This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.
Introduction
Qualitative properties such as existence, uniqueness, stability and controllability for various types of stochastic differential equations have been extensively studied by many researchers (see [4, 6, 8, 17] and references therein). Many fundamental problems of control theory such as pole-assignment, stabilizability and optimal control may be solved under the assumption that the system is controllable. The controllability problem for an evolution equation also consists of driving the solution of the system to a prescribed final target state (exactly or in some approximate way) in a finite time interval. As an area of application oriented mathematics, the control problem has been studied extensively in the fields of infinite dimensional nonlinear systems [10]. The theory of semigroups of bounded linear operators is closely related to the solution of differential equations. In recent years, this theory has been applied to a large class of nonlinear differential equations in Banach spaces. Using the method of semigroups, various types of solutions of semilinear evolution equations have been discussed by Pazy in [20]. Semigroup theory gives a unified treatment of a wide class of stochastic parabolic, hyperbolic and functional differential equations, and much effort has been devoted to the study of controllability results for such evolution equations.
Motivated by the above works, in this paper we address sufficient conditions to ensure the controllability of neutral stochastic integrodifferential equations with infinite delays in a Hilbert space described by
where A is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators T (t),t ≥ 0, on a separable Hilbert space H with inner product (⋅, ⋅) and norm ∥ ⋅ ∥. Let K be another separable Hilbert space with inner product (⋅, ⋅)K and norm ∥ ⋅ ∥K. Suppose {w(t)}t≥0 is a given K-valued Brownian motion or Wiener process with a finite trace nuclear covariance operator Q ≥ 0. We are also employing the same notation ∥ ⋅ ∥ for the norm L(K, H), where L(K, H) denotes the space of all bounded linear operators from K into H. The histories xt belongs to some abstract phase space $\begin{array}{}
\displaystyle
\mathfrak{B}
\end{array}$ defined axiomatically (see Section 2); $\begin{array}{}
\displaystyle
F, h : J \times \mathfrak{B} \rightarrow H
\end{array}$ are the measurable mappings in H-norm, and $\begin{array}{}
\displaystyle
G : J \times J \times \mathfrak{B} \rightarrow L_Q(K , H ) (L_Q (K, H)
\end{array}$ denotes the space of all Q-Hilbert-Schmidt operators from K into H, which is going to be defined below) is a measurable mapping in LQ(K, H)-norm. The control function u(⋅) taking values in L2(J, U) of admissible control functions for a separable Hilbert space U, C is a bounded linear operator from U into H, and ϕ(t) is a $\begin{array}{}
\displaystyle
\mathfrak{B}
\end{array}$-valued random variable independent of Brownian motion {w(t)} with finite second moment.
The aim of our paper is to present some results on the controllability of (1) based on the Nussbaum fixed point theorem combined with theories of resolvent operators for integrodifferential equations. Our main results concerning (1), rely essentially on techniques using strongly continuous family of operators {R(t), t ≥ 0}, defined on the Hilbert space H and called their resolvent. The resolvent operator is similar to the semigroup operator for abstract differential equations in Banach spaces. However, the resolvent operator does not satisfy semigroup properties (see, for instance, [11]), and our objective in this paper is to apply the theories of resolvent operators, which was proposed by Grimmer [2].
The rest of this paper is organized as follows. In Section 2, we recall some basic definitions, notations, and lemmas which will be needed in the sequel. In Section 3, the controllability of neutral stochastic integrodifferential equations with infinite delay is studied in Hilbert spaces. Section 4 is devoted to an application which illustrates the main results.
Preliminaries
Basic Concepts of Stochastic Analysis
For more details on this section, the reader is referred to Da Prato and Zabczyk [5], Gard [12], and the references therein. Throughout the paper, H and K denote real separable Hilbert spaces.
