Approximate controllability of second-order distributed implicit functional systems

In this paper we study the approximate controllability of control systems with states and controls in Hilbert spaces, and described by a second-order semilinear abstract functional differential equation with infinite delay. Initially we establish a characterization for the approximate controllability of a second-order abstract linear system and, in the last section, we compare the approximate controllability of a semilinear abstract functional system with the approximate controllability of the associated linear system.     

Relative approximate controllability of abstract functional systems with infinite delay and delayed control

This paper is concerned with the relative approximate controllability of functional systems with infinite delay and delayed control in Hilbert spaces. In particular, we begin with studying some criteria of the controllability of linear systems. Based on those results, sufficient conditions are derived for the relative approximate controllability of nonlinear functional systems. Finally, an example is included to illustrate the effectiveness of the proposed methods. Copyright © 2016 John Wiley & Sons, Ltd.     

On the approximate controllability for fractional evolution hemivariational inequalities

In this paper, we focus on the approximate controllability of control systems described by a large class of fractional evolution hemivariational inequalities. Firstly, we introduce the concept of mild solutions and present the existence of mild solutions for this kind of problems. Next, we show the approximate controllability of the corresponding linear control system. Finally, the approximate controllability of the fractional evolution hemivariational inequalities is formulated and proved under some appropriate conditions. An example demonstrates the applicability of our results. Copyright © 2015 John Wiley & Sons, Ltd.     

On the approximate controllability of stochastic stokes systems

In this paper, we analyze the approximate controllability in quadratic mean of some systems governed by stochastic partial differential equations of the Stokes kind. When the noise is state-independent, we obtain satisfactory results, similar to those known for the corresponding deterministic system. In the more complicate case of a multiplicative noise, we are able to give (only) partial results. More precisely, we prove in this case that approximate controllability is equivalent to the unique continuation property for a particular backward (adjoint) stochastic system     

Approximate controllability of second‐order distributed systems

In this paper, we study the approximate controllability of control systems with state and control in Banach spaces and described by a second‐order semilinear abstract differential equation. We compare the approximate controllability of the system with the approximate controllability of an associated discrete system. Copyright © 2013 John Wiley & Sons, Ltd.     

Skew-product semi-flows and non-autonomous control systems

In this paper we give a necessary and sufficient conditions for exact and approximate controllability of a wide class of linear infinite-dimensional non-autonomous control systems. This is done by employing skew-product semi-flow technique. Finally, we apply these results to prove the controllability of a broad class of non-autonomous reaction diffusion equations in Hilbert spaces.     

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 a class of hemivariational inequalities

In this paper, we deal with the approximate controllability for control systems described by a class of hemivariational inequalities. Firstly, we introduce the concept of mild solutions for hemivariational inequalities. Then the approximate controllability is formulated and proved by utilizing a fixed-point theorem of multivalued maps and properties of generalized Clarke subdifferential.     

Approximate controllability of impulsive fractional evolution equations of order 1<α<2$$ 1&amp;lt;\alpha &amp;lt;2 $$ with state-dependent delay in Banach spaces

This article deals with the approximate controllability problem for fractional evolution equations involving noninstantaneous impulses and state-dependent delay. In order to derive sufficient conditions for the approximate controllability of our problem, we first consider the linear-regulator problem and find the optimal control in the feedback form. By using this optimal control, we develop the approximate controllability of the linear fractional control system. Further, we obtain sufficient conditions for the approximate controllability of the nonlinear problem. In the end, we provide a concrete example to support the applicability of the derived results.     

Approximate controllability of a fractional stochastic partial integro-differential systems via noncompact operators

Objective: in this article, we discuss the approximate controllability problems of a new class of fractional impulsive stochastic partial integro-differential systems in separable Hilbert spaces. Methods: by applying the fractional calculus, the measure of noncompactness, properties of fractional resolvent operators and fixed point theorems. Results: we prove our main results without the hypotheses of compactness on the operator generated by the linear part of systems. Instead we suppose that the nonlinear term only satisfies a weakly compactness condition. Conclusion: the approximate controllability for the control systems with noncompact operators is established. Finally, an example is given for the illustration of the obtained theoretical results.     

Approximate Controllability for Integrodifferential Equations with Multiple Delays

This paper considers the approximate controllability for a class of control systems governed by semilinear delay integrodifferential equations with multiple delays. Sufficient conditions for approximate controllability are established by using the Schauder fixed-point theorem. The results obtained improve some analogous existing results. Several examples are provided to illustrate the application of the approximate controllability result.     

