Semilinear Fractional Systems Research Articles

Regional observability of Caputo semilinear fractional systems

AbstractThis article studies the problem of regional observability of an important class of time‐fractional evolution systems involving the Caputo time‐fractional derivative. The main goal here is to check the regional observability of the considered system in the desired subregion, , of the evolution domain, . We assume that the linear part of the studied system is approximately regionally observable in the desired subregion , and we add some reasonable hypotheses on the system's dynamic, , as well as on the nonlinear operator . Then, with the help of all these assumptions, we attempt to reconstruct the initial state of our system in the desired subregion . To do that, we propose an extension of the Hilbert uniqueness method (HUM) to semilinear fractional systems. The main advantage of the HUM approach is that it transforms the reconstruction problem into a solvability one, which is more practical. We also present an algorithm for reconstructing the initial state in , which leads to some successful numerical simulations.

On p–Laplacian boundary value problems involving Caputo–Katugampula fractional derivatives

In this paper, we study the existence and uniqueness of solutions for a p–Laplacian boundary value problem defined by semilinear fractional system that involves Caputo–Katugampola fractional derivatives. Our main results rely on the implementation of the Banach and Schauder fixed point theorems. An example is introduced to expose the applicability of the theoretical findings.

Non-existence for a semi-linear fractional system with Sobolev exponents via direct method of moving spheres

The aim of this article is to consider the semi-linear fractional system with Sobolev exponents q = frac{{n + alpha}}{{n - beta}} and p = frac{{n + beta}}{{n - alpha}} (alpha nebeta): {(−Δ)α/2u(x)=k(x)vq(x)+f(v(x)),(−Δ)β/2v(x)=j(x)up(x)+g(u(x)),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\left \\{ \\textstyle\\begin{array}{l} {( - \\Delta)^{\\alpha/2}}u(x) = k(x){v^{q}}(x) + f(v(x)),\\\\ {( - \\Delta)^{\\beta/2}}v(x) = j(x){u^{p}}(x) + g(u(x)), \\end{array}\\displaystyle \\right . $$\\end{document} where 0 < alpha, beta < 2. We first establish two maximum principles for narrow regions in the ball and out of the ball by the iteration technique, respectively. Based on these principles, we use the direct method of moving spheres to prove the non-existence of positive solutions to the above system in the whole space and bounded star-shaped domain. As a consequence, the monotonic decreasing properties of W(x) = { vert x vert ^{frac{{n - alpha}}{2}}}u(x) and {W_{1}}(x) = { vert x vert ^{frac{{n - beta}}{2}}}v(x) along the radial direction in the whole space are obtained.

Non-existence results for cooperative semi-linear fractional system via direct method of moving spheres

In this article, we consider the cooperative semi-linear fractional system \begin{document}$ (-\Delta)^{\frac{\alpha}{2}}\vec {u}(x) = \vec {h}(x,\vec {u}(x)), $\end{document} where \begin{document}$ 0 , \begin{document}$ \vec u $\end{document} and \begin{document}$ \vec h $\end{document} stand for \begin{document}$ k $\end{document} -dimentional vector-valued functions, and \begin{document}$ \vec {h}(x,\vec {u}(x)) $\end{document} is locally Lipschitz in \begin{document}$ \vec {u} $\end{document} . We first establish two narrow region principles for different cases. Based on these principles, we use the direct method of moving spheres to prove the non-existence of positive solutions of the above system in bounded star-shaped domains and the whole space.

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.

Cone-type constrained relative controllability of semilinear fractional systems with delays

The paper presents fractional-order semilinear, continuous, finite-dimensional dynamical systems with multiple delays both in controls and nonlinear function $f$. The constrained relative controllability of the presented semilinear system and corresponding linear one are discussed. New criteria of constrained relative controllability for the fractional semilinear systems with delays under assumptions put on the control values are established and proved. The conical type constraints are considered. The results are illustrated by an example.

Controllability of retarded semilinear fractional system with non-local conditions

Controllability of retarded semilinear fractional system with non-local conditions

Semi-linear fractional systems with Caputo type multi-step differences

A new class of fractional linear and semi-linear discrete-time systems described by the collection of state equations is introduced. The solution to the system is derived by the recursive formula. Presented results concern the fractional h-difference operator.