Let (Ω, $\begin{array}{}
\displaystyle
\mathfrak{F}
\end{array}$, P) be a complete probability space furnished with a complete family of right continuous increasing sub σ-algebras $\begin{array}{}
\displaystyle
\lbrace \mathfrak{F}_{t}, t \in J\rbrace
\end{array}$ satisfying $\begin{array}{}
\displaystyle
\mathfrak{F}_t \subset \mathfrak{F}
\end{array}$. A H-valued random variable is an $\begin{array}{}
\displaystyle
\mathfrak{F}
\end{array}$-measurable function x(t) : Ω → H, and a collection of random variables S = {x(t, w) : Ω → H|t ∊ J} is called a stochastic process. Usually, we suppress the dependence on ω ∊ Ω and write x(t) instead of x(t, ω) and x(t) : J → H in the place of S. Let βn(t)(n = 1, 2,...) be a sequence of real-valued one-dimensional standard Brownian motions mutually independent over (Ω, $\begin{array}{}
\displaystyle
\mathfrak{F}
\end{array}$, P). Set
where λn ≥ 0(n = 1, 2,...) are nonnegative real numbers and {ζn}(n = 1, 2,...) is complete orthonormal basis in K. Let Q ∊ L(K, K) be an operator defined by Qζn = λnζn with finite $\begin{array}{}
\displaystyle
TrQ = \Sigma _{n = 1}^\infty {\lambda _n} <\ \infty
\end{array}$ (Tr denotes the trace of the operator). Then the above K-valued stochastic process w(t) is called a Q-Wiener process. We assume that $\begin{array}{}
\displaystyle
\mathfrak{F}_t = \sigma(w(s) : 0 \leq s \leq t)
\end{array}$ is the σ-algebra generated by w and $\begin{array}{}
\displaystyle
\mathfrak{F}_T = \mathfrak{F}
\end{array}$. Let φ ∊ L(K, H) and define
If ∥φ∥Q < ∞, then φ is called a Q-Hilbert-Schmidt operator. Let LQ(K, H) denote the space of all Q-Hilbert-Schmidt operators φ : K → H. The completion LQ(K, H) of L(K, H) with respect to the topology induced by the norm ∥ ⋅ ∥Q where $\begin{array}{}
\displaystyle
\| \varphi \|^2_Q = \langle \varphi, \varphi \rangle
\end{array}$ is a Hilbert space with the above norm topology.
In this work, we will employ an axiomatic definition of the phase space $\begin{array}{}
\displaystyle
\mathfrak{B}
\end{array}$ introduced by Hale and Kato [13]. The axioms of the space $\begin{array}{}
\displaystyle
\mathfrak{B}
\end{array}$ are established for $\begin{array}{}
\displaystyle
\mathfrak{F}_0
\end{array}$-measurable functions from J0 into H, endowed with a seminorm $\begin{array}{}
\displaystyle
\| \cdot \|_{\mathfrak{B}}
\end{array}$. We will assume that $\begin{array}{}
\displaystyle
\mathfrak{B}
\end{array}$ satisfies the following axioms:
(ai) If x : (−∞, a) → H, a > 0, is continuous on [0, a) and x0 in $\begin{array}{}
\displaystyle
\mathfrak{B}
\end{array}$, then for every t ∊ [0, a) the following conditions hold:
xt is in $\begin{array}{}
\displaystyle
\mathfrak{B}
\end{array}$,
$\begin{array}{}
\displaystyle
\| x_t \|_{\mathfrak{B}} \leq \Gamma(t)\sup\lbrace \| x(s)\| : 0\leq s\leq t \rbrace +N(t) \| x_0 \|_{\mathfrak{B}}
\end{array}$, where L > 0 is a constant; Γ, N : [0, ∞) → [0, ∞), Γ is continuous, N is locally bounded, and L, Γ, N are independent of x(⋅).
(aii) For the function x(⋅) in (ai), xt is a $\begin{array}{}
\displaystyle
\mathfrak{B}
\end{array}$-valued function [0, a).
(aiii) The space $\begin{array}{}
\displaystyle
\mathfrak{B}
\end{array}$ is complete.
