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Our main concern is the existence of a new P C 2 - v -mild solution for Hilfer fractional impulsive evolution equations of order α ∈ (1 , 2) and β ∈ [ 0 , 1 ] as well as the approximate controllability problem in Banach spaces. Firstly, under the condition that the operator A is the infinitesimal generator of a cosine family, we give a new representation of P C 2 - v -mild solution for the objective equations by the Laplace transform and probability density function. Secondly, we rely on the Banach contraction mapping principle to discuss a new existence and uniqueness result of P C 2 - v -mild solution when the sine family is compact. Thirdly, a sufficient condition for the approximate controllability result of impulsive evolution equations is formulated and proved under the assumptions that the nonlinear item is uniformly bounded and the corresponding fractional linear system is approximately controllable. Finally, two examples are given to illustrate the validity of the obtained results in the application. [ABSTRACT FROM AUTHOR]

This paper focuses on the approximate controllability of Sobolev-type fuzzy Hilfer fractional neutral integro-differential inclusions involving the Clarke subdifferential. We establish the approximate controllability of this inclusion problem using fuzzy set theory, Sobolev-type methods, Fractional calculus, and Clarke subdifferentials. These techniques help to derive the necessary conditions for approximate controllability of the system. Dhage’s fixed point theorem is applied to prove the results related to approximate controllability. Finally, an example is provided to illustrate the discussion. [ABSTRACT FROM AUTHOR]

We study a class of dynamic control systems described by nonlinear fractional stochastic degenerate evolution equations in Hilbert spaces. We investigate the possibility of replacing a degenerate Cauchy problem by an equivalent problem in the factor space H ⊥ = H \ Ker (L) . The core of this paper is to factorize the degenerate Cauchy problem, to derive a mild solution of stochastic system based on the perturbation theory for linear operators and to study approximate controllability results. Using fixed point technique, fractional calculations, stochastic analysis technique and methods adopted directly from deterministic control problems, a new set of sufficient conditions for approximate controllability of fractional stochastic degenerate evolution equations is formulated and proved. In particular, we discuss the approximate controllability of nonlinear fractional stochastic degenerate control system under the assumption that the corresponding linear system is approximately controllable. As an application, the abstract results are illustrated by the application in complex media electromagnetic. In addition, the fractional stochastic partial differential equations (fractional SPDEs) are discussed and approximate controllability results are proved by verifying main assumptions. [ABSTRACT FROM AUTHOR]

Stochastic differential equations with Poisson jumps become very popular in modeling the phenomena arising in various fields, for instance in financial mathematics, where the jump processes are widely used to describe the asset and commodity price dynamics. The objective of this paper is to investigate the approximate controllability for a class of control systems represented by second-order stochastic differential equations with time delay and Poisson jumps. The main technique is the theory of fundamental solution constructed through Laplace transformation. By employing the so-called resolvent condition, theory of cosine operators and stochastic analysis, we formulate and prove some sufficient conditions for the approximate controllability of the considered system. In the end an example is given and discussed to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we aim to establish a set of sufficient conditions for the existence of an integral form mild solution and approximate controllability for a class of Sobolev-type Hilfer fractional stochastic differential systems driven by the Rosenblatt process and Poisson jumps. In the proposed control problem, we deal with a system that is under non-instantaneous impulsive effect. The sufficient condition for the existence of a mild solution for the proposed nonlinear control system has been established by using Schauder’s fixed point theorem. The approximate controllability results for the proposed control problem have been established under the consideration that the corresponding linear system is approximate controllable. By utilising stochastic analysis, the theory of resolvent operator, fractional calculus, and the fixed point technique, sufficient conditions have been established. At the end, an example is given to illustrate the abstract results. [ABSTRACT FROM AUTHOR]

In this paper, the approximate controllability problem is discussed for second-order neutral differential inclusions with damping in Banach spaces. The study establishes the existence of mild solutions for the given system by employing sine and cosine operators and applying Bohnenblust-Karlin’s fixed point theorem for multivalued maps. By introducing a new set of sufficient conditions, it has been proven that the given system is approximately controllable. An example is provided to illustrate the theory. [ABSTRACT FROM AUTHOR]

