Sufficient Conditions For Approximate Controllability Research Articles

Approximate Controllability via Resolvent Operators of Sobolev-Type Fractional Stochastic Integrodifferential Equations with Fractional Brownian Motion and Poisson Jumps

Using fractional calculus, stochastic analysis theory, and fixed point theorems with the properties of analytic \(\alpha \)-resolvent operators, sufficient conditions for approximate controllability of Sobolev-type fractional stochastic integrodifferential equations with fractional Brownian motion and Poisson jumps are established. Finally, an example is given to illustrate the obtained results.

Approximate controllability of semilinear fractional stochastic control system

The objective of this paper is to present some sufficient conditions for approximate controllability of semilinear fractional stochastic control system with delay. The results hold when the nonlinear function is Lipschitz continuous. Sufficient conditions are obtained by separating the given fractional semilinear stochastic system into two systems namely a semilinear fractional system and a fractional linear stochastic system. To prove our results, the Schauder fixed point theorem is applied. At the end, one example is given to illustrate the results.

Approximate Controllability of Non-autonomous Evolution System with Nonlocal Conditions

In this article, we are concerned with the existence of mild solutions as well as approximate controllability for a class of non-autonomous evolution system of parabolic type with nonlocal conditions in Banach spaces. Sufficient conditions of existence of mild solutions and approximate controllability for the desired problem are presented by introducing a new Green’s function and constructing a control function involving Gramian controllability operator. Some sufficient conditions of approximate controllability are formulated and proved here by using the resolvent operator condition. The discussions are based on Schauder’s fixed-point theorem as well as the theory of evolution family. An example is also given to illustrate the feasibility of our theoretical results.

Approximate controllability for abstract semilinear impulsive functional differential inclusions based on Hausdorff product measures

A second order semilinear impulsive functional differential inclusion in a separable Hilbert space is considered. Without imposing hypotheses of the compactness on the cosine families of operators, some sufficient conditions of approximate controllability are formulated in the case where the multivalued nonlinearity of the inclusion is a completely continuous map dominated by a function. By the use of resolvents of controllability Gramian operators and developing appropriate computing techniques for the Hausdorff product measures of noncompactness, the results of approximate controllability for position and velocity are derived. An example is also given to illustrate the application of the obtained results.

Exact and approximate controllability of nonlinear dynamic systems in infinite time: The Green's function approach

AbstractA systematic method for establishment of exact and approximate controllability of dynamic systems with nonlinear state constraints within infinite time is developed. Using the recently developed Green's function approach, we derive necessary and sufficient conditions for exact controllability, as well as sufficient conditions for approximate controllability. Conditions for null‐controllability and lack of controllability are established as well. Several possibilities of deriving explicit form for resolving controls are described. Possibilities of heuristic determination of resolving controls is studied in details. The advantages and drawbacks of the method are revealed on specific examples of nonlinear wave and diffusion equations in unbounded domains.

Approximate Controllability of a Neutral Stochastic Fractional Integro-Differential Inclusion with Nonlocal Conditions

This paper discusses the approximate controllability of a neutral functional integro-differential inclusion involving Caputo fractional derivative in a Hilbert space under the assumptions that the corresponding linear system is approximately controllable. A new set of sufficient conditions for approximate controllability of neutral fractional stochastic functional integro-differential inclusions are formulated and established by utilizing stochastic analysis theory, fractional calculus and the technique of fixed point theorem with analytic compact resolvent operator. An example is also considered for illustrating the discussed theory.

Approximate controllability of semilinear non-autonomous evolution systems with state-dependent delay

The controllability of non-autonomous evolution systems is an important and difficult topic in control theory. In this paper, we study the approximate controllability of semilinear non-autonomous evolution systems with state-dependent delay. The theory of linear evolution operators is used instead of \begin{document}$C_0-$\end{document} semigroup to discuss the problem. Some sufficient conditions of approximate controllability are formulated and proved here by using the resolvent operator condition. Finally, two examples are provided to illustrate the applications of the obtained results.

Approximate controllability of Sobolev type fractional evolution systems with nonlocal conditions

In this paper, we study the approximate controllability of Sobolev-type fractional evolution systems with non-local conditions in Hilbert spaces. Sufficient conditions of approximate controllability of the desired problem are presented by supposing an approximate controllability of the corresponding linear system. By constructing a control function involving Gramian controllability operator, we transform our problem to a fixed point problem of nonlinear operator. Then the Schauder Fixed Point Theorem is applied to complete the proof. An example is given to illustrate our theoretical results.

Approximate Controllability for Functional Equations with Riemann-Liouville Derivative by Iterative and Approximate Method

We discuss the functional control systems governed by differential equations with Riemann-Liouville fractional derivative in general Banach spaces in the present paper. First we derive the uniqueness and existence of mild solutions for functional differential equations by the approach of fixed point and fractional resolvent under more general settings. Then we present new sufficient conditions for approximate controllability of functional control system by means of the iterative and approximate method. Our results unify and generalize some previous works on this topic.

Approximate Controllability of Fractional Semilinear Stochastic System of Order α∈ (1,2

A family of dynamical control systems described by nonlinear fractional of order (1,2] stochastic differential equations in Lp spaces is considered. We discussed the approximate controllability of stochastic semilinear fractional control system of order α∈(1,2] under the assumption that the corresponding linear system is approximately controllable. A new set of sufficient conditions for approximate controllability of system are obtained by the theory of strongly continuous α-order cosine family, fixed point theorem, and stochastic analysis techniques. At the end, an example is given to illustrate the theory.

