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In this paper, we apply the resolvent operator theory and an approximating technique to derive the existence and controllability results for nonlocal impulsive neutral integro‐differential equations with finite delay in a Hilbert space. To establish the results, we take the impulsive functions as a continuous function only, and we assume that the nonlocal initial condition is Lipschitz continuous function in the first case and continuous functions only in the second case. The main tools applied in our analysis are semigroup theory, the resolvent operator theory, an approximating technique, and fixed‐point theorems. Finally, we illustrate the main results with the help of two examples. [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]

In this paper, we study the concept of approximate controllability of retarded network systems of neutral type. On one hand, we reformulate such systems as free-delay boundary control systems on product spaces. On the other hand, we use the rich theory of infinite-dimensional linear systems to derive necessary and sufficient conditions for the approximate controllability. Moreover, we propose a rank condition for which we can easily verify the conditions of controllability. Our approach is mainly based on the feedback theory of regular linear systems in the Salamon-Weiss sense. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the approximate controllability problems of certain Sobolev type differential equations. Here, we obtain sufficient conditions for the approximate controllability of a semilinear Sobolev type evolution system in Banach spaces. In order to establish the approximate controllability results of such a system, we have employed the resolvent operator condition and Schauder's fixed point theorem. Finally, we discuss a concrete example to illustrate the efficiency of the results obtained. [ABSTRACT FROM AUTHOR]

The present study delineates the exact controllability at time T of infinite dimensional hyperbolic system with bilinear distributed controls. It consists in providing suitable controls that allow exact controllability of such a system to a target state with non null speed. It is an extension of results developed for null-speed target states Ouzahra (2014), to non null-speed target states. The approach is designed following two steps; first we give a control that ensures the approximate controllability to a target state at an intermediate time T1

In this manuscript, our primary focus on the approximate controllability outcomes for non‐densely defined Sobolev‐type Hilfer fractional neutral differential system with infinite delay. By applying the findings and facts associated with fractional theory and the fixed‐point method, the principal discussions are demonstrated. First, we focus on the existence and then discuss the approximate controllability. At last, we provide an example for the demonstration of theory. [ABSTRACT FROM AUTHOR]

By using fractional calculus, stochastic analysis theory and fixed point theorems, sufficient conditions for approximate controllability of nonlocal Sobolev-type neutral fractional stochastic differential equations with fractional Brownian motion and Clarke subdifferential are established. Finally, an example is given to illustrate the obtained results. [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]

In this paper, we establish the existence of solutions and the approximate controllability for the semilinear neutral differential control system under natural assumptions such as the local Lipschitz continuity of nonlinear term. First, we deal with the regularity of solutions of the neutral control system using fractional powers of operators. We also consider the approximate controllability for the semilinear neutral control equation, with a control part in place of a forcing term, using conditions for the range of the controller without the inequality condition as in previous results. [ABSTRACT FROM AUTHOR]

In this paper, we are considered with approximate controllability for a class of non-autonomous stochastic evolution equations of parabolic type with discrete nonlocal initial conditions. Some new results about existence of mild solutions as well as approximate controllability are established under more natural conditions on nonlinear functions and control operator by introducing a new Green function and using the theory of evolution family, Schauder fixed point theorem and the resolvent operator condition. At last, as a sample of application, these results are applied to a class of non-autonomous stochastic partial differential equation of parabolic type with discrete nonlocal initial conditions. The results obtained in this paper is a supplement to the existing literature and essentially extends some existing results in this area. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of non-autonomous nonlinear evolution equations in separable reflexive Banach spaces. First, we consider a linear problem and establish the approximate controllability results by finding a feedback control with the help of an optimal control problem. We then establish the approximate controllability results for a semilinear differential equation in Banach spaces using the theory of linear evolution systems, properties of resolvent operator and Schauder's fixed point theorem. Finally, we provide an example of a non-autonomous, nonlinear diffusion equation in Banach spaces to validate the results we obtained. [ABSTRACT FROM AUTHOR]

Abstract The objective of this paper is to investigate the approximate controllability of a semilinear impulsive stochastic system with nonlocal conditions and Poisson jumps in a Hilbert space. Nonlocal initial condition is a generalization of the classical initial condition and is motivated by physical phenomena. The results are obtained by using Sadovskii’s fixed point theorem. Finally, an example is provided to illustrate the effectiveness of the obtained result.

In this article, we study the approximate controllability for quasilinear differential equations with deviating arguments in a Hilbert space. We establish sufficient conditions for the existence of a mild solution. We also obtain sufficient conditions for the controllability of quasilinear equations. Further, we use it to establish the approximate controllability for quasilinear differential equations with deviating arguments. We discuss examples in which the analytical results are applied.