This article explores the existence and approximate controllability of integral solutions for Hilfer fractional evolution equations in a non-dense domain. Leveraging the well-known generalized Banach contraction theorem, we establish both the existence and uniqueness of the integral solution. Furthermore, we adopt a sequential approach to derive results related to approximate controllability, without relying on the compactness of semigroups or the uniform boundedness of nonlinear functions. To validate our findings, we present and discuss an illustrative example. [ABSTRACT FROM AUTHOR]

In this paper, we show the approximate controllability for a class of semilinear fractional stochastic systems in abstract space with the Riemann–Liouville fractional derivative. The key of the proof is the existence of the mild solution for the proposed problem. These results are based on new properties of the operator obtained by the subordination principle, compact semigroup and Schauder fixed point theorem. Here we obtain the compactness of the solution operator by using Arzelà–Ascoli theorem. As an application, we establish the approximate controllability of the stochastic Rayleigh–Stokes problem for a generalized second grade fluid. [ABSTRACT FROM AUTHOR]

In this paper, we aim to find a mild solution for a delay control system described by nonlinear fractional evolution differential equations in Banach spaces while being subjected to nonlocal conditions. Further, we explore the sufficient conditions for the approximate controllability of the nonlinear fractional control system, assuming that the associated linear system is approximately controllable. We provide some applications at the end to demonstrate our proposed results. [ABSTRACT FROM AUTHOR]

In this work, we study the approximate controllability problem for a system of nonlocal integro-differential equations with impulsive effects. We start by investigating the existence and uniqueness of solutions for this system. The results are derived using the theory of resolvent operators combined with fixed point theory in a generalized Banach space. Next, we examine approximate controllability without necessarily requiring the nonlinear terms to be uniformly bounded. In particular, we do not impose here the compactness condition for either the resolvent operator or the state-dependent function in the nonlocal condition, as is commonly found in the literature. Finally, we provide an example to demonstrate the abstract results of this work. [ABSTRACT FROM AUTHOR]

We study the approximate controllability of the discrete fractional systems of order 1 < α < 2 (∗) C ∇ α u n = A u n + B v n + f (n , u n) , n ≥ 2 , subject to the initial states u 0 = x 0 , u 1 = x 1 , where A is a closed linear operator defined in a Hilbert space X, B is a bounded linear operator from a Hilbert space U into X , f : N 0 × X → X is a given sequence and C ∇ α u n is an approximation of the Caputo fractional derivative ∂ t α of u at t n : = τ n , where τ > 0 is a given step size. To do this, we first study resolvent sequences { S α , β n } n ∈ N 0 generated by closed linear operators to obtain some subordination results. In addition, we discuss the existence of solutions to (∗) and next, we study the existence of optimal controls to obtain the approximate controllability of the discrete fractional system (∗) in terms of the resolvent sequence { S α , β n } n ∈ N 0 for some α , β > 0. Finally, we provide an example to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, the approximate controllability of semilinear integrodifferential degenerate Sobolev equations with nonlocal conditions is investigated in the sense of integral solution in Hilbert spaces. Some sufficient and necessary conditions are obtained. Firstly, the existence and uniqueness of integral solutions of semilinear integrodifferential degenerate Sobolev equations with nonlocal conditions are considered by GE-evolution operator theory and Sadovskii's fixed point theorem, the existence and uniqueness theorem of solutions is given. Secondly, the approximate controllability of semilinear integrodifferential degenerate Sobolev equations with nonlocal conditions is studied in the sense of integral solution. The criterion for approximate controllability is provided. The obtained results have important theoretical and practical value for the study of controllability of semilinear integrodifferential degenerate Sobolev equations with nonlocal conditions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the approximate controllability for Hilfer fractional stochastic hemivariational inequalities of order 1 < μ < 2 in Hilbert spaces. Initially, we define the concept of a mild solution for our problem in terms of fractional calculus, cosine families, stochastic analysis, and generalized Clarke subdifferential. Then, the existence and approximate controllability for Hilfer fractional stochastic evolution hemivariational inequalities are formulated and proven under appropriate conditions using fixed point theorems for multivalued maps. Finally, an example is presented to illustrate the theory. [ABSTRACT FROM AUTHOR]

The main study of this article is approximate controllability results for neutral stochastic hemivariational inequalities in Hilbert spaces. The mild solution is initially derived by using the theories on the semigroup, Clarke's subdifferential operators and fixed point theorem method. Initially we proved the existence of the mild solution for the neutral stochastic hemivariational inequalities and provided the approximate controllability of the considered system. Finally, a simple application also provided to illustrate the theory obtained. [ABSTRACT FROM AUTHOR]

