁𝑀²₃𝑀²₄×^𝑡₀5𝑥₁²+5𝑀²₁2𝐸𝜙(0)²+2(−𝐴)^−𝛽2𝑐₁𝜙²+𝑐₂+5(−𝐴)^−𝛽2𝑐₁𝑞^′2+𝑐₂+5^𝑏₀𝑐_1−𝛽(𝑏−𝑠)^1−𝛽2𝑑𝑠^𝑏₀𝑐₁𝑦_𝑠+𝜙_𝑠²+𝑐₂𝑑𝑠+5𝑀²₁sup_{𝑥∈[0,𝑞′]}𝜓𝑥^2𝑏₀𝑝(𝑠)𝑑𝑠𝜂𝑑𝜂≤5𝑀²₁(−𝐴)^−𝛽2𝑐₁𝜙²+𝑐₂+5(−𝐴)^−𝛽2𝑐₁𝑞^′2+𝑐₂𝑐+5^21−𝛽𝑏^2𝛽𝑐2𝛽−1¹𝑞^′2+𝑐₂+5𝑀²₁sup_{𝑥∈[0,𝑞′]}𝜓𝑥^2𝑏₀𝑝(𝑠)𝑑𝑠+5𝑏²𝑀²₁𝑀²₃𝑀²₄×5𝐸𝑥₁²+5𝑀²₁2𝐸𝜙(0)²+2(−𝐴)^−𝛽2𝑐₁𝜙²+𝑐₂+5(−𝐴)^−𝛽2𝑐₁𝑞^′2+𝑐₂𝑐+5^21−𝛽𝑏^2𝛽𝑐2𝛽−1¹𝑞^′2+𝑐₂+5𝑀²₁sup_{𝑥∈[0,𝑞′]}𝜓𝑥^2𝑏₀𝑝(𝑠)𝑑𝑠=Λ², (33) then for each 𝜌∈𝐿₁(𝔅_𝑞), we have 𝜌_𝑏≤Λ. (34)
Step  4. 𝔏₁ has a closed graph.
Let 𝑦^(𝑛)→𝑦^∗, 𝜌_𝑛∈𝔏₁𝑦^(𝑛), and 𝜌_𝑛→𝜌_∗. We will prove that 𝜌_∗∈𝔏₁𝑦^∗. Indeed, 𝜌_𝑛∈𝔏₁𝑦^(𝑛) means that there exists 𝑔_𝑛∈𝑆_{𝐺,𝑦(𝑛)}, such that 𝜌_𝑛(𝑡)=−𝑇(𝑡)𝑓0,𝜙+𝑓𝑡,𝑦_𝑡^(𝑛)+𝜙_𝑡+^𝑡₀𝐴𝑇(𝑡−𝑠)𝑓𝑠,𝑦_𝑠^(𝑛)+𝜙_𝑠+𝑑𝑠_𝑡0𝑇(𝑡−𝑠)𝑔_𝑛(𝑠)(𝑠)+𝑑𝑤_𝑡0𝑇𝑡−𝜂𝐵𝑢_𝑦(𝑛)𝜂𝑑𝜂,𝑡∈𝐽, (35) where 𝑢_𝑦(𝑛)(𝑡)=𝑊⁻¹𝑥₁−𝑇(𝑏)𝜙(0)−𝑓0,𝜙−𝑓𝑏,𝑦_𝑏+𝜙_𝑏−^𝑏₀𝐴𝑇(𝑏−𝑠)𝑓𝑠,𝑦_𝑠+𝜙_𝑠−𝑑𝑠_𝑏0𝑇(𝑏−𝑠)𝑔_𝑛(𝑠)𝑑𝑤(𝑠)(𝑡). (36) We must prove that there exists 𝑔_∗∈𝑆_𝐺,𝑦∗ such that 𝜌_∗(𝑡)=−𝑇(𝑡)𝑓0,𝜙+𝑓𝑡,𝑦^∗_𝑡+𝜙_𝑡+^𝑡₀𝐴𝑇(𝑡−𝑠)𝑓𝑠,𝑦^∗_𝑠+𝜙_𝑠+𝑑𝑠_𝑡0𝑇(𝑡−𝑠)𝑔_∗(𝑠)(𝑠)+𝑑𝑤_𝑡0𝑇𝑡−𝜂𝐵𝑢_𝑦∗𝜂𝑑𝜂,𝑡∈𝐽, (37) where 