Suppose x(t) : Ω → H, t ≤ a, is a continuous $\begin{array}{}
\displaystyle
\mathfrak{F}_t
\end{array}$-adapted H-valued stochastic process. We can associate with another process $\begin{array}{}
\displaystyle
x_t : \Omega \rightarrow \mathfrak{B}, t \geq 0
\end{array}$ by setting xt = {x(t + s)(w) : s ∊ (−∞, 0]}. This is regarded as a $\begin{array}{}
\displaystyle
\mathfrak{B}
\end{array}$-valued stochastic process.
The collection of all strongly measurable, square-integrable H-valued random variables, denoted by $\begin{array}{}
\displaystyle
{L_2}(\Omega ,,P;H) \equiv {L_2}(\Omega ;H)
\end{array}$, is a Banach space equipped with norm
where the expectation $\begin{array}{}
\displaystyle
\mathbb{E}
\end{array}$ is defined by $\begin{array}{}
\displaystyle
\mathbb{E}(h) = \int_\Omega h(\omega)dP
\end{array}$.
Let J1 = (−∞, b] and C(J1, L2(Ω;H)) be the Banach space of all continuous maps from J1 into L2(Ω;H) satisfying the condition $\begin{array}{}
\sup\nolimits_{t \in {J_1}} \mathbb{E}\| x (t)\|{^2} \lt \infty
\end{array}$. An important subspace is given by $\begin{array}{}
\displaystyle
L_2^0(\Omega, H) = \{f \in L_2(\Omega, H) : f
\end{array}$ is $\begin{array}{}
\displaystyle
\mathfrak{F}_0
\end{array}$ − measurabale}.
Let Z be the closed subspace of all continuous process x that belong to the space C(J1, L2(Ω;H)) consisting of $\begin{array}{}
\displaystyle
\mathfrak{F}_t
\end{array}$-adapted measurable processes such that the $\begin{array}{}
\displaystyle
\mathfrak{F}_0
\end{array}$-adapted processes $\begin{array}{}
\displaystyle
\phi \in L_2(\Omega;\mathfrak{B})
\end{array}$. Let ∥ ⋅ ∥Z be a seminorm in Z defined by
$\begin{array}{}
\displaystyle
\overline{N} = \sup\nolimits_{t\in J}\{N(t)\}, \overline{\Gamma} = \sup\nolimits_{t\in J} \{\Gamma(t)\}
\end{array}$. It is easy to verify that Z furnished with the norm topology as defined above is a Banach space.
In the present section, we recall some definitions, notations and propreties needed in the sequel. In what follows, H will denote a Banach space, A and B(t) are closed linear operators on H. Y represents the Banach space $\begin{array}{}
\displaystyle
\mathscr{D}(A)
\end{array}$, the domain of operator A, equiped with the graph norm
$$\begin{array}{}
\displaystyle
\| y\|_{Y} := \| Ay \| + \| y \| \, \text{for} \quad y \in Y.
\end{array}$$
The notation C([0, +∞); Y) stands for the space of all continuous function from [0, +∞) into Y. We then consider the following Cauchy problem
[2] A resolvent operator of Eq. (2) is a bounded linear operator valued function $\begin{array}{}
\displaystyle
R(t)\in \mathscr{L}(H)
\end{array}$ for t ≥ 0, satisfying the following propreties:
R(0) = I and $\begin{array}{}
\displaystyle
\| R(t)\| \leq \tilde{N}e^{\beta t}
\end{array}$ for some constant $\begin{array}{}
\displaystyle
\tilde{N}
\end{array}$ and β.
For each x ∊ H, R(t)x is strongly continuous for t ≥ 0.
For x ∊ Y, R(.)x ∊ C1([0, +∞);H) ∩ C([0, +∞); Y) and
For additional details on resolvent operators, we refer the reader to [2]. The resolvent operator plays an important role to study the existence of solutions and to establish a variation of constants formula for nonlinear systems. For this reason, we need to know when the linear system (2) possesses a resolvent operator. Theoreml below provides a satisfactory answer to this problem.
