Controllability Of Semilinear Systems Research Articles

Existence, Stability and Optimality of a Semilinear Control System Involving Instantaneous and Non-instantaneous Impulses

Existence, Stability and Optimality of a Semilinear Control System Involving Instantaneous and Non-instantaneous Impulses

On Partial Exact Controllability of Fractional Control Systems in Conformable Sense

In this work, we investigate the partial exact controllability of fractional semilinear control systems in the sense of conformable derivatives. Initially, we establish the existence and uniqueness of the mild solution for this type of fractional control systems. Then, by employing a contraction mapping principle, we obtain sufficient conditions for the conformable fractional semilinear system to be partially exactly controllable, assuming that its associated linear part is partially exactly controllable. To demonstrate the efficacy of the theoretical findings, a typical example is provided at the end.

Approximate Controllability of Semilinear Control System with State Delay Using Tikhonov Regularization

Approximate Controllability of Semilinear Control System with State Delay Using Tikhonov Regularization

Results on partial approximate controllability of fractional control systems in Hilbert spaces with conformable derivatives

This paper investigates partial approximate controllability of fractional linear and semi-linear control systems involving the conformable derivative. First, we derive sufficient and necessary conditions for partial approximate controllability of fractional linear systems over a small interval. Then, we apply a new approach, different from typical methods involving complexity inequalities and fixed point theorems, to obtain sufficient conditions for partial approximate controllability of semi-linear fractional control systems, assuming that the linear part is partially approximately controllable. The effectiveness of these results is demonstrated through an illustrative example.

Impulsive second order control differential equations: Existence and approximate controllability

The primary focus of this manuscript is on the approximate controllability of second-order semilinear control systems with impulses. There have been two sets of necessary requirements discussed. Combining the theories of the sine and cosine families, as well as the compactness of the cosine operator along with the fixed point technique (FPT) yields the first set of results. The following discussion avoids the apply of the compactness and fixed point approach of the cosine function and is shown instead using Gronwall’s inequality. The existence and uniqueness of the mild solution are also established. We have also provided a case study of theoretical outcomes that have been confirmed the theory.

Compactness of Fixed Point Maps and the Ball-Marsden-Slemrod Conjecture

Given a parameter dependent fixed point equation $x = F(x,u)$, we derive an abstract compactness principle for the fixed point map $u \mapsto x^*(u)$ under the assumptions that (i) the fixed point equation can be solved by the contraction principle and (ii) the map $u \mapsto F(x,u)$ is compact for fixed $x$. This result is applied to infinite-dimensional, semi-linear control systems and their reachable sets. More precisely, we extend a non-controllability result of Ball, Marsden, and Slemrod [1] to semi-linear systems. First we consider $L^p$-controls, $p>1$. Subsequently we analyze the case $p=1$.

Results on the Approximate Controllability of Hilfer Type fractional Semilinear Control Systems

Results on the Approximate Controllability of Hilfer Type fractional Semilinear Control Systems

Interior approximate controllability of second-order semilinear control systems

In our manuscript, we investigate the interior approximate controllability for the subsequent semilinear second-order system in L 2 ( Ω ) s ″ ( σ ) + A 0 s ( σ ) = B 0 u ( σ ) + ξ ( σ , s ( σ ) ) , 0 ≤ σ ≤ ϱ s ( 0 ) = s 0 , s ′ ( 0 ) = s 1 , where A 0 is the linear and unbounded operator, B 0 is a bounded linear operator and ξ is a nonlinear operator defined on appropriate spaces. The proposed problem can be converted into an equivalent first-order semilinear control system, then the approximate controllability results for the proposed system are obtained from the study of the approximate controllability of the reduced first-order system. The Leray–Schauder alternative theorem and principle of contraction are used in the proof of our main theorems.

An analysis on approximate controllability of Atangana–Baleanu fractional semilinear control systems

Abstract The article deals with the approximate controllability of Atangana–Baleanu semilinear control systems. The outcomes are derived by applying Gronwall’s inequality and Cauchy sequence, and avoid the use of the fixed point theorem. We have also included an example for the validation of theoretical results.

Tikhonov solutions of approximately controllable second-order semilinear control systems

Tikhonov solutions of approximately controllable second-order semilinear control systems

Approximate controllability of semi-linear stochastic integrodifferential system with multiple delays and Poisson jumps in control

The objective of this paper is to interpret the approximate controllability of a semi-linear stochastic integrodifferential system with multiple delays and Poisson jumps in control in infinite-dimensional spaces. Sufficient conditions for the approximate controllability of semi-linear control system have been established. The results are obtained using the Banach fixed point theorem and the theory of resolvent operator developed in Grimmer [Resolvent operator for integral equations in Banach spaces, Trans. Am. Math. Soc. 273 (1982), pp. 333– 349.]. An example is introduced to show the effectiveness of the result.