𝑢_𝑦∗(𝑡)=𝑊⁻¹𝑥₁−𝑇(𝑏)𝜙(0)−𝑓0,𝜙−𝑓𝑏,𝑦^∗_𝑏+𝜙_𝑏−^𝑏₀𝐴𝑇(𝑏−𝑠)𝑓𝑠,𝑦^∗_𝑠+𝜙_𝑠−𝑑𝑠_𝑏0𝑇(𝑏−𝑠)𝑔_∗(𝑠)𝑑𝑤(𝑠)(𝑡). (38) Set 𝑢_𝑦(𝑡)=𝑊⁻¹𝑥₁−𝑇(𝑏)𝜙(0)−𝑓0,𝜙−𝑓𝑏,𝑦_𝑏+𝜙_𝑏−^𝑏₀𝐴𝑇(𝑏−𝑠)𝑓𝑠,𝑦_𝑠+𝜙_𝑠𝑑𝑠(𝑡). (39) Since 𝑓, 𝑊⁻¹ are continuous, then 𝑢_𝑦(𝑛)(𝑡)→𝑢_𝑦∗(𝑡) for 𝑡∈𝐽, 𝜌_𝑛(𝑡)+𝑇(𝑡)𝑓0,𝜙−𝑓𝑡,𝑦_𝑡^(𝑛)+𝜙_𝑡−^𝑡₀𝐴𝑇(𝑡−𝑠)𝑓𝑠,𝑦_𝑠^(𝑛)+𝜙_𝑠−𝑑𝑠_𝑡0𝑇𝑡−𝜂𝐵𝑢_𝑦(𝑛)𝜂−𝑑𝜂𝜌_∗(𝑡)+𝑇(𝑡)𝑓0,𝜙−𝑓𝑡,𝑦^∗_𝑡+𝜙_𝑡−^𝑡₀𝐴𝑇(𝑡−𝑠)𝑓𝑠,𝑦^∗_𝑠+𝜙_𝑠−𝑑𝑠_𝑡0𝑇𝑡−𝜂𝐵𝑢_{𝑦∗(𝜂)}𝜂𝑑𝜂⟶0,as𝑛⟶∞. (40) Consider the linear continuous operator Γ∶𝐿²()(),𝑔(𝑡)=𝐽,𝐻⟶𝐶𝐽,𝐻𝑔⟶Γ_𝑡0𝑇()𝑔(𝑠)(𝑠)−𝑡−𝑠𝑑𝑤_𝑡0𝑇(𝑡−𝑠)𝐵𝑊^−1𝑏₀𝑇(𝑏−𝜏)𝑔(𝜏)𝑑𝑤(𝜏)𝑑𝑠. (41) From Lemma 5, it follows that 𝑇∘𝑆_𝐺 is a closed graph. Moreover, we have 𝜌_𝑛(𝑡)+𝑇(𝑡)𝑓0,𝜙−𝑓𝑡,𝑦_𝑡^(𝑛)+𝜙_𝑡−^𝑡₀𝐴𝑇(𝑡−𝑠)𝑓𝑠,𝑦_𝑠^(𝑛)+𝜙_𝑠−𝑑𝑠_𝑡0𝑇𝑡−𝜂𝐵𝑢_𝑦(𝑛)𝜂𝑑𝜂∈Γ𝑆_{𝐺,𝑦(𝑛)}. (42) Since 𝑦^(𝑛)→𝑦^∗, it follows from Lemma 5 that 𝜌_∗(𝑡)+𝑇(𝑡)𝑓0,𝜙−𝑓𝑡,𝑦^∗_𝑡+𝜙_𝑡−^𝑡₀𝐴𝑇(𝑡−𝑠)𝑓𝑠,𝑦^∗_𝑠+𝜙_𝑠−𝑑𝑠_𝑡0𝑇𝑡−𝜂𝐵𝑢_𝑦∗𝜂=𝑑𝜂_𝑡0𝑇(𝑡−𝑠)𝑔_∗(𝑠)(𝑠)−𝑑𝑤_𝑡0𝑇(𝑡−𝑠)𝐵𝑊^−1𝑏₀𝑇(𝑏−𝜏)𝑔(𝜏)𝑑𝑤(𝜏)𝑑𝑠 (43) for some 𝑔_∗∈𝑆_𝐺,𝑦∗.
Step 5. Multivalued function 𝔏₁∶𝑈→𝒫_{𝑐𝑝,𝑐𝑣}(𝔅⁰_𝑏) is a condensing operator.
