Sufficient Conditions For Approximate Controllability Research Articles

A study on approximate controllability of evolution systems with nonlocal conditions

The purpose of this paper is to present the existence of mild solutions and approximate controllability for a class of non-autonomous measure driven evolution systems with nonlocal conditions in Banach spaces. Firstly, we obtain the existence of mild solutions for the concerned problem by the semigroup theory and Schauder's theorem. Secondly, some sufficient conditions of approximate controllability are proved. At the end, an example is also given to illustrate the feasibility of our theoretical results.

A study on approximate controllability of linear impulsive equations in Hilbert spaces

In this paper, we study an approximate controllability for the impulsive linear evolution equations in Hilbert spaces. We give a representation of solution in terms of semigroup and impulsive operators. We present the necessary and sufficient conditions for approximate controllability of linear impulsive evolution equation in terms of impulsive resolvent operator. An example is provided to illustrate the application of the obtained results.

Approximate controllability of impulsive evolution stochastic functional differential equations driven by a fractional Brownian motion

In this paper, we study the approximate controllability of certain class of impulsive evolution stochastic functional differential equations, with variable delays, driven by a fractional Brownian motion in a separable real Hilbert space. We derive a new set of sufficient conditions for approximate controllability using a stochastic analysis of fractional Brownian motion with Hurst parameter H ∈ (1/2,1) and a Schaefer's fixed point theorem. An example is considered at the end of the paper to illustrate the obtained abstract results.

Study approximate controllability and null controllability of neutral delay Hilfer fractional stochastic integrodifferential system with Rosenblatt process

In this paper, we investigate the sufficient conditions for approximate controllability and null controllability of neutral delay stochastic Hilfer fractional integrodifferential system with Rosenblatt process are established. The required results are obtained based on a fixed point technique, fractional calculus and stochastic analysis. Finally, an example is given to illustrate the obtained results.

Controllability under positivity constraints of a size‐structured population model with delayed birth process

In this paper, we study the approximate controllability, under positivity constraints on the control and state, of a size‐structured population system with time‐delayed behavior. In this model, the birth term is non‐local in size and time, which describes the time‐delay in the recruitment process in newborn individuals population. Our control function is localized at birth and corresponds to a supply of newborn individuals. First, we establish the well‐posedness and positivity property of the model. Second, and more importantly, we derive necessary and sufficient conditions for approximate controllability under positivity constraints. The method relies on the feedback theory of infinite‐dimensional positive systems.

Approximate Controllability of Non-autonomous Second Order Impulsive Functional Evolution Equations in Banach Spaces

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.

Approximate controllability of neutral delay integro-differential inclusion of order $ \alpha\in (1, 2) $ with non-instantaneous impulses

<p style='text-indent:20px;'>This paper aims to establish the approximate controllability results for fractional neutral integro-differential inclusions with non-instantaneous impulse and infinite delay. Sufficient conditions for approximate controllability have been established for the proposed control problem. The tools for study include the fixed point theorem for discontinuous multi-valued operators with the <inline-formula><tex-math id="M3">\begin{document}$ \alpha- $\end{document}</tex-math></inline-formula>resolvent operator. Finally, the proposed results are illustrated with the help of an example.</p>

Hilfer fractional neutral stochastic partial differential equations with delay driven by Rosenblatt process

This manuscript is mainly focusing on the approximate controllability and the null controllability for fractional neutral stochastic partial differential equations with delay driven by Rosenblatt process. We discuss the sufficient conditions for approximate controllability and null controllability for Hilfer fractional neutral stochastic partial differential equations driven by Rosenblatt process. Finally, we provide two examples to verify the obtained results.

Approximate controllability of network systems

<p style='text-indent:20px;'>In this paper, the rich feedback theory of regular linear systems in the Salamon-Weiss sense as well as some advanced tools in semigroup theory are used to formulate and solve control problems for network systems. In fact, we derive necessary and sufficient conditions for approximate controllability of such systems. These criteria, in some particular cases, are given by the well-known Kalman's controllability rank condition.</p>

Controllability for abstract semilinear control systems with homogeneous properties

This paper considers the approximate controllability for a class of abstract semilinear control systems using a method called surjectivity theorems for nonlinear operator equations under restrictive assumptions. The sufficient conditions for approximate controllability are derived under the natural assumptions on the nonlinear terms, which is an odd homogeneous and the nonlinear inverse considered as a multivalued mapping is bounded. An example is given to illustrate our main results.