The paper is devoted to study the solvability of a solution for a Dirichlet problem for a class of time-fractional partial parabolic differential system, by using the energy inequality method for the primal problem. Then, by the Tikhonov regularization method, we can study the approximate controllability of the dual problem. We were therefore able to study the solvability of a solution and controlability of this problem. [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]

We consider the 3D Navier--Stokes system driven by an additive finite-dimensional control force. The purpose of this paper is to show how the approximate controllability of this system can be derived from the approximate controllability of the Euler system linearized around some suitable trajectory. The proof presented here is shorter than the previous ones obtained by Lie algebraic methods and gives some new information about the structure of the control. The dimension of the control space provided by this approach is larger, but it is still uniform with respect to the viscosity. [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]

In this paper, we suggest a proof method for the exact controllability of semilinear systems. Previously, it was applied in the context of the approximate controllability. In comparison to the traditional proof methods, the suggested method is easy because it does not require heavy estimations. Also, it is effective because it does not need some unnatural conditions, which are required for application of fixed‐point theorems. Using these advantages of the suggested method, we prove that a semilinear system in a Hilbert space, which satisfies general conditions for the existence of its solution, steers any initial state to any final state from the domain of the system operator if its linear part is exactly controllable at every noninitial time moment, the nonlinear term is bounded, and the norm of the inverse of the Gramian increases as time goes to zero not faster than the reciprocal function. The obtained result is demonstrated on examples. [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]

In this paper, a class of time-space fractional stochastic delay control problems with fractional noises and Poisson jumps in a bounded domain is considered. The proper function spaces and assumptions are proposed to discuss the existence of mild solutions. In particular, approximate strategy is used to obtain the existence of mild solutions for the problem with linear fractional noises; fixed point theorem is used to achieve the existence of mild solutions for the problem with nonlinear fractional noises. Finally, the approximate controllability of the problems with linear and nonlinear fractional noises is proved by the property of mild solutions. [ABSTRACT FROM AUTHOR]

In our manuscript, we organize a group of sufficient conditions of neutral integro-differential inclusions of Sobolev-type with infinite delay via resolvent operators. By applying Bohnenblust-Karlin's fixed point theorem for multivalued maps, we proved our results. Lastly, we present an application to support the validity of the study. [ABSTRACT FROM AUTHOR]

In this paper, we consider a parabolic PDE on a torus of arbitrary dimension. The nonlinear term is a smooth function of polynomial growth of any degree. In this general setting, the Cauchy problem is not necessarily well posed. We show that the equation in question is approximately controllable by only a finite number of Fourier modes. This result is proved by using some ideas from the geometric control theory introduced by Agrachev and Sarychev. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the approximate controllability of fractional evolution inclusions with hemivariational inequalities of order 1 < r < 2 . The main results of this paper are verified by using the fractional theories, multivalued analysis, cosine families, and fixed-point approach. At first, we discuss the existence of the mild solution for the class of fractional systems. After that, we establish the approximate controllability of linear and semilinear control systems. Finally, an application is presented to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider a control system for semilinear integrodifferential equations in Hilbert spaces with Lipschitz continuous nonlinear term. Our method is to find the equivalence of approximate controllability for the given semilinear system and the linear system excluded the nonlinear term, which is based on results on regularity for the mild solution. Finally, we give a simple example to which our main result can be applied. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Jilin University (Science Edition) / Jilin Daxue Xuebao (Lixue Ban) is the property of Zhongguo Xue shu qi Kan (Guang Pan Ban) Dian zi Za zhi She and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this study, we establish the approximate controllability of the Hilfer fractional neutral differential inclusions with infinite delay. The main result of this article is proven by applying some ideas of semigroup theory with fractional order and Dhage's fixed point theorem. Firstly, we prove the approximate controllability of the Hilfer fractional system. Next, we extend our study to the system with nonlocal conditions. Finally, theoretical and practical applications are presented to assist in the effectiveness of our discussion. [ABSTRACT FROM AUTHOR]

This paper addresses approximate controllability for a class of control systems represented by second order stochastic differential equations driven by Teugels martingales associated with a Lévy process. The main technique is the fundamental solution theory constructed through Laplace transformation. By using the so-called resolvent condition and cosine family of linear operators, stochastic analysis, and the technique of stochastic control theory, a new set of sufficient conditions for the approximate controllability of the considered second order stochastic differential system are formulated and proved. Due to the fundamental solution theory, the nonlinear terms are only required to be partly uniformly bounded. As an illustration of the applications of the obtained results, an example is also provided in the end. [ABSTRACT FROM AUTHOR]

This paper investigates the nonnegative approximate controllability for the one-dimensional degenerate heat equation governed by bilinear control. Both non-controllability and approximate controllability are studied for the system. If the control is restricted to act on a fixed domain, it is not controllable. If the control is allowed to mobile, it is approximately controllable. [ABSTRACT FROM AUTHOR]

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

In this article, we determine a set of appropriate conditions for approximate controllability of second‐order Sobolev‐type impulsive neutral differential evolution inclusions with infinite delay. We verify the main results by applying ideas about the cosine function, sine functions, and fixed point approach. Next, we continue the discussion to the nonlocal second‐order Sobolev‐type differential system. In the end, we provide a theoretical application to assist in the effectiveness of our discussion. [ABSTRACT FROM AUTHOR]