This paper discusses the approximate controllability of Hilfer fractional stochastic differential system involving non-instantaneous impulses with Rosenblatt process and Poisson jumps. By utilising stochastic analysis, semigroup theory, fractional calculus, and Krasnoselskii's fixed point theorem, we prove our primary outcomes. Firstly, we prove the approximate controllability of the Hilfer fractional system. As a final step, we provide an example to highlight our discussion. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the approximate controllability of mild solutions for second-order differential systems. Using principles and ideas from the theory of the cosine family of operators and the fixed-point approach, we verify the existence of mild solutions for the given system. A new set of sufficient conditions is formulated and proved for the approximate controllability of second-order differential systems under the assumption that the associated linear part of the system is approximately controllable. In addition, we extend our system with nonlocal conditions. Our research on approximate controllability was also extended by utilizing impulse systems. To demonstrate the theory of the primary outcomes, an application is shown. [ABSTRACT FROM AUTHOR]

In this paper, we study the approximate controllability of damped elastic systems with initial conditions without the assumptions that the corresponding linear system is approximately controllable. Firstly, the existence of mild solution is obtained by means of contraction mapping principle and operator semigroup theory. Secondly, using the sequence method, a new set of sufficient conditions for approximate controllability of damped elastic systems are formulated and proved. The results in this paper are generalization and continuation of the recent results on this issue. Finally, as the application of abstract results, an example is given to illustrate our main results. [ABSTRACT FROM AUTHOR]

This paper presents the controllability of a family of linear and semilinear systems with control delay in Hilbert spaces. Firstly, the approximate controllability of the linear control delay system is proved by assuming that the linear system without control delay is completely controllable. Then, Nemytskii operators have been constructed associated to control operators and the nonlinear function. The approximate controllability of the retarded semilinear system is established by using the Rothe type fixed point theorem. The applications of results are explained through examples of parabolic and hyperbolic partial differential equations. [ABSTRACT FROM AUTHOR]

This manuscript is devoted to the approximate controllability of semilinear infinite-delayed stochastic differential inclusions involving impulses. The presence of a mild solution for the aforementioned system is shown by employing the theory of stochastic analysis and the fixed point theorem due to Covitz and Nadler. Then, we construct a group of sufficient conditions and utilize Grownwall's inequality to establish the approximate controllability results. Finally, two examples are constructed to exemplify the feasibility of derived outcomes. [ABSTRACT FROM AUTHOR]

This paper is concerned with the approximate controllability of a class of semi-linear neutral integro-differential equation with nonlocal condition. The basic tools of this study are the theory of resolvent operators of linear neutral integro-differential equation and fractional powers. The fundamental solution of linear neutral integro-differential equations is also constructed to deal with the non-uniform boundedness of the extra linear term. Sufficient conditions of approximate controllability are then obtained through the resolvent condition. Finally, an example is provided to illustrate the applications of the obtained results. [ABSTRACT FROM AUTHOR]

This manuscript is about infinite-delayed semilinear fractional differential systems. The considered system is of weighted type so that the initial condition may be non null at time zero. The principle contribution is the approximate controllability of the considered system. The state space is not required to be reflexive and the non linear function is not supposed to be Lipschitz. More than that, the nonlinear function in this work can involve spatial derivatives which enlarges the area of application of the obtained result. [ABSTRACT FROM AUTHOR]

In this article, we presented some sufficient conditions for the approximate controllability of a class of fractional differential systems with fixed delay in Banach space. The existence of a mild solution is discussed with the help of the fixed point theorem. Controllability results are obtained using generalized Gronwall's inequality, Cauchy sequence, and the basics of functional analysis. As a last note, we have included an example for the illustration of the obtained theory. [ABSTRACT FROM AUTHOR]

This manuscript considers a second-order semilinear control system in Hilbert space. By using Tikhonov regularization, we first construct the regularized control for the corresponding linear system and prove its approximate controllability. We then establish approximate controllability results using Gronwall's Lemma for the semilinear system. In addition, the error estimates and convergence rates involved in the regularization procedure are also obtained. Further, we give an example to demonstrate the results. [ABSTRACT FROM AUTHOR]