From the decomposition of operator 𝔏₁, we define two operators 𝑇,𝑆 on space 𝔅⁰_𝑏. For operator 𝑇, 𝑇∶𝑈→𝒫_{𝑐𝑝,𝑐𝑣}(𝔅⁰_𝑏), for each 𝑦, 𝑇𝑦 defined by the set of ℎ₁(𝑡)∈𝔅⁰_𝑏 such that ℎ₁(𝑡)=(],(𝑡)𝑓0,𝑡∈−∞,0−𝑇0,𝜙+𝑓𝑡,𝑦_𝑡+𝜙_𝑡+∫^𝑡₀𝐴𝑇(𝑡−𝑠)𝑓𝑠,𝑦_𝑠+𝜙_𝑠𝑑𝑠𝑡∈𝐽. (44)
For operator 𝑆, 𝑆∶𝑈→𝒫_{𝑐𝑝,𝑐𝑣}(𝔅⁰_𝑏), for each 𝑦, 𝑆𝑦 defined by the set of ℎ₂(𝑡)∈𝔅⁰_𝑏 such that ℎ₂(𝑡)=(],∫0,𝑡∈−∞,0_𝑡0𝑇(𝑡−𝑠)𝑔(𝑠)𝑑𝑤(𝑠)+∫^𝑡₀𝑇𝑡−𝜂𝐵𝑊⁻¹×𝑥₁−𝑇(𝑏)𝜙(0)−𝑓0,𝜙−𝑓𝑏,𝑦_𝑏+𝜙_𝑏−∫^𝑏₀𝐴𝑇(𝑏−𝑠)𝑓𝑠,𝑦_𝑠+𝜙_𝑠−∫𝑑𝑠_𝑏0𝑇(𝑏−𝑠)𝑔(𝑠)𝑑𝑤(𝑠)𝜂𝑑𝜂,𝑡∈𝐽. (45) So, operator 𝔏₁𝑦=𝑇𝑦+𝑆𝑦. Next, we will prove that operator 𝑇 is contraction operator and 𝑆 is completely continuous operator.
First, we prove operator 𝑇 is contractive. For each 𝑦¹,𝑦²∈𝔅⁰_𝑏, we have 𝐸ℎ¹₁(𝑡)−ℎ²₁(𝑡)²=𝐸𝑓𝑡,𝑦¹_𝑡+𝜙_𝑡−𝑓𝑡,𝑦²_𝑡+𝜙_𝑡+^𝑡₀𝐴𝑇(𝑡−𝑠)𝑓𝑠,𝑦¹_𝑠+𝜙_𝑠−𝑑𝑠_𝑡0𝐴𝑇(𝑡−𝑠)𝑓𝑠,𝑦²_𝑠+𝜙_𝑠𝑑𝑠²≤2𝐸𝑓𝑡,𝑦¹_𝑡+𝜙_𝑡−𝑓𝑡,𝑦²_𝑡+𝜙_𝑡²+2𝐸^𝑡₀𝐴𝑇(𝑡−𝑠)𝑓𝑠,𝑦¹_𝑠+𝜙_𝑠−𝑑𝑠_𝑡0𝐴𝑇(𝑡−𝑠)𝑓𝑠,𝑦²_𝑠+𝜙_𝑠𝑑𝑠²≤2(−𝐴)^−𝛽2𝑞²𝑦¹_𝑡−𝑦²_𝑡²+2𝐸^𝑡₀(−𝐴)^1−𝛽𝑇()×)𝑡−𝑠(−𝐴^𝛽𝑓𝑠,𝑦¹_𝑠+𝜙_𝑠−(−𝐴)^𝛽𝑓𝑠,𝑦²_𝑠+𝜙_𝑠𝑑𝑠²≤2(−𝐴)^−𝛽2𝑞²𝑦¹_𝑡−𝑦²_𝑡²+2^𝑡₀(−𝐴)^1−𝛽𝑇(𝑡−𝑠)²×𝑑𝑠_𝑡0𝐸(−𝐴)^𝛽𝑓𝑠,𝑦¹_𝑠+𝜙_𝑠−(−𝐴)^𝛽𝑓𝑠,𝑦²_𝑠+𝜙_𝑠²)𝑑𝑠≤2(−𝐴^−𝛽2𝑞²𝑦¹_𝑡−𝑦²_𝑡²+2^𝑡₀𝑐_1−𝛽(𝑡−𝑠)^1−𝛽2𝑑𝑠^𝑡₀𝑞²𝑦¹_𝑡−𝑦²_𝑡²)𝑑𝑠≤2(−𝐴^−𝛽2𝑞²𝑦¹_𝑡−𝑦²_𝑡²+2𝑐²_1−𝛽1𝑏2𝛽−1^2𝛽−1𝑞²sup_{[]𝑠∈0,𝑡}𝑦¹_𝑡−𝑦²_𝑡²≤2(−𝐴)^−𝛽2𝑞²+2𝑐²_1−𝛽1𝑏2𝛽−1^2𝛽−1𝑞²𝑙²𝑦¹−𝑦²²=𝑄𝑦¹−𝑦²². (46) So operator 𝑇 is contraction operator.