This manuscript is mainly focusing on the approximate controllability of Hilfer fractional neutral stochastic integro‐differential equation with infinite delay. We study our primary outcomes by utilizing the theoretical concepts related to the fractional calculus and Krasnoselskii's fixed point theorem. First, we prove the approximate controllability of the fractional evolution system. Then, we extend our results to the concept of nonlocal conditions. Last, we provide a theoretical application to assist in the effectiveness of the discussion. [ABSTRACT FROM AUTHOR]

This paper is concerned with existence, uniqueness, and approximate controllability of mild solutions to second-order non-autonomous stochastic impulsive differential systems. The impulsive differential systems may provide mathematical models to the phenomena having discontinuous jumps. The main results are carried out by using the generalized Banach contraction principle, evolution family, and stochastic analysis. Finally, an example is given to illustrate the results. [ABSTRACT FROM AUTHOR]

In the paper approximate controllability of second order infinite dimensional system with damping is considered. Applying linear operators in Hilbert spaces general mathematical model of second order dynamical systems with damping is presented. Next, using functional analysis methods and concepts, specially spectral methods and theory of unbounded linear operators, necessary and sufficient conditions for approximate controllability are formulated and proved. General result may be used in approximate controllability verification of second order dynamical system using known conditions for approximate controllability of first order system. As illustrative example using Green function approach approximate controllability of distributed dynamical system is also discussed. [ABSTRACT FROM AUTHOR]

An explicit saturating set consisting of eigenfunctions of Stokes operator in general 3D Cylinders is proposed. The existence of saturating sets implies the approximate controllability for Navier–Stokes equations in Cylinders under Lions boundary conditions. [ABSTRACT FROM AUTHOR]

In the paper, the problems of controllability and approximate controllability are studied for the control system wt = wxx, wx(0,⋅) = u, x > 0, t ∈(0,T), where u ∈ L∞(0,T) is a control. It is proved that each initial state of the system is approximately controllable to each target state in a given time T. A necessary and sufficient condition for controllability in a given time T is obtained in terms of solvability of a Markov power moment problem. It is also shown that there is no initial state which is null-controllable in a given time T. Orthogonal bases are constructed in H1 and H1. Using these bases, numerical solutions to the approximate controllability problem are obtained. The results are illustrated by examples. [ABSTRACT FROM AUTHOR]

In this article, we study an Euler-Bernoulli viscoelastic control system which is dissipative due to the presence of the viscoelastic term. The main feature which distinguishes this paper from other related works lies in the fact that we no longer impose traditional conditions such as complete monotonicity and decay property on the kernel function g. Without loss of generality, we study the system in the case of $g\equiv 1$. By means of the duality principle and the Hahn-Banach theorem, we show that the system with g=1 is approximately controllable in the appropriate function space.

In this study, time-delayed stochastic dynamical systems of linear and nonlinear equations are discussed. The existence and uniqueness of the stochastic semilinear time-delay system in finite dimensional space is investigated. Introducing the delay Gramian matrix, we establish some sufficient and necessary conditions for the relative approximate controllability of time-delayed linear stochastic dynamical systems. In addition, by applying the Banach fixed point theorem, we establish some sufficient relative approximate controllability conditions for semilinear time-delayed stochastic differential systems. Finally, concrete examples are given to illustrate the main results.

Abstract A shape optimization approach for the design of cavities with a specified wall shear stress profile is presented. Applications are the design of spin pack geometries with low and uniform residence times and without dead spaces to prevent polymer degradation for sensitive materials. The optimization uses a Surrogate Model based on the Newtonian Stokes equation as a simplification. An indirect objective based on wall shear stress is used to improve the residence time. The results are then validated on a realistic spin pack with the Full Model based on the non-Newtonian Navier–Stokes equation.

Abstract This paper is concerned with the approximate controllability and complete controllability of semilinear fractional functional differential systems with control involving Caputo fractional derivative. By using the operator semigroup theory and the fixed point theorem, we establish sufficient conditions for each of these types of controllability. The results are obtained under the assumption that the corresponding linear system is approximately controllable and completely controllable, respectively. In the end, an example is presented to illustrate the obtained theory.

Abstract Existence and controllability results for nonlinear Hilfer fractional differential equations are studied. Sufficient conditions for existence and approximate controllability for Sobolev-type impulsive fractional differential equations are established, where the time fractional derivative is the Hilfer derivative. An example for Sobolev-type Hilfer fractional delay partial differential equation with impulsive condition is considered.

In this paper, we study the approximate controllability of a class of semilinear systems governed by coupled degenerate parabolic equations. The equations may be weakly degenerate and strongly degenerate on a portion of the lateral boundary. We prove that the control systems are approximately controllable by using the Kakutani fixed point theorem and the controls can be taken to be of quasi bang-bang form. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the multidimensional heat equation governed by a bilinear control. We first investigate the finite-time stability and null controllability properties. Then, we study the global and partial approximate controllability. These results are further applied to establish the exact controllability of the system at hand. Nonnegative controllability results are also discussed. The approaches are based on linear semigroup theory and the controllability of the heat equation with additive controls. An other essential tool that we use here is the classical maximum principle. [ABSTRACT FROM AUTHOR]