In what follows we suppose the following assumptions:
(H1)A is the infinitesimal generator of a C0— semigroup (T(t))t≥0 on H
(H2) For all t ≥ 0, B(t) is a continuous linear operator from (Y, ∥ ⋅ ∥Y) into (H, ∥ ⋅ ∥H).
Moreover, there exists an integrable function $\begin{array}{}
\displaystyle
\,c:[0,+\infty)\rightarrow \mathbb{R}^{+}
\end{array}$ such that for any y ∊ Y, t ↦ B(t)y belongs to W1,1 ([0, +∞), H) and
$$\begin{array}{}
\displaystyle
\| \frac{d}{dt}B(t)y\|_{H} \leq c(t)\| y \|_Y\; \mbox{for}\; y \in Y \;\mbox{and}\, t\geq 0.
\end{array}$$
We recall that $\begin{array}{}
\displaystyle
W^{k,p}(\mathbb{O})=\{\tilde{\omega} \in L^{p}(\mathbb{O}):D^{\alpha}\tilde{\omega} \in L^{p}(\mathbb{O}), \forall \|\alpha \| \leq k\}
\end{array}$, where $\begin{array}{}
\displaystyle
D^{\alpha}\tilde{\omega}
\end{array}$ is the weak α-th partial derivative of $\begin{array}{}
\displaystyle
\tilde{\omega}
\end{array}$.
Theorem 1.
[2] Assume that hypotheses (H1) and (H2) hold. Then the Eq. (2) admits a resolvent operator (R(t))t≥0.
Lemma 2.
[11] Let hypotheses (H1) and (H2) be satisfied. Then there exists a constant L = L(T) such that
$$\begin{array}{}
\displaystyle
\|R(t+\varepsilon)-R(\varepsilon)R(t)\|\;\leq L\varepsilon,\;\;\;\;\forall 0\leq \varepsilon\leq t\leq T.
\end{array}$$
Theorem 3.
[11] Assume that hypotheses (H1) and (H2) hold. Let T(t) be a compact operator for t > 0. Then, the corresponding resolvent operator R(t) of Eq. (2) is continuous for t > 0 in the operator norm, namely, for t0 > 0, it holds that limh→0 ∥R(t0 + h) – R(t0)∥ = 0.
In the sequel, we recall some results on the existence of solutions for the following integro-differentiel equation
Remark 1. From this definition we deduce that $\begin{array}{}
\displaystyle
\nu (t) \in {\mathscr D}(A)
\end{array}$, and the function B(t − s)ν(s) is integrable, for all t > 0 and s ∊ [0, +∞).
Theorem 4.
[2] Assume that hypotheses (H1) and (H2) hold. If v is a stict solution of Eq. (3), then the following variation of constant formula holds
$$\begin{array}{}
\displaystyle
\nu (t) = R(t){\nu _0} + \int_0^t R (t - s)q(s)ds\;\;\;{\rm{for}}\quad t \ge 0.
\end{array}$$
Accordingly, we can establish the following definiton.
Definition 3.
A function ν : [0, +∞) → H is called mild solution of Eq. (3), for ν0 ∊ H, if ν satisfies the variation of constants formula (4).
The next theorem provides sufficient conditions ensuring the regularity of solutions of Eq.(3).
Theorem 5.
Let q ∊ C1([0, +∞);H) and ν be defined by (4). If$\begin{array}{}
\displaystyle
\nu_0 \in \mathscr{D}(A)
\end{array}$, then ν is a strict solution of the Eq. (3).
Definition 4.