Next, we will prove that operator 𝑆 is completely continuous.
For 𝑡₁,𝑡₂∈𝐽, 0<𝑡₁<𝑡₂<𝑏, if 𝑦∈𝑈, ℎ₂∈𝑆𝑦, then there exist 𝑓∈𝑆_𝐹,𝑦 such that for arbitrary 𝑡∈[0,𝑏], ℎ₂(𝑡)=^𝑡₀𝑇(𝑡−𝑠)𝑔(𝑠)𝑑𝑤(𝑠)+^𝑡₀𝑇𝑡−𝜂𝐵𝑊⁻¹×𝑥₁−𝑇(𝑏)𝜙(0)−𝑓0,𝜙−𝑓𝑏,𝑦_𝑏+𝜙_𝑏−^𝑏₀𝐴𝑇(𝑏−𝑠)𝑓𝑠,𝑦_𝑠+𝜙_𝑠𝑑𝑠−^𝑏₀𝑇(𝑏−𝑠)𝑔(𝑠)𝑑𝑤(𝑠)𝜂𝑑𝜂,𝑡∈𝐽. (47)
Thus 𝐸ℎ₂𝑡₂−ℎ₂𝑡₁²=𝐸^𝑡2₀𝑇𝑡₂−𝑠−𝑇𝑡₁𝑔(𝑠)(𝑠)+−𝑠𝑑𝑤_𝑡2𝑡1𝑇𝑡₁𝑔(𝑠)(𝑠)+−𝑠𝑑𝑤_𝑡20𝑇𝑡₂−𝜂−𝑇𝑡₁−𝜂×𝐵𝑊⁻¹𝑥₁−𝑇(𝑏)𝜙(0)−𝑓0,𝜙−𝑓𝑏,𝑦_𝑏+𝜙_𝑏−^𝑏₀𝐴𝑇(𝑏−𝑠)𝑓𝑠,𝑦_𝑠+𝜙_𝑠−𝑑𝑠_𝑏0𝑇()𝑔(𝑠)(𝑠𝜂+𝑏−𝑠𝑑𝑤)𝑑𝜂_𝑡2𝑡1𝑇𝑡₁−𝜂𝐵𝑊⁻¹×𝑥₁−𝑇(𝑏)𝜙(0)−𝑓0,𝜙−𝑓𝑏,𝑦_𝑏+𝜙_𝑏−^𝑏₀𝐴𝑇(𝑏−𝑠)𝑓𝑠,𝑦_𝑠+𝜙_𝑠𝑑𝑠−^𝑏₀𝑇(𝑏−𝑠)𝑔(𝑠)𝑑𝑤(𝑠)𝜂𝑑𝜂²≤4𝐸^𝑡2₀𝑇𝑡₂−𝑠−𝑇𝑡₁−𝑠𝑔(𝑠)𝑑𝑤(𝑠)²+4𝐸^𝑡2_𝑡1𝑇𝑡₁−𝑠𝑔(𝑠)𝑑𝑤(𝑠)²+4𝐸^𝑡2₀𝑇𝑡₂−𝜂−𝑇𝑡₁−𝜂𝐵𝑊⁻¹×𝑥₁−𝑇(𝑏)𝜙(0)−𝑓0,𝜙−𝑓𝑏,𝑦_𝑏+𝜙_𝑏−^𝑏₀𝐴𝑇(𝑏−𝑠)𝑓𝑠,𝑦_𝑠+𝜙_𝑠−𝑑𝑠_𝑏0𝑇(𝑏−𝑠)𝑔(𝑠)𝑑𝑤(𝑠)𝜂𝑑𝜂²+4𝐸^𝑡2_𝑡1𝑇𝑡₁−𝜂𝐵𝑊⁻¹×𝑥₁−𝑇(𝑏)𝜙(0)−𝑓0,𝜙−𝑓𝑏,𝑦_𝑏+𝜙_𝑏−^𝑏₀𝐴𝑇(𝑏−𝑠)𝑓𝑠,𝑦_𝑠+𝜙_𝑠−𝑑𝑠_𝑏0𝑇(𝑏−𝑠)𝑔(𝑠)𝑑𝑤(𝑠)𝜂𝑑𝜂²≤4𝐸^𝑡2₀𝑇𝑡₂−𝑠−𝑇𝑡₁−𝑠𝑔(𝑠)𝑑𝑤(𝑠)²₂+4𝐸^𝑡2_𝑡1𝑇𝑡₁−𝑠𝑔(𝑠)𝑑𝑤(𝑠)²₂+4𝐸^𝑡2₀𝑇𝑡₂−𝜂−𝑇𝑡₁−𝜂𝐵𝑊⁻¹×𝑥₁−𝑇(𝑏)𝜙(0)−𝑓0,𝜙−𝑓𝑏,𝑦_𝑏+𝜙_𝑏−^𝑏₀𝐴𝑇(𝑏−𝑠)𝑓𝑠,𝑦_𝑠+𝜙_𝑠−𝑑𝑠_𝑏0𝑇(𝑏−𝑠)𝑔(𝑠)𝑑𝑤(𝑠)𝜂𝑑𝜂²+4𝐸^𝑡2_𝑡1𝑇𝑡₁−𝜂𝐵𝑊⁻¹×𝑥₁−𝑇(𝑏)𝜙(0)−𝑓0,𝜙−𝑓𝑏,𝑦_𝑏+𝜙_𝑏−^𝑏₀𝐴𝑇(𝑏−𝑠)𝑓𝑠,𝑦_𝑠+𝜙_𝑠−𝑑𝑠_𝑏0𝑇(𝑏−𝑠)𝑔(𝑠)𝑑𝑤(𝑠)𝜂𝑑𝜂²≤4^𝑡2₀𝑇𝑡₂−𝑠−𝑇𝑡₁−𝑠²𝐸𝑔(𝑠)²₂𝑑𝑠+4^𝑡2_𝑡1𝑇𝑡₁−𝑠²𝐸𝑔(𝑠)²₂𝑑𝑠+4𝑏^𝑡2₀𝑇𝑡₂−𝜂−𝑇𝑡₁−𝜂²𝑀²₃𝑀²₄×𝐸𝑥₁−𝑇(𝑏)𝜙(0)−𝑓0,𝜙−𝑓𝑏,𝑦_𝑏+𝜙_𝑏−^𝑏₀𝐴𝑇(𝑏−𝑠)𝑓𝑠,𝑦