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We establish a class of nonlinear fractional differential systems with distributed time delays in the controls and impulse effects. We discuss the controllability criteria for both linear and nonlinear systems. The main results required a suitable Gramian matrix defined by the Mittag–Leffler function, using the standard Laplace transform and Schauder fixed-point techniques. Further, we provide an illustrative example supported by graphical representations to show the validity of the obtained abstract results.

This paper investigates numerical solution of generalized space-time fractional Klein–Gordon equations (GSTFKGE) by using Gegenbauer wavelet method (GWM). The developed method makes use of fractional order integral operator (FOIO) for Gegenbauer wavelet, which is constructed by employing the definition of Riemann–Liouville fractional integral (RLFI) operator and Laplace transformation. The present algorithm is based on Gegenbauer wavelet jointly with FOIO to convert a GSTFKGE into a system of equations which is solved by using Newton’s technique. Additionally, the upper bound of error norm of the proposed method is calculated to validate the theoretical authenticity of the developed method. The comparison of numerical outcomes with the existing results in the literature and graphical illustrations show the accuracy and reliability of our method.

The aim of this work is to investigate the controllability of a class of switched Hilfer neutral fractional systems with non-instantaneous impulses in the finite-dimensional spaces. We construct a new class of control function that controls the system at the final time of the time-interval and controls the system at each of the impulsive points i.e. we give the so-called total controllability results. Also, we extend these results to the corresponding integro-system. We mainly use the fixed point theorem, Laplace transformation, Mittag-Leffler function, Gramian type matrices and fractional calculus to establish these results. In the end, we provide a simulated example to verify the obtained analytical results.

We study a class of final value problems for time fractional wave equations involvingCaputo’s fractional derivative of order 1 α 2 and a symmetric uniformly elliptic operator in a bounded domain Ω ⊂ R N , N = 1 , 2 , 3 . Firstly, we analyze the difficulties involved in the final value problem under consideration. Secondly, we establish the existence and uniqueness of the mild solution. Finally, we investigate the regularity of the mild solution in some special spaces.

This paper deals with a set of three partial differential equations involving time-fractional derivatives and nonlinear diffusion operators. This model helps us to understand the HIV spread and transmission into the patient. First, we prove the existence and uniqueness of weak solutions to the mathematical model. Then, the Galerkin finite element scheme is implemented to approximate the solution of the model. Further, a-priori error bounds and convergence estimates for the fully-discrete problem are derived. The second order convergence for the proposed scheme is also proved. Numerical tests are shown to validate the theoretical studies.

In this work, we present and discuss the approaches, that are used for modeling and surveillance of dynamics of infectious diseases by considering the early stage asymptomatic and later stage symptomatic infections. We highlight the conceptual ideas and mathematical tools needed for such infectious disease modeling. We compute the basic reproduction number of the proposed model and investigate the qualitative behaviours of the infectious disease model such as, local and global stability of equilibria for the non-delayed as well as delayed system. At the end, we perform numerical simulations to validate the effectiveness of the derived results.

In this article, exact traveling wave solutions of a Wick-type stochastic nonlinear Schr\"{o}dinger equation and of a Wick-type stochastic fractional Regularized Long Wave-Burgers (RLW-Burgers) equation have been obtained by using an improved computational method. Specifically, the Hermite transform is employed for transforming Wick-type stochastic nonlinear partial differential equations into deterministic nonlinear partial differential equations with integral and fraction order. Furthermore, the required set of stochastic solutions in the white noise space is obtained by using the inverse Hermite transform. Based on the derived solutions, the dynamics of the considered equations are performed with some particular values of the physical parameters. The results reveal that the proposed improved computational technique can be applied to solve various kinds of Wick-type stochastic fractional partial differential equations., Comment: This is a preprint of a paper whose final and definite form is with 'Chaos, Solitons & Fractals', ISSN 0960-0779 [https://doi.org/10.1016/j.chaos.2019.109542]. Submitted 19-Sept-2019; Revised 14-Nov-2019; Accepted for publication 18-Nov-2019. This version includes minor corrections detected while reading galley proofs