An $\begin{array}{}
\displaystyle
\mathfrak{F}_t
\end{array}$-adapted stochastic process x(t) : J1 → H is a mild solution of the abstract Cauchy problem (1) if $\begin{array}{}
\displaystyle
x_0 = \phi \in \mathfrak{B}
\end{array}$ on J0 satisfying $\begin{array}{}
\displaystyle
\| \phi \|^2_{\mathfrak{B}} <\ \infty
\end{array}$. The restriction of x(⋅) to the interval [0, b) is a continuous stochastic process such that the following equation is satisfied
The nonlinear neutral stochastic integrodifferential equation (1) is said to be controllable on the interval J, if for every continuous initial stochastic process $\begin{array}{}
\displaystyle
\phi \in \mathfrak{B}
\end{array}$ defined on J0, there exists a stochastic control u ∊ L2(J, U) that is adapted to the filtration $\begin{array}{}
\displaystyle
\{ \mathscr{F}_t\}_{t\geq 0}
\end{array}$ such that the solution x(⋅) of (1) satisfies x(b) = x1, where x(b) is a random variable which is $\begin{array}{}
\displaystyle
\,\mathscr{F}_{b}
\end{array}$-measurable, x1 and b are preassigned terminal state and time, respectively.
As a key tool for developing the controllability in this work, the consideration of this paper is based on the following fixed point theorem due Nussbaum [17]. Throughout the paper, $\begin{array}{}
\displaystyle
B_r[x] \subset L_2(\Omega, \mathfrak{B})
\end{array}$ is the closed ball centered at x with radius r > 0.
Theorem 6.
(Nussbaum Fixed Point Theorem). Let S be a closed, bounded, and convex subset of a Banach space X. Let Φ1, Φ2be continuous mappings from S into X such that
(Φ1 + Φ2)S ⊂ S.
∥Φ1x1 − Φ1x2∥ ≤ k∥x1 − x2∥ for all x1, x2 ∊ S, where k is a constant and 0 ≤ k < 1.
To investigate the controllability of system (1), we assume the following conditions:
(H3) the resolvent operator R(t) is compact with ∥R(t)∥ ≤ M, for all t ≥ 0;
(H4) the linear operator W from L2(J, U) into L2(Ω;H), defined by
$$\begin{array}{}
\displaystyle
W = \int_0^b R(b- s)(Cu)(s)ds
\end{array}$$
has an induced inverse operator W−1 that takes values in L2(J, U)/KerW (see Carmichael and Quinn [7]) and there exist positive constants MC,MW such that
(H5) $\begin{array}{}
\displaystyle
F: J \times \mathfrak{B} \rightarrow H
\end{array}$ is a continuous function, and there exist a constant MF > 0 such that the function F satisfies the Lipschitz condition:
(H6) F and $\begin{array}{}
\displaystyle
h: J \times \mathfrak{B} \rightarrow H\,
\end{array}$ are continuous and there exists nonnegative constants $\begin{array}{}
\displaystyle
\,\overline{M}_{F}
\end{array}$, Mh such that
(H7) the function $\begin{array}{}
\displaystyle
g : J \times J \times \mathfrak{B} \rightarrow L(K, H)
\end{array}$ is continuous and there exists Mg ≥ 0 such that
exists and it is continuous. Further, there exists $\begin{array}{}
\displaystyle
\overline{M}_g
\end{array}$ such that $\begin{array}{}
\displaystyle
\| l(t)\|_Q \leq \overline{M}_g
\end{array}$;
Theorem 7.
In addition to hypotheses (H1)-(H8), assume that the following conditions are also satisfied
Let $\begin{array}{}
\displaystyle
\mathfrak{B}_b
\end{array}$ be the space of all functions x : (−∞, b] → H such that $\begin{array}{}
\displaystyle
x_0 \in \mathfrak{B}
\end{array}$ and the restriction x : J → H is continuous.
Let ∥ ⋅ ∥b be the seminorm in $\begin{array}{}
\displaystyle
\mathfrak{B}_b
\end{array}$ defined by
For any $\begin{array}{}
\displaystyle
z\in \mathfrak{B}_b^0
\end{array}$, we have
$$\begin{array}{}
\displaystyle
\| z \|_b= \| z_0\|_{\mathfrak{b}}+ \sup\{ \| z(s)\| :0\leq s \leq b\}= \{\sup \{ \| z(s) \|: 0\leq s \leq b \}
\end{array}$$
Thus if $\begin{array}{}
\displaystyle
Z_0^b = C(J_1, L_2(\Omega; \mathfrak{B}_0^b)
\end{array}$, then $\begin{array}{}
\displaystyle
(Z^b_0, \| \cdot \|_b)
\end{array}$ is a Banach space. Set
then $\begin{array}{}
\displaystyle
B_q \subseteq Z_b^0
\end{array}$ is uniformly bounded. For z(⋅) ∊ Bq, from axiom (ai) and hypothesis (H8), we remark that
where $\begin{array}{}
\displaystyle
\,\varepsilon=\| y_t - \phi \|_{\mathfrak{B}}^2.
\end{array}$
Thus, zt + yt ∊ Br[ϕ] for all 0 ≤ t ≤ b. Let the operator $\begin{array}{}
\displaystyle
\mathscr{Q} : Z_b^0 \rightarrow Z_b^0
\end{array}$ be defined by $\begin{array}{}
\displaystyle
\mathscr{Q}z
\end{array}$, by
$$\begin{array}{}
\displaystyle
{\mathscr {Q}}z(t) = \left\{ \begin{array}{*{20}{l}}
0\qquad {\kern 1pt} t \in {J_0} \\
R(t)F(0,\phi ) - F(t,{z_t} + {y_t}) + \int_0^t R (t - \eta )Cu_{z + y}^b(\eta )d\eta \\
\quad + \int_0^t R (t - s)\left[ {h(s,{z_s} + {y_s}) + l(s) + \int_0^s g (s,\tau ,{z_\tau } + {y_\tau })dw(\tau )} \right]ds,\quad t \in J. \\
\end{array}\right.
\end{array}$$
Obviously, the operator Φ has a fixed point is equivalently to prove that $\begin{array}{}
\displaystyle
\mathscr{Q}
\end{array}$ has a fixed point. For each positive number q, let
$$\begin{array}{}
\displaystyle
B_q= \{z \in Z_b^0 :z(0)=0, \| z\|^2_b \leq q, 0 \leq t \leq b \}\quad \text{ for some}\;\; q\geq 0
\end{array}$$
then for each q, $\begin{array}{}
\displaystyle
B_q \subseteq Z_b^0
\end{array}$ is clearly a bounded closed convex set. In addition to the familiar Young, Hölder, and Minkowskii inequalities, the inequality of the form $\begin{array}{}
\displaystyle
{\left( {\Sigma _{i = 1}^n{a_i}} \right)^m} \le {n^m}\Sigma _{i = 1}^na_i^m
\end{array}$ where ai are nonnegative constants (i = 1, 2,...,n) and $\begin{array}{}
\displaystyle
m, n \in \mathbb{N}
\end{array}$ is helpful to establishing various estimates. The Hölder inequality yields the following relation :
Next, we will show that the operator $\begin{array}{}
\displaystyle
\mathscr{Q}
\end{array}$ has a fixed point on Bq, which implies equation (1) has a mild solution. To this end, we decompose $\begin{array}{}
\displaystyle
\mathscr{Q}
\end{array}$ as $\begin{array}{}
\displaystyle
\mathscr{Q} = \mathscr{Q}_1 + \mathscr{Q}_2
\end{array}$, where the operators $\begin{array}{}
\displaystyle
\mathscr{Q}_1, \mathscr{Q}_2
\end{array}$ are defined on Bq, respectively, by
for t ∊ J. In order to apply the Nussbaum fixed point theorem for the operator $\begin{array}{}
\displaystyle
\mathscr{Q}
\end{array}$, we prove the following assertions:
$\begin{array}{}
\displaystyle
\mathscr{Q}_1
\end{array}$ and $\begin{array}{}
\displaystyle
\mathscr{Q}_2
\end{array}$ are well defined;
Hence $\begin{array}{}
\displaystyle
\mathscr{Q}B_q \subseteq B_q
\end{array}$. Next, we shall prove that the operator $\begin{array}{}
\displaystyle
\mathscr{Q}_1
\end{array}$ satisfies the Lipschitz condition, we take z(1), z(2) ∊ Bq, then for each t ∊ J and by condition (H3), equations (3.3) and (3.5), we have
and so $\begin{array}{}
\displaystyle
\mathscr{Q}_1
\end{array}$ satisfies Lipschitz condition with L0 < 1.