This article addresses the boundary controllability problem for a class of third order in time PDE, known as Moore–Gibson–Thompson equation, with a control supported on the boundary. It is shown that it is not spectrally controllable, which means that nontrivial finite linear combination of eigenvectors can be driven to zero in finite time. This implies that the Moore–Gibson–Thompson equation is not exact and null controllable. However, the approximate controllability will be proved. [ABSTRACT FROM AUTHOR]

We consider the approximate controllability for a class of second-order impulsive neutral stochastic differential equations with state-dependent delay and Poisson jumps in a real separable Hilbert space. Under the sufficient conditions, we obtain approximate controllability results by virtue of the theory of a strongly continuous cosine family of bounded linear operators combined with stochastic inequality technique and the Sadovskii fixed point theorem. Finally, we illustrate the main results by an example. [ABSTRACT FROM AUTHOR]

This article investigates the approximate controllability of second order non-autonomous functional evolution equations involving non-instantaneous impulses and nonlocal conditions. First, we discuss the approximate controllability of second order linear system in detail, which lacks in the existing literature. Then, we derive sufficient conditions for approximate controllability of our system in separable reflexive Banach spaces via linear evolution operator, resolvent operator conditions, and Schauder’s fixed point theorem. Moreover, in this paper, we define proper identification of resolvent operator in Banach spaces. Finally, we provide two concrete examples to validate our results. [ABSTRACT FROM AUTHOR]

This article is primarily targeting the approximate controllability of nonlocal Atangana–Baleanu fractional derivative by Mittag-Leffler kernel to stochastic differential systems. In particular, we obtain a new set of sufficient conditions for the approximate controllability of nonlinear Atangana–Baleanu fractional stochastic differential inclusions under the assumption that the corresponding linear system is approximately controllable. In addition, we establish the approximate controllability results for the Atangana–Baleanu fractional stochastic control system with infinite delay. The results are obtained with the help of the fixed-point theorem for multivalued operators and fractional calculus. At last, an example is included to show the applicability of our results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Korteweg-de Vries equation posed on the periodic domain T . We show that the Korteweg-de Vries equation is globally approximately controllable by a two dimensional external force. The proof is based on the Agrachev-Sarychev approach in geometric control theory. [ABSTRACT FROM AUTHOR]

This paper investigates the issue of existence and approximate controllability results for impulsive fractional differential inclusions with delay of order r ∈ (1 , 2 ] in Banach space. To begin, we analyze existence results for impulsive fractional evolution inclusions with delay using fractional calculations, the r-order cosine family, multivalued maps, and Martelli’s fixed point theorem. The approximate controllability results for impulsive fractional evolution inclusions with delay were then derived using Gronwall’s inequality and the sequence method. Then, we investigate the Sobolev fractional integrodifferential inclusions with finite delay. Moreover, we develop the nonlocal conditions in a given system. Finally, an example is presented to illustrate the main results. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the approximate controllability for mixed type non-autonomous fractional differential equations. By using the Schauder fixed point theorem and two parameters β - resolvent family, we prove the approximate controllability of the control system. Last of all, the application of our main results is given. [ABSTRACT FROM AUTHOR]

This paper deals with the approximate controllability problem for a class of linear neutral evolution systems with infinite delay in Banach spaces. Since the concerned system involves additional linear terms, the fundamental solution theory is used to describe the mild solutions. Based on the spectral analysis and the adjoint system, sufficient conditions for the approximate controllability of the concerned system are established. The rank condition is also shown when the control space is finite-dimensional. A practical example is given to show the application of the obtained results. [ABSTRACT FROM AUTHOR]

In this manuscript, the approximate controllability is encountered for a system involving instantaneous and non-instantaneous impulses. The theory of semigroup and Sadovskii's fixed point theorem are used to prove our main results. In the direction of controllability for a functional differential equations, a problem consists of either instant impulses or non-instant impulses separately. This is the reason that makes the study of the considered problem more interesting. We also make an observation related to high blood sugar level that is quite natural to relate with our problem. In the end, an example is given to validate the developed theory. [ABSTRACT FROM AUTHOR]