In this paper, we utilize fractional calculus, theory of semigroup and fixed point approach to prove existence and approximate controllability results for a class of fractional differential equations (FDEs) with noninstantaneous impulses by constructing a suitable composite control function, imposing that the associated linear problem is approximately controllable on the terminal subinterval and dividing our global task into many subtasks on each subinterval. Next, we apply P-type iterative learning control (ILC) updating law to generate a sequence of control functions to find a desired control function to guarantee the error between the output and the desired reference trajectories tending to zero via a suitable norm in the sense of uniform convergence. As a result, the limit of the sequence of control functions is the required control to guarantee the above problem is approximately controllable. In addition, a numerical example is illustrated to demonstrate ILC scheme to solve approximate controllability of time fractional impulsive PDEs via tracking the given continuous and discontinuous trajectory.

We introduce a time-fractional Keller–Segel model with Dirichlet conditions on the boundary and Caputo fractional derivative for the time. The main result shows the existence theorem of the proposed model using the Faedo–Galerkin method with some compactness arguments. Moreover, we prove the Mittag–Leffler stability of solutions of the considered model.

We investigate time optimal control of a system governed by a class of non-instantaneous impulsive differential equations in Banach spaces. We use an appropriate linear transformation technique to transfer the original impulsive system into an approximate one, and then we prove the existence and uniqueness of their mild solutions. Moreover, we show the existence of optimal controls for Meyer problems of the approximate. Further, in order to solve the time optimal control problem for the original system, we construct a sequence of Meyer approximations for which the desired optimal control and optimal time are well derived.

The goal of this paper is to consider a new class of φ-Hilfer fractional differential equations with impulses and nonlocal conditions. By using fractional calculus, semigroup theory, and with the help of the fixed point theorem, the existence and uniqueness of mild solutions are obtained for the proposed fractional system. Symmetrically, we discuss the existence of optimal controls for the φ-Hilfer fractional control system. Our main results are well supported by an illustrative example.

We introduce a class of pairs of operators defining a linear homogeneous degenerate evolution fractional differential equation in a Banach space. Reflexive Banach spaces are represented as the direct sums of the phase space of the equation and the kernel of the operator at the fractional derivative. In a sector of the complex plane containing the positive half-axis, we construct an analytic family of resolving operators that degenerate only on the kernel. The results are used in the study of the solvability of initial-boundary value problems for partial differential equations containing fractional time-derivatives and polynomials in the Laplace operator with respect to the spatial variable.

This paper introduces a new concept called impulsive control inclusion condition, i.e., the impulsive condition is presented, in the first time, as inclusion related to multivalued maps and controls. The notion of approximate controllability of a class of semilinear Hilfer fractional differential control inclusions in Banach spaces is established. For the main results, we use fractional calculus, fixed point technique, semigroup theory and multivalued analysis. An appropriate set of sufficient conditions for the considered system to be approximately controllable is studied. Finally, we give an illustrated example to provide the obtained theory.

We try to seek a representation of solution to an initial value problem for impulsive fractional differential equations (IFDEs for short) involving Riemann–Liouvill (RL for short) fractional derivative, then prove an interesting existence result, and introduce Ulam type stability concepts of solution for this kind of equations by introducing some differential inequalities. In addition, we study iterative learning control (ILC for short) problem for system governed by IFDEs via a varying iterative state that does not coincide with a given initial state and apply proportional type learning principle involving the original learning condition to generate each output to following the final path in a finite time interval, then give a convergence result. Numerical examples are reported to check existence and stability of solutions and display the error for different iterative times.

In this paper, a new class of stochastic impulsive differential equations involving Bernoulli distribution is introduced. For tracking the random discontinuous trajectory, a modified tracking error associated with a piecewise continuous variable by zero-order holder is defined. In the sequel, a new random ILC scheme by adopting global and local iteration average operators is designed too. Sufficient conditions to guarantee the convergence of modified tracking error are obtained by using the tools of mathematical analysis via an impulsive Gronwall inequality. Finally, two illustrative examples are presented to demonstrate the performance and the effectiveness of the averaging ILC scheme to track the random discontinuous trajectory.