Finally, we prove that $\begin{array}{}
\displaystyle
\mathscr{Q}_2
\end{array}$ is relatively compact in Bq. To prove this, first we shall show that $\begin{array}{}
\displaystyle
\mathscr{Q}_2
\end{array}$ maps Bq into a precompact subset of $\begin{array}{}
\displaystyle
\mathscr{Q}
\end{array}$. We now show that for every fixed t ∊ J the set $\begin{array}{}
\displaystyle
V (t)= \{ (\mathscr{Q}_2z)(t) : z \in B_q\}
\end{array}$ is precompact in H.
Obviously for t = 0, $\begin{array}{}
\displaystyle
V(0)= \{ \mathscr{Q}(0)\}
\end{array}$. Let 0 < t ≤ b be fixed and ε be a real number satisfying ε ∊ (0, t). For z ∊ Bq, we define the operators
By Lemma 2 and the compactness of the operator R(ε), the set $\begin{array}{}
\displaystyle
V_\varepsilon ^*(t) = \{ (\mathfrak{Q}_2^{*\varepsilon }z)(t):z \in {B_q}\}
\end{array}$ is relatively compact in H, for every ε, ε ∊ (0; t). Moreover, also by Lemma 2, Hölder’s inequality, for each z ∊ Bq, we obtain
We obtain that the set $\begin{array}{}
\displaystyle
\,\,\tilde{V}^{*}_{\varepsilon} (t)= \{ (\mathscr{\tilde{Q}}^{*\varepsilon}_2z)(t) : z \in B_q\}
\end{array}$ is precompact in H by using the total boundedness.
when ε → 0, and there are precompact sets arbitrarily close to the set $\begin{array}{}
\displaystyle
\, \{ (\mathscr{Q}_2z)(t) : z \in B_q\}
\end{array}$. Thus the set $\begin{array}{}
\displaystyle
\,\{(\mathscr{Q}_2z)(t) : z \in B_q\}\,
\end{array}$ is precompact in H.
We now show that the image of $\begin{array}{}
\displaystyle
B_q, \mathscr{Q}(B_{q} )= \{ \mathscr{Q}z : z \in B_q\}
\end{array}$ is an equicontinuous family of functions. To do this, let ε > 0 small, 0 < t1 < t2, then from (10), we have
we see that $\begin{array}{}
\displaystyle
\|( \mathscr{Q}_2z)(t_1) - (\mathscr{Q}_2z)(t2)\|^2_Z
\end{array}$ tends to zero independently of z ∊ Bq as t2 → t1, with ε sufficiently small since the compactness of T (t) for t > 0 implies the continuity in the uniform operator topology. Hence, $\begin{array}{}
\displaystyle
\mathscr{Q}_2
\end{array}$ maps Bq into a equicontinuous family of functions.
Also $\begin{array}{}
\displaystyle
\mathscr{Q}_2(B_q)
\end{array}$ is bounded in Z and so by the Arzela–Ascoli theorem, $\begin{array}{}
\displaystyle
\mathscr{Q}_2(B_q)
\end{array}$ is precompact. Hence it follows from the Nussbaum fixed point theorem there exists a fixed point z(⋅) for $\begin{array}{}
\displaystyle
\mathscr{Q}
\end{array}$ on Bq such that $\begin{array}{}
\displaystyle
\mathscr{Q}z(t)=z(t)
\end{array}$. Since we have x(t) = z(t) + y(t), it follows that x(t) is a mild solution of (1) on J satisfying x(b) = x1. Thus the system (1) is controllable on J.