_𝑠+𝜙_𝑠−𝑑𝑠_𝑏0𝑇(𝑏−𝑠)𝑔(𝑠)𝑑𝑤(𝑠)²𝜂𝑡𝑑𝜂+4₂−𝑡₁^𝑡2_𝑡1𝑇𝑡₁−𝜂²𝑀²₃𝑀²₄×𝐸𝑥₁−𝑇(𝑏)𝜙(0)−𝑓0,𝜙−𝑓𝑏,𝑦_𝑏+𝜙_𝑏−^𝑏₀𝐴𝑇(𝑏−𝑠)𝑓𝑠,𝑦_𝑠+𝜙_𝑠𝑑𝑠−^𝑏₀𝑇(𝑏−𝑠)𝑔(𝑠)𝑑𝑤(𝑠)²𝜂𝑑𝜂≤4^𝑡2₀𝑇𝑡₂−𝑠−𝑇𝑡₁−𝑠²𝐸𝑔(𝑠)²₂𝑑𝑠+4^𝑡2_𝑡1𝑇𝑡₁−𝑠²𝐸𝑔(𝑠)²₂𝑑𝑠+4𝑏^𝑡2₀𝑇𝑡₂−𝜂−𝑇𝑡₁−𝜂²𝑀²₃𝑀²₄×5𝐸𝑥₁²+5𝑀²₁×2𝐸𝜙(0)²+2(−𝐴)^−𝛽2𝑐₁𝜙²_𝔅ℎ+𝑐₂+5(−𝐴)^−𝛽2𝑐₁𝑦_𝑏+𝜙_𝑏²_𝔅ℎ+𝑐₂+5𝑏^𝑏₀𝐸𝐴𝑇(𝑏−𝑠)𝑓𝑠,𝑦_𝑠+𝜙_𝑠²𝑑𝑠+5^𝑏₀𝐸𝑇(𝑏−𝑠)𝑔(𝑠)²₂𝜂𝑡𝑑𝑠𝑑𝜂+4₂−𝑡₁𝑀²₃𝑀²₄^𝑡2_𝑡1𝑇𝑡₁−𝜂²×5𝐸𝑥₁²+5𝑀²₁×2𝐸𝜙(0)²+2(−𝐴)^−𝛽2𝑐₁𝜙²_𝔅ℎ+𝑐₂+5(−𝐴)^−𝛽2𝑐₁𝑦_𝑏+𝜙_𝑏𝔅ℎ+𝑐₂+5𝑏^𝑏₀𝐸𝐴𝑇(𝑏−𝑠)𝑓𝑠,𝑦_𝑠+𝜙_𝑠²𝑑𝑠+5^𝑏₀𝐸𝑇(𝑏−𝑠)𝑔(𝑠)²₂𝑑𝑠𝜂𝑑𝜂≤4^𝑡2₀𝑇𝑡₂−𝑠−𝑇𝑡₁−𝑠²𝐸𝑔(𝑠)²₂𝑑𝑠+4^𝑡2_𝑡1𝑇𝑡₁−𝑠²𝐸𝑔(𝑠)²₂𝑑𝑠+4𝑏^𝑡2₀𝑇𝑡₂−𝜂−𝑇𝑡₁−𝜂²𝑀²₃𝑀²₄×5𝐸𝑥₁²+5𝑀²₁×2𝐸𝜙(0)²+2(−𝐴)^−𝛽2𝑐₁𝜙²_𝔅ℎ+𝑐₂+5(−𝐴)^−𝛽2𝑐₁𝑦_𝑏+𝜙_𝑏²_𝔅ℎ+𝑐₂+5𝑏^𝑏₀(−𝐴)^1−𝛽𝑇(𝑏−𝑠)²×𝑐₁𝑦_𝑠+𝜙_𝑠²_𝔅ℎ+𝑐₂𝑑𝑠+5𝑀²₁^𝑏₀𝐸𝑔(𝑠)²₂𝜂𝑡𝑑𝑠𝑑𝜂+4₂−𝑡₁𝑀²₃𝑀²₄^𝑡2_𝑡1𝑇𝑡₁−𝜂²×5𝐸𝑥₁²+5𝑀²₁×2𝐸𝜙(0)²+2(−𝐴)^−𝛽2𝑐₁𝜙²_𝔅ℎ+𝑐₂+5(−𝐴)^−𝛽2𝑐₁𝑦_𝑏+𝜙_𝑏²_𝔅ℎ+𝑐₂+5𝑏^𝑏₀(−𝐴)^1−𝛽𝑇(𝑏−𝑠)²𝑐₁𝑦_𝑠+𝜙_𝑠²_𝔅ℎ+𝑐₂𝑑𝑠+5𝑀²₁^𝑏₀𝐸𝑔(𝑠)²₂𝑑𝑠𝜂𝑑𝜂. (48) The right-hand side of the previous inequality is independent of 𝑦 and tends to zero as 𝑡₂→𝑡₁. It is easy to prove that 𝑆 maps bounded set into bounded set, so operator 𝑆 is completely continuous.
Hence, it follows from Theorem 2 that the operator 𝔏₁ has a fixed-point 𝑦^∗∈𝔅⁰_𝑏. Let 𝑥(𝑡)=𝑦^∗(𝑡)+𝜙(𝑡), 𝑡∈(−∞,𝑏]. Then 𝑥 is a fixed-point of the operator 𝔏 which is a mild solution of problem (1); then system (1) is controllable on 𝐽.

Remark 9. Sakthivel et al. [28] considered approximate controllability for nonlinear stochastic systems with resolvent operators and unbounded delay. Approximate controllability results are being established for a class of nonlinear stochastic differential equations with resolvent operators in a real separable Hilbert spaces by using the resolvent operators and fixed-point technique.

4. Conclusion

In this paper, we have investigated the controllability of stochastic neutral functional differential inclusions in Hilbert spaces. Based on a fixed-point theorem with regard to condensing map, sufficient conditions for the controllability of the stochastic neutral functional differential inclusions have been derived.

Acknowledgments

This work was supported by the grant of Chongqing municipal educational commission (no. KJ120609), Natural Science Foundation Project of CSTC, 2010BB9318 and CQ CSTC, 2009BB3057, and the Ph.D. Foundation of Chongqing Normal University under Grant no. 09XLB007.