To characterize the ultra-slow diffusion processes with time-dependent logarithmical law attenuation, the distributed order fractional diffusion equation is needed. This paper discusses an analysis of approximate controllability from the exterior of distributed order fractional diffusion problem with the fractional Laplace operator subject to the non-zero exterior condition. We first establish some well-posedness results, such as the existence, uniqueness and regularity of the solutions allowing the weighted function μ that may be non-continuous. Especially, we show that the solutions can be represented by the series for the integral of a real-valued function. After giving the unique continuation property of the adjoint system, approximate controllability of the system is also included. [ABSTRACT FROM AUTHOR]

The main motivation of our conversation is the approximate controllability of Hilfer fractional neutral evolution hemivariational inequalities. Using fractional calculus, the theory of operators semigroup and the probability density function, we first construct a new C 1 - β mild solution for Hilfer fractional differential inclusion. Secondly, we prove the approximate controllability of Hilfer fractional evolution hemivariational inequalities to linear and semilinear systems using characteristic solution operators and fundamental features via a fixed point theorem for multi-valued mappings. Finally, two examples are provided to demonstrate our theory. [ABSTRACT FROM AUTHOR]

This paper is devoted to a class of nonlocal stochastic differential equations involving conformable fractional time derivative operator. Existence of mild solution for the nonlocal conformable stochastic differential equation is proved. Sufficient conditions for controllability and approximate controllability of the nonlocal conformable stochastic differential equation with impulsive conditions are established. An example is given to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

This work is to study a control problem for a nonlinear Volterra nonautonomous evolution inclusion subjected to a local initial condition. The evolution inclusion involves a family of (possibly unbounded) operators. It is proved that the solution set for the control problem has Aronszajn-type regularity (also called Rδ-structure). Moreover, the reachability set for the control problem is invariant under nonlinear perturbations. Then, we obtain that the control problem is approximately controllable under some supplementary conditions. We finally present an example to illustrate the applicability of the abstract results. [ABSTRACT FROM AUTHOR]

The three concepts of exact, null and approximate controllabilities are analyzed from the exterior of the Moore–Gibson–Thompson equation associated with the fractional Laplace operator subject to the nonhomogeneous Dirichlet type exterior condition. Assuming that b > 0 and α - τ c 2 b > 0 , we show that if 0 < s < 1 and Ω ⊂ R N ( N ≥ 1 ) is a bounded domain with a Lipschitz continuous boundary ∂ Ω , then there is no control function g such that the following system τ u ttt + α u tt + c 2 (- Δ) s u + b (- Δ) s u t = 0 in Ω × (0 , T) , u = g χ O in (R N \ Ω) × (0 , T) , u (· , 0) = u 0 , u t (· , 0) = u 1 , u tt (· , 0) = u 2 in Ω , is exactly or null controllable in time T > 0 . However, we prove that for 0 < s < 1 , the system is approximately controllable for every g ∈ H 1 ((0 , T) ; L 2 (O)) , where O ⊂ R N \ Ω ¯ is an arbitrary non-empty open set. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the approximate controllability for second-order impulsive differential inclusions with infinite delay. For proving the main results, we use the results related to the cosine and sine function of operators, Martelli's fixed point theorem, and the results when combined with the properties of differential inclusions. Firstly, we prove the approximate controllability for second-order impulsive differential inclusions with initial conditions. Then, we extend the discussion to the second-order impulsive system with nonlocal conditions. Finally, we provide an example for the illustration of the obtained theoretical results. [ABSTRACT FROM AUTHOR]

By a procedure of successive projections, the authors decompose a coupled system of wave equations into a sequence of sub-systems. Then, they can clarify the indirect controls and the total number of controls. Moreover, the authors give a uniqueness theorem of solution to the system of wave equations under Kalman's rank condition. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the existence of mild solutions and approximate controllability for measure driven evolution system with nonlocal conditions. First, we obtain the existence of mild solutions and approximate controllability for the concerned problem by constructing new Green's function and control function involving Gramian controllability operator. This research relies on Schauder's fixed point theorem and the theory of operator semigroup. At the end, two examples are also given to illustrate the feasibility of our theoretical results. [ABSTRACT FROM AUTHOR]

This work studies the approximate controllability of a class of second-order retarded semilinear differential equations with nonlocal conditions and with delays in control. First, we deduce the existence of mild solutions using cosine family and fixed point approach. For this, the nonlinear function is supposed to be locally Lipschitz. Controllability of the system is shown using an approximate and iterative technique. The results are illustrated using an example. [ABSTRACT FROM AUTHOR]