Approximate Controllability of Fractional Differential Equations with State-Dependent Delay

The systems governed by delay differential equations come up in different fields of science and engineering but often demand the use of non-constant or state-dependent delays. The corresponding model equation is a delay differential equation with state-dependent delay as opposed to the standard models with constant delay. The concept of controllability plays an important role in physics and mathematics. In this paper, first we study the approximate controllability for a class of nonlinear fractional differential equations with state-dependent delays. Then, the result is extended to study the approximate controllability fractional systems with state-dependent delays and resolvent operators. A set of sufficient conditions are established to obtain the required result by employing semigroup theory, fixed point technique and fractional calculus. In particular, the approximate controllability of nonlinear fractional control systems is established under the assumption that the corresponding linear control system is approximately controllable. Also, an example is presented to illustrate the applicability of the obtained theory.     

Relative controllability of stochastic nonlinear systems with delay in control

The existence of solutions to systems is a natural premise to carry our study about controllability. Under the basic and readily verified conditions to guarantee the existence of the solutions to a system, in this paper, we prove the relative controllability (approximate controllability ) of the stochastic differential systems with delay in control. Sufficient conditions are given firstly for the relative controllability and relative approximate controllability in finite dimensional spaces, and these results are then generalized to infinite-dimensional Hilbert spaces. Finally, examples are given to illustrate the effectiveness of the proposed methods.     

Approximate controllability of impulsive semilinear stochastic system with delay in state

Many practical systems in physical and biological sciences have impulsive dynamical behaviors during the evolution process that can be modeled by impulsive differential equations. This article studies the approximate controllability of impulsive semilinear stochastic system with delay in state in Hilbert spaces. Assuming the conditions for the approximate controllability of the corresponding deterministic linear system, we obtain the sufficient conditions for the approximate controllability of the impulsive semilinear stochastic system with delay in state. The results are obtained by using Banach fixed point theorem. Finally, two examples are given to illustrate the developed theory.     

Controllability of Some Nonlinear Systems in Hilbert Spaces

In this paper, several abstract results concerning the controllability of semilinear evolution systems are obtained. First, approximate controllability conditions for semilinear systems are obtained by means of a fixed-point theorem of the Rothe type; in this case, the compactness of the linear operator is assumed. Next, the exact controllability of semilinear systems with nonlinearities having small Lipschitz constants is derived by means of the Banach fixed-point theorem; in this case, the compactness of the operators is not assumed. In both cases, it is proven that the controllability of the linear system implies the controllability of the associated semilinear system. Finally, these abstract results are applied to the controllability of the semilinear wave and heat equations.     

Controllability of impulsive neutral functional evolution integrodifferential systems with infinite delay

In this paper we establish sufficient conditions for the approximate controllability of impulsive neutral functional evolution integrodifferential systems in Hilbert spaces. Also we study the exact controllability of the same system. The conditions are obtained by using Schauder’s fixed point theorem when the operator is compact and the Banach fixed point theorem when the operator is not compact. The results are obtained by using the evolution operators.     

Controllability of second-order infinite-dimensional systems

In the paper, the approximate controllability of linear abstractsecond-order infinite-dimensional dynamical systems is considered.Using the frequency-domain method, it is proved that approximatecontrollability of such a system follows from that of a suitablydefined first-order system. The general results are then appliedto investigating the approximate controllability of a vibratorydynamical system modelling a flexible mechanical structure.Some special case are also considered. Moreover, remarks andcomments on the relationships between different concepts ofcontrollability are given. The paper generalizes earlier resultson second-order abstract dynamical systems.     

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.     

Approximate boundary controllability of Sobolev-type stochastic differential systems

The objective of this paper is to investigate the approximate boundary controllability of Sobolev-type stochastic differential systems in Hilbert spaces. The control function for this system is suitably constructed by using the infinite dimensional controllability operator. Sufficient conditions for approximate boundary controllability of the proposed problem in Hilbert space is established by using contraction mapping principle and stochastic analysis techniques. The obtained results are extended to stochastic differential systems with Poisson jumps. Finally, an example is provided which illustrates the main results.     

Approximate Controllability of Fractional Impulsive Partial Neutral Stochastic Differential Inclusions with State-Dependent Delay and Fractional Sectorial Operators

In this article, the approximate controllability of fractional impulsive partial neutral stochastic differential inclusions with state-dependent delay and fractional sectorial operators in Hilbert spaces is studied. By using the stochastic analysis, the fractional sectorial operators and a fixed point theorem for multi-valued maps combined with approximation techniques, we discuss a new set of su?cient conditions for the approximate controllability of the systems under the mixed Lipschitz and Carathéodory conditions. An example is provided to illustrate the obtained theory.