In this paper, we examine an optimal control problem of a coupled nonlinear parabolic system with cross-diffusion operators. The system describes the density of tumor cells, effector-immune cells, circulating lymphocyte population and chemotherapy drug concentration. The distributed control has been taken for drug concentration to control the amount of drug to be injected and to evade the side effects of the drug. We prove the existence of a weak solution of the direct problem. Then, the existence of control for the proposed control problem is proved. Further, we derive the optimality conditions and also the existence of a solution of the adjoint problem. The finite element numerical method is implemented for the proposed control problem. Then, theoretical results are illustrated with the help of numerical experiments. Finally, the importance of control function and the cross-diffusion effect are studied for the proposed control problem using numerical computations.

This article studies the existence and uniqueness of a weak solution of the time-fractional cancer invasion system with nonlocal diffusion operator. Existence and uniqueness results are ensured by adapting the Faedo-Galerkin method and some a priori estimates. Further, finite element numerical scheme is implemented for the considered system. Finally, various numerical computations are performed along with the convergence analysis of the scheme.

[EN] The aim of this paper is to study a generalization of fractional Airy differential equations whose input data (coefficient and initial conditions) are random variables. Under appropriate hypotheses assumed upon the input data, we construct a random generalized power series solution of the problem and then we prove its convergence in the mean square stochastic sense. Afterwards, we provide reliable explicit approximations for the main statistical information of the solution process (mean, variance and covariance). Further, we show a set of numerical examples where our obtained theory is illustrated. More precisely, we show that our results for the random fractional Airy equation are in full agreement with the corresponding to classical random Airy differential equation available in the extant literature. Finally, we illustrate how to construct reliable approximations of the probability density function of the solution stochastic process to the random fractional Airy differential equation by combining the knowledge of the mean and the variance and the Principle of Maximum Entropy., This work has been partially supported by the Ministerio de Economia y Competitividad grant MTM2017-89664-P. The authors express their deepest thanks and respect to the editors and reviewers for their valuable comments.

In this paper, we prove the existence, stability and controllability results for fractional damped differential system with non-instantaneous impulses. The results are obtained by using Banach fixed-point theorem, nonlinear functional analysis, Mittag-Leffler matrix function and controllability Grammian matrix. At last, some numerical examples are given to illustrate the obtained theory.

In this article, we establish some sufficient conditions for total controllability of a neutral fractional differential system with impulsive conditions in the finite-dimensional spaces. This type of controllability concerns the controllability problem not only at the final time but also at the impulse time. We use Mittag-Leffler matrix function, nonlinear functional analysis, controllability Grammian type matrices and fixed point theorem due to Krasnoselskii’s to establish these results. Also, we extend these results for the studied problem with the integro term. At last, some simulated examples are given to validate the obtained results.

A time-fractional diffusion equation involving the Dirichlet energy is considered with nonlocal diffusion operator in the space which has dimension d ∈ { 2 , 3 } and the Caputo sense fractional derivative in time. Further, nonlocal term in diffusion operator is of Kirchhoff type. We discretize the space using the Galerkin finite elements and time using the finite difference scheme on a uniform mesh. First, we prove the existence and uniqueness of a fully discrete numerical solution of the problem using the Brouwer fixed point theorem. Then, we give a priori bounds and convergence estimates in L 2 and L ∞ norms for fully-discrete problem. A more delicate analysis in the error provides the second order convergence for the proposed scheme. Numerical results are provided to validate the theoretical analysis.

In this manuscript, we investigate existence theory as well as stability results to the biological model of HIV (human immunodeficiency virus) disease. We consider the proposed model under Caputo-Fabrizio derivative (CFD) with exponential kernel. We investigate the suggