Controllability of Impulsive Semilinear Stochastic Heat Equation with Delay

LaSalle wrote the following: “it is never possible to start the system exactly in its equilibrium state, and the system is always subject to outside forces not taken into account by the differential equations. 'e system is disturbed and is displaced slightly from its equilibrium state. What happens? Does it remain near the equilibrium state? 'is is stability. Does it remain near the equilibrium state and in addition tend to return to the equilibrium? 'is is asymptotic stability.” Continuing with what LaSalle said, we conjecture that real-life systems are always under the influence of impulses, delays, memory, nonlocal conditions, and noises, which are intrinsic phenomena no taken into account by the mathematical model that is representing by a differential equation. For many control systems in real life, delays, impulses, and noises are natural properties that do not change their behavior. So, we conjecture that, under certain conditions, the abrupt changes, delays, and noises as perturbations of a system do not modify certain properties such as controllability. In this regard, we prove the interior S∗-controllability of the semilinear stochastic heat equation with impulses and delay on the state variable, and this is done by using new techniques avoiding fixed point theorems employed by Bashirov et al.


Introduction
In this paper, we prove the interior approximate S * -controllability of the semilinear stochastic heat equation with multiplicative noise, impulses, and delay on the state variable. is is done by using the result from Leiva [1]; Acosta where Γ is a bounded domain in R d (d � 1, 2, 3), zΓ denotes its boundary, θ is an open nonempty subset of Γ, 1 θ denotes the characteristic function of the set θ, u is a control processes R-valued, the noise _ m is a colored noise R valued on [0, τ] × R d with spatial correlation q(ξ, η) given by q(ξ, η) � (t∧s) − 1 E(m(t, ξ)m(s, η)), ξ, η ∈ Γ, t, s ∈ [0, τ], (2) and ϕ(·, x) is a R-valued F m 0 -measurable random variable with respect to filtration e nonlinear terms f, g, I k : [0, τ] × R × R ⟶ R are smooth enough such that system (1) admits unique mild solutions for each control u and satisfies the following property: e term white-noise is denoted by _ m � (zm/zt), where m is a Gaussian process m � (t, A): t ∈ [0, τ], A ∈ B b (R d ) with zero mean and covariance given by (2). e noise m behaves as a Brownian motion with respect to the time variable, and it has a correlated spatial covariance.
ere are many practical examples of impulsive control systems which are modeled by impulsive differential equations (for more information, see the monographs: Samoilenko and Perestyuk [6]; Franco and Nieto [7]; Sun and Zhang [8]; Lakshmikanthan, Bainov and Simeonov [9]; He and Yu [10]; Luo and Shen [11]). e controllability of impulsive evolution equations has been studied recently for several authors, but most them study the exact controllability only (to mention, Radhakrishnan and Balachandran [12]; Chalishajar [13]; Selvi and Mallika [14]). To our knowledge, there are a few works on approximate controllability of impulsive semilinear evolution equations (to mention, Chen and Li [15] and Sakthivel and Anandhi [16]). Recently, in the study of Carrasco, Leiva, Sanchez, and Tineo [17]; Leiva [1]; Leiva and Merentes [18], the approximate controllability of semilinear evolution equations with impulses has been studied applying Rothe's fixed point theorem. Contrained controllability of finite-dimensional semilinear systems with delayed controls has been studied by Klamka [19,20] where the author gives sufficient conditions for contrained local relative controllability applying a generalized open mapping theorem. Also, Klamka [21] gave necessary and sufficient conditions for different kinds of stochastic relative controllability in a given time interval which are proved for stochastic finite-dimensional linear systems with multiple delays in control. e existence of solutions for impulsive evolution equations with delays has been studied by Hernandez, Sakthivel, and Tamaka [22]; Abada, Benchohra, and Hammouche [23]; Shikharchan and Baburao [24] and Chang [25,26]. Besides, impulsive and stochastic effects appear in real-life systems. Moreover, a lot of dynamical systems have structure variables subject to stochastic abrupt changes, which may result from sudden phenomena such as stochastic failures and repair of components, quick environmental changes, and changes in the interconnections of subsystems (see Mao [27]). In the stochastic context, we can mention some papers related to impulsive and delay stochastic systems: Lijuan, Junping, and Jitao [28]; Sakthivel [16]; Sukavanam and Kumar [29]; Parthasarathy and Sathya [30]. e exact and approximate controllability is known for determinist systems; but the exact controllability was introduced as a concept for linear finite-dimensional systems by Kalman in the 50s. Nevertheless, the extension of this concept to infinite dimensional systems is too strong. erefore, the approximate controllability was introduced as a weakened version of the exact controllability. However, the exact and approximate controllability cannot be a property of stochastic systems, and this needs to be a weaker concept than the approximate controllability concepts in order to extend them to the stochastic systems. en, the concept of the S-controllability is introduced. A control system is S-controllable, if given an arbitrary ϵ > 0, it is possible to steer from the point z 0 to within a distance � ε √ from all points in the state space Z at time τ with probability close to one. e approximate controllability and S-controllability concepts are equivalent for the linear system but are different for nonlinear stochastic systems. is concept and generalization are defined in Bashirov et al. [5,31]. In this context, we used the S * -controllability which is a weaker version of S-controllability.
e main objective of this article is to prove the interior S * -controllability of the semilinear stochastic heat equation with impulses, delay, and multiplicative noises (1) simultaneously, under appropriate conditions presented above. For this, we apply the new technique presented in Bashirov et al. [3,4,31,32]. In the literature, S-controllability for such systems, only a few works such as Bashirov and an article by Sukavanam and Kumar [29], has been reported.