In this paper, noninstantaneous impulsive conformable fractional stochastic delay integro-differential system driven by Rosenblatt process is studied. Sufficient conditions for approximate controllability and null controllability for the considered problem are established. Finally, an example is introduced to explain the obtained results. [ABSTRACT FROM AUTHOR]

We study the approximate controllability of parabolic-elliptic coupled systems involving the fractional Laplacian (− Δ) s , s ∈ (0 , 1) . The control is located on a non-empty open subset O of the complementary of the open bounded domain Ω in R N . For this reason, the approximate controllability is said to be exterior. This concept was recently introduced by Warma M. for the fractional diffusion equation. For this purpose, we first prove the existence and uniqueness of the series solution of the studied systems and their dual. Then we state a unique continuation principle for the dual equation which follows from a unique continuation property for the eigenvalues of (− Δ) s with the homogeneous exterior Dirichlet condition. Finally, we show that for any s ∈ (0 , 1) and any control in D (O × (0 , T)) , under certain conditions on the coefficients, the approximate controllability at any time T > 0 holds. [ABSTRACT FROM AUTHOR]

This article studies the approximate controllability for a class of fractional control system with analytic semigroup governed by differential equations with Hilfer fractional derivatives of order δ ∈ (0 , 1) and type ζ ∈ [ 0 , 1 ] in a Banach space. The existence and uniqueness of the mild solution is established with the help of semigroup theory, fractional power of operators and a generalized contraction type fixed point theorem. Further, a set of sufficient conditions is formulated for the approximate controllability of the system under consideration. The result obtained holds irrespective of whether the generated semigroup is compact or non-compact. [ABSTRACT FROM AUTHOR]

The approximate controllability of second-order integro-differential evolution control systems using resolvent operators is the focus of this work. We analyze approximate controllability outcomes by referring to fractional theories, resolvent operators, semigroup theory, Gronwall's inequality, and Lipschitz condition. The article avoids the use of well-known fixed point theorem approaches. We have also included one example of theoretical consequences that has been validated. [ABSTRACT FROM AUTHOR]

In this paper, we study the approximate control problem from the exterior of a nonlocal equation of Sobolev–Galpern type, specifically the Barenblatt–Zheltov–Kochina equation, involving the fractional Laplace operator of order . We prove that the system under consideration is approximate controllable at any time . [ABSTRACT FROM AUTHOR]

This paper deals with the controllability of the second-grade fluids, a class of non-Newtonian of differential type, on a two-dimensional torus. Using the method of Agrachev and Sarychev (J. Math Fluid Mech., 7(1):108–52 (2005)), Agrachev and Sarychev (Commun Math Phys., 265(3):673–97 (2006)), and of Shirikyan (Commun Math Phys., 266(1):123–51 (2006)), we prove that the system of second-grade fluids is approximately controllable by a finite-dimensional control force. [ABSTRACT FROM AUTHOR]

In this work, we study the approximate controllability for a class of semilinear second-order control systems with finite delay. Sufficient conditions for approximate controllability are established by constructing fundamental solutions and using the resolvent condition and techniques on cosine family of linear operators. To illustrate the applications of the obtained results, an example is provided in the end. [ABSTRACT FROM AUTHOR]

This paper addresses some results about mild solution to time-fractional Navier–Stokes equations with bounded delay existed in the convective term and the external force. Based on the Schauder's fixed point theorem, we establish the sufficient conditions for the existence and uniqueness of the global mild solution in Banach spaces. Moreover, the polynomial decay estimate of the mild solution is given. Furthermore, the approximate controllability for the mild solution is obtained by constructing a Cauchy sequence. [ABSTRACT FROM AUTHOR]

This paper is devoted to a studying of the controllability properties for the Coleman–Gurtin-type equation, which is a class of multidimensional integral–differential equations. The goal is to prove the existence of a control function which steers the state variable and the integral term to the neighborhood of two given final configurations at the same time, respectively. This new approximate controllability is defined by imposing some additional integral-type constraints on the usual approximate controllability, ensuring that the whole process reaches the neighborhood of the equilibrium. We also provide a characterization of the initial values, which can be driven to zero by a distributed control. The later is a supplement of non-null controllability for the Coleman–Gurtin model in the square integrable space. [ABSTRACT FROM AUTHOR]