"Feedback null controllability for a class of semilinear control systems"

"For a class of semilinear control systems we get null controllability results and estimates for the minimum time function by considering appropriate feedback laws."

An analysis on approximate controllability of semilinear control systems with impulsive effects

The article uses semigroup theory and Gronwall’s inequality to investigate the approximate controllability of impulsive semilinear control systems. A weaker Lipschitz condition on nonlinearity is enforced to reach the main conclusions. Two sets of assumptions have been used to discuss the key findings. As noted before by various researchers in the accessible literature, application of standard fixed point theorems such as Banach, Schauder, Sadovskii, and others is avoided. Finally, in order to verify the theoretical findings, one case study based on the presented topic is discussed.

Approximate controllability of the semilinear population dynamics system with diffusion

This paper aims to study the approximate controllability of semilinear population dynamics system with diffusion using semigroup theory. The semilinear population dynamical model with the nonlocal birth process is transformed into a standard abstract semilinear control system by identifying the state, control, and the corresponding function spaces. The state and control spaces are assumed to be Hilbert spaces. The semigroup theory is developed from the properties of the population operators and Laplacian operators. Then the approximate controllability results of the system are obtained using ‐semigroup approach and some other simple conditions on the nonlinear term and operators involved in the model.

Results on impulsive semilinear differential equations with control functions

The article deals with the sufficient conditions for the approximate controllability for the impulsive semilinear control systems. We have considered two different sets of sufficient conditions. In the first set, we derived the results by applying the theories related to the compactness of semigroup and Schauder's fixed point theorem. In the second set, we derived the results by applying Gronwall's inequality and avoids the use of fixed point theorem and compactness of semigroup. Under suitable assumptions, the discussion on the existence and uniqueness of the impulsive system is also proved. We have also included an example for the validation of theoretical results.

New discussion concerning to optimal control for semilinear population dynamics system in Hilbert spaces

The objective of our paper is to investigate the optimal control of semilinear population dynamics system with diffusion using semigroup theory. The semilinear population dynamical model with the nonlocal birth process is transformed into a standard abstract semilinear control system by identifying the state, control, and the corresponding function spaces. The state and control spaces are assumed to be Hilbert spaces. The semigroup theory is developed from the properties of the population operators and Laplacian operators. Then the optimal control results of the system are obtained using the C0-semigroup approach, fixed point theorem, and some other simple conditions on the nonlinear term as well as on operators involved in the model.

Regional boundary controllability of semilinear subdiffusion Caputo fractional systems

In this paper we considers the Caputo fractional semi linear control systems, then we obtain sufficient conditions for regional boundary controllability of a board class of considered system with order (0<α≤1). The conditions are obtained using fractional calculus and Schauder’s fixed point theorem, that leads to an important approach based on a link between internal and boundary controllability which will be summarized as an algorithm. At the end, some simulations are given to show the application of our result.

Results on exact controllability of second-order semilinear control system in Hilbert spaces

In our manuscript, we extend the controllability outcomes given by Bashirov (Math. Methods Appl. Sci. 44(9):7455–7462, 2021) for a family of second-order semilinear control system by formulating a sequence of piecewise controls. This approach does not involve large estimations which are required to apply fixed point theorems. Therefore, we avoid the use of fixed point theory and the contraction mapping principle. We establish that a second-order semilinear system drives any starting position to the required final position from the domain of the system. To achieve the required results, we suppose that the linear system is exactly controllable at every non-initial time period, the norm of the inverse of the controllability Grammian operator increases as the time approaches zero with the slower rate in comparison to the reciprocal of the square function, and the nonlinear term is bounded. Finally, an example has been presented to validate the results.

Results on the approximate controllability of fractional hemivariational inequalities of order 1< r

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.

Controllability for Retarded Semilinear Neutral Control Systems of Fractional Order in Hilbert Spaces

In this paper, we discuss the approximate controllability for a class of retarded semilinear neutral control systems of fractional order by investigating the relations between the reachable set of the semilinear retarded neutral system of fractional order and that of its corresponding linear system. The research direction used here is to find the conditions for nonlinear terms so that controllability is maintained even in perturbations. Finally, we will show a simple example to which the main result can be applied.