Preliminaries
In this section, we introduce notations, definitions, and preliminaries which are used to write (1) as an abstract differential equation.
Let Z, U, and K be separable Hilbert spaces and (Ω, F, P) be a complete probability space with a probability { } be a Wiener processes with values in K and covariance nonnegative operator Q ∈ L(K) (L(K) is the space of bounded linear operators on K). If the control system is stochastic, we denote by F m t the smallest σ-field generated by m(s): 0 ≤ s ≤ τ { }. We assume that there exists a complete orthonormal set ξ n , n � 1, 2, . . . , in K and a bounded sequence of nonnegative real numbers ρ n such that Qξ n � ρ n ξ n with Tr(Q) � ∞ n�1 ρ n < ∞. Let β n (t), n � 1, 2, . . . , be a sequence of real-valued one-dimensional standard Brownian motions mutually independent over (Ω, F, P) such that E · { } denotes the expectation of a random variable and E(x|·) denotes the conditional expectation of x. Let denote the Hilbert space of all square-integrable and F m t -measurable processes with values in Z with topology given by the norm: is the family U-valued measurable and F m t -adapted processes with norm topology given by We consider the function We shall denote by C the set consisting of all F m 0 -measurable bounded random processes ϕ with value in Z: satisfying sup − r≤s≤0 When the control system is stochastic and completely observable, then F m t is a natural filtration of Bashirov et al. [5]). So, we shall consider the following notation: where Definition 2 (see [5]). e stochastic semilinear control system (1) is S * -controllable if and only if, for every initial

Abstract Formulation of the Problem
is section is devoted to set system (1) as an abstract control system in a suitable Hilbert space. To this end, we recall that the operator A � − Δ with Dirichlet boundary condition in Z � L 2 (Γ) has the following spectral decomposition 0 < λ 1 < λ 2 < · · · < λ j ⟶ ∞, where λ j denotes the eigenvalues of A, each one with finite multiplicity c j equal to the dimension of the corresponding eigenspace. erefore, the following properties for A hold: where 〈·, ·〉 is the inner product in Z, ϕ j,k is a complete orthonormal set of eigenvectors of A, and So, E j is a family of complete orthogonal pro- erefore, system (1) can be written as abstract functional differential equations with impulses and noses (see Acosta-Leiva [33].
International Journal of Differential Equations 3

Approximate Controllability of the Linear Heat Equation
Since the associated linear stochastic heat equation is approximately controllable in any interval of the form [τ − δ, τ], with 0 < δ ≤ τ, we shall recall some properties and characterizations of the approximate controllability of linear deterministic evolution equations and linear stochastic evolution equations. In this regard, we consider the corresponding linear stochastic heat equation: Note that, for all z 0 random variable F 0 -measurable and admits only one mild solution given by We also consider the deterministic system corresponding to (20), and for all y 0 ∈ Z and u ∈ L 2 ([0, τ], U), the initial value problem admits only one mild solution given by Definition 3. e stochastic linear system (20) is said to be approximately controllable on [0, τ] if for every initial state z 0 ∈ F 0 and final state z 1 ∈ L 2 (Ω, F τ , Z) and any ϵ > 0 there exists a control u ∈ L F 2 ([0, τ], U), Z � U � L 2 (Γ), such that the mild solution of (20) z(·) corresponding to u verifies where It is known that approximately controllability of the stochastic linear system (20) and deterministic linear system (22) for linear infinite dimensional systems are equivalent (see Mahmudov [34]). Now, we define the following operator.
Definition 4 (see [33]). For system (22), we define the following concept: the controllability maps satisfy the following relation: 4 International Journal of Differential Equations e controllability operators Q δ : Z ⟶ Z are given by e following lemma holds in general for a linearbounded operator G: W ⟶ Z between Hilbert spaces W and Z(see Bashirov et al. [5]; Curtain and Pritchard [35]; Curtain and Zwart [36] and Leiva et al. [37]).

Lemma 1 e following statements are equivalent to the approximate controllability of the linear system (20) on
where u α � G * τδ (αI + Q τδ ) − 1 z, α ∈ (0, 1]. So, lim α⟶0 G τδ u α � z and the error E τδ z of this approximation is given by the formula: and the family of linear operators Γ ατδ : Z ⟶ W, defined for 0 < α ≤ 1 by is an approximate inverse for the right of the operator G τδ , in the sense that in the strong topology. (22) is approximately controllable on [τ − δ, τ]. Moreover, given an initial state y 0 ∈ Z and a final state z 1 , we can find a sequence of controls

Lemma 2. Q τδ > 0 if and only if linear system
such that the solutions y(t) � y(t, τ − δ, y 0 , u α ) of the initial value problem i.e.,

S * -Controllability of the Semilinear Stochastic System
In this section, we shall prove the main result of this paper, the interior S * -controllability of the heat equation with impulses, delay, and multiplicative noise (1), which is equivalent to prove the S * -controllability of system (16). To this end, for all ϕ ∈ C and u ∈ L F 2 ([0, τ], U), the initial value problem International Journal of Differential Equations 5 admits only one mild solution given by with 0 ≤ t ≤ τ.
Proposition 2 (see [37]). If Range(G) � Z, then Now, we are ready to present and prove the main result of this paper, the S * -controllability of the semilinear heat equation with impulses, delay, and multiplicative noise.