A study on impulsive fractional hybrid evolution equations using sequence method

In this paper, we introduce a new concept called $$\alpha $$ -order cosine-resolvent family, by using the theory of fractional calculus, the concepts of measure of noncompactness and Hybrid fixed point theorem, we consider the existence of PC-mild solutions for a class of impulsive fractional hybrid evolution equation in a Banach space . Furthermore, we obtain some sufficient conditions for approximate controllability of our concern problem. At the end, an example is given to illustrate the feasibility of our results.

Approximate controllability of second order nonlocal neutral differential evolution inclusions

Abstract In our manuscript, we organize a group of sufficient conditions of approximate controllability for second order nonlocal neutral differential evolution inclusions. Next, we develop the result to analyze approximate controllability of impulsive systems. Lastly, a model is presented for illustration of theory.

Approximate controllability of semi-linear stochastic integro-differential equations with infinite delay

Abstract In this work, by constructing fundamental solutions and using the theory of resolvent operators and fractional powers of operators, we study the approximate controllability of a class of semi-linear stochastic integro-differential equations with infinite delay in $L_p$ space ($2<p<\infty $). Sufficient conditions for approximate controllability of the discussed equations are obtained under the assumption that the associated deterministic linear system is approximately controllable. An example is provided to illustrate the obtained results.

Approximate controllability of semilinear impulsive functional differential systems with non-local conditions

AbstractThis paper is concerned with the approximate controllability of semilinear impulsive functional differential systems in Hilbert spaces with non-local conditions. We establish sufficient conditions for approximate controllability of such systems via resolvent operator and Schauder’s fixed point theorem. An application involving the impulse effect associated with delay and non-local conditions is presented to verify our claimed results.

Neutral fractional stochastic partial differential equations with Clarke subdifferential

ABSTRACT 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.

APPROXIMATE CONTROLLABILITY OF SECOND-ORDER SEMILINEAR EVOLUTION SYSTEMS WITH STATE-DEPENDENT INFINITE DELAY

In this article, we study the problem of approximate controllability for a class of semilinear second-order control systems with state-dependent delay. We establish some sufficient conditions for approximate controllability for this kind of systems by constructing fundamental solutions and using the resolvent condition and techniques on cosine family of linear operators. Particularly, theory of fractional power operators for cosine families is also applied to discuss the problem so that the obtained results can be applied to the systems involving derivatives of spatial variables. To illustrate the applications of the obtained results, two examples are presented in the end.

Approximate controllability for semilinear retarded control equations using surjectivity results

Sufficient conditions for approximate controllability of semilinear retarded functional control equations are obtained under general conditions on nonlinear terms. To get the main result, we transform the controllability problem into surjectivity results similar to Fredholm alternative for nonlinear operators under restrictive assumptions. Finally, a simple example to which our main result can be applied is given.

Approximate controllability of fractional delay evolution inclusions with noncompact semigroups

ABSTRACTThis paper is devoted to investigate the control systems governed by nonlinear fractional delay evolution inclusions (FDEIs) with noncompact semigroups. Under suitable assumptions, sufficient conditions for approximate controllability of FDEIs are established. The approximate controllability results are further extended to FDEIs with nonlocal conditions, where the compactness or Lipschitz continuity on the nonlocal functions is not required. Finally, two examples are given to illustrate the effectiveness of our conclusions.

Controlling Power Consumption in a Heterogeneous Population Model of TCLs with Diffusion: The Green’s Function Approach

We consider a control problem for a diffusive PDE model of heterogeneous population of thermostatically controlled loads (TCLs) aiming to balance the aggregate power consumption within a given amount of time. Using the Green’s function approach, the problem is formulated as an approximate controllability problem for a residue depending on control parameters nonlinearly. A sufficient condition for approximate controllability is derived in terms of initial temperature distribution, operation time of TCLs and threshold value of the aggregate power consumption. Case studies allow to reveal the advantages of the proposed solution from numerical calculations point of view.

On the Approximate Controllability for Hilfer Fractional Evolution Hemivariational Inequalities

The paper is concerned with the approximate controllability of some Hilfer fractional evolution hemivariational inequalities. Using two classes of operators and their fundamental properties, we derive sufficient conditions for approximate controllability of linear and semilinear controlled systems via a fixed point theorem for multivalued maps. Finally, an example is given to illustrate our theory.