Approximate controllability of infinite dimensional system with internal damping dependent on fractional powers of system operator

In this study, a class of models governed by a second-order (in time) evolution differential equation with damping and positive definite, self-adjoint system operator is studied. Damping term has a special, but practically important, form of a finite sum of fractional powers of the system operator. This type of damping operator can be used to model a variety of physical phenomena related to dissipation of energy empirically observed in physical systems. It may be used to model dissipation mechanisms resulting from air damping, internal structural damping, internal viscous damping, etc. Using spectral theory of linear unbounded operators and semi-group theory necessary and sufficient conditions of approximate controllability for second order infinite dimensional system with damping are formulated and proved. Some important, from the practical point of view, remarks and comments are also provided. In particular, necessary condition for approximate controllability is formulated and discussed. Finally, an illustrative example related to approximate controllability of distributed parameter system described by the partial differential equation with higher order spatial differential operators is presented. The example refers to a system describing dynamical behaviour of damped Kirchhoff–Love plate. The study extends earlier results on approximate controllability of damped second-order abstract evolution dynamical systems.

Schauder’s fixed-point theorem in approximate controllability problems

Abstract The main objective of this article is to present the state of the art concerning approximate controllability of dynamic systems in infinite-dimensional spaces. The presented investigation focuses on obtaining sufficient conditions for approximate controllability of various types of dynamic systems using Schauder’s fixed-point theorem. We describe the results of approximate controllability for nonlinear impulsive neutral fuzzy stochastic differential equations with nonlocal conditions, impulsive neutral functional evolution integro-differential systems, stochastic impulsive systems with control-dependent coefficients, nonlinear impulsive differential systems, and evolution systems with nonlocal conditions and semilinear evolution equation.

Approximate controllability of semilinear non-autonomous evolutionary systems with nonlocal conditions

We consider approximate controllability of semilinear non-autonomous evolutionary systems with nonlocal conditions. In this study, we use the theory of fractional powers and ?-norms, so our results can be applied to systems where nonlinear terms include derivatives of spatial variables. We formulate and prove sufficient conditions for approximate controllability. We also give a sample application of our results.

Approximate controllability of neutral functional differential systems with state-dependent delay

This paper deals with the approximate controllability of semilinear neutral functional differential systems with state-dependent delay. The fractional power theory and α-norm are used to discuss the problem so that the obtained results can apply to the systems involving derivatives of spatial variables. By methods of functional analysis and semigroup theory, sufficient conditions of approximate controllability are formulated and proved. Finally, an example is provided to illustrate the applications of the obtained results.

Approximate Controllability of Semilinear Fractional Control Systems of Order $${{\alpha \in (1, 2]}}$$ α ∈ ( 1 , 2 ] with Infinite Delay

The objective of this paper is to present some sufficient conditions for approximate controllability of semilinear fractional control system of order \({\alpha \in (1, 2]}\) with infinite delay. The results are obtained by the theory of strongly continuous \({\alpha}\)-order cosine family and sequence method. At the end, an example is given to illustrate the theory.

Approximate controllability of semilinear system with state delay using sequence method

The objective of this paper is to present some sufficient conditions for approximate controllability of semilinear system with state delay using sequence method. Instead of a C0-semigroup associated with the mild solution of the system, we use the so-called fundamental solution. Controllability results are obtained by sequential approach and operator semigroup theory. At the end, an example is given to illustrate the theory.

Approximate controllability of fractional impulsive evolution systems involving nonlocal initial conditions

This work is concerned with the approximate controllability of a nonlinear fractional impulsive evolution system under the assumption that the corresponding linear system is approximate controllable. Using the fractional calculus, the Krasnoselskii fixed point theorem, and the technique of controllability theory, some new sufficient conditions for approximate controllability of fractional impulsive evolution equations are obtained. The results in this paper are generalizations and continuations of the recent results on this issue. At the end, an example is given to illustrate the effectiveness of the main results.

On Approximate Controllability of Second-Order Neutral Partial Stochastic Functional Integrodifferential Inclusions with Infinite Delay and Impulsive Effects

We discuss the approximate controllability of second-order impulsive neutral partial stochastic functional integrodifferential inclusions with infinite delay under the assumptions that the corresponding linear system is approximately controllable. Using the fixed point strategy, stochastic analysis, and properties of the cosine family of bounded linear operators combined with approximation techniques, a new set of sufficient conditions for approximate controllability of the second-order impulsive partial stochastic integrodifferential systems are formulated and proved. The results in this paper are generalization and continuation of the recent results on this issue. An example is provided to show the application of our result.

APPROXIMATE CONTROLLABILITY OF FRACTIONAL ORDER STOCHASTIC VARIATIONAL INEQUALITIES DRIVEN BY POISSON JUMPS

This paper proposes the sufficient conditions of approximate controllability for a class of fractional order stochastic variational inequalities driven by Poisson jumps. The possibilities of finding the approximate controllability of a given problem of this type introduce the smoothing system corresponding to the fractional order stochastic variational inequalities driven by Poisson jumps. The results are achieved upon the Moreau-Yosida approximation of subdifferential operator. Sufficient conditions for the approximate controllability of smoothing system are discussed under the boundedness condition on control operator. The results are formulated and proved by using the fractional calculus, semigroup theory, stochastic analysis techniques. An example is provided to illustrate the obtained theory.

Approximate Controllability for a Semilinear Evolution System with Infinite Delay

In this work, we study the approximate controllability for a class of control systems governed by semilinear equations with infinite delay in Hilbert spaces. Sufficient conditions for approximate controllability are established by constructing fundamental solution and using resolvent condition and techniques on fractional power operators. As an illustration of the application of the obtained results, an example is also provided.