Example
In this section an example is presented for the controllability results to the following partial neutral stochastic integrodifferential equation:
To rewrite (12) into the abstract form of (1), we consider H = K = U = L2([0, π]) with the norm ∥⋅∥. Let $\begin{array}{}
\displaystyle
\,\, e_{n}:=\sqrt{\frac{2}{\pi} }\sin(nx),\;
\end{array}$, (n = 1, 2, 3, ···) denote the completed orthonormal basis in H and $\begin{array}{}
\displaystyle
{\kern 1pt} {\kern 1pt} w: = \Sigma _{n = 1}^\infty \sqrt {{\lambda _n}} {\beta _n}(t){e_n}({\lambda _n} > 0)
\end{array}$, where βn(t) are one dimensional standard Brownian motion mutually independent on a usual complete probability space $\begin{array}{}
\displaystyle
(\Omega ,{\mathscr F},{\{ {{\mathscr F}_t}\} _{t \ge 0}},).
\end{array}$
Define A : H → H by $\begin{array}{}
\displaystyle
\,A=\frac{\partial^{2}}{\partial z^{2}},\,
\end{array}$, with domain $\begin{array}{}
\displaystyle
D(A)=H^{2}([0,\pi])\cap H_{0}^{1}([0,\pi])\,
\end{array}$ where
Then $\begin{array}{}
\displaystyle
Ah = - \Sigma _{n = 1}^\infty {n^2} \lt \ h,{e_n} \gt {e_n},\:\:\:h \in D(A)\ ,
\end{array}$, where en, n = 1, 2, 3, ···, is also the orthonormal set of eigenvectors of A.
It is well-known that A is the infinitesimal generator of a strongly continuous semigroup on H, thus (H1) is true.
Let B : D(A) ⊂ H → H be the operator defined by B(t)(z) = γ(t)Az for t ≥ 0 and z ∊ D(A).
Here we take the phase space $\begin{array}{}
\displaystyle
\mathfrak{B}= C_0 \times L^2(q; H)
\end{array}$, which contains all classes of functions ϕ : J0 → H such that ϕ is $\begin{array}{}
\displaystyle
\mathfrak{F}_0
\end{array}$-measurable and q(⋅)∥ϕ(⋅)∥2 is integrable on J0 where q : (−∞, 0) → ℝ is a positive integrable function. The seminorm in $\begin{array}{}
\displaystyle
\mathfrak{B}
\end{array}$ is defined by
The general form of phase space $\begin{array}{}
\displaystyle
\mathfrak{B}= C_r \times L^p(q; H), r\geq 0, 1\leq p<\ \infty
\end{array}$ has been discussed in Hino et al. [15] (here in particular, we are taking r=0, p= 2). From Hino et al. [15], under some conditions, $\begin{array}{}
\displaystyle
(\mathfrak{B}, \|\phi \|_{\mathfrak{B}})
\end{array}$ is a Banach space that satisfies (i)-(iii) with
We suppose γ is a bounded and C1 function such that γ′ is bounded and uniformly continuous, which implies that the operator B(t) satisfies (H2). Consequently by Theorem 1, we deduce that Eq. (2) has a resolvent operator (R(t))t≥0 on H. Moreover, for $\begin{array}{}
\displaystyle
0\le s_1,s_2 \leq b, \psi_1, \psi_2 \in L_2(J, \mathfrak{B})
\end{array}$, we have from (i) by using Hölder inequality the following estimation
Similarly we can verify under conditions (ii) that F, G and h satisfy respectively the hypotheses (H6)-(H8). Therefore, under the above assumptions, the stated conditions of Theorem 7 are satisfied, the system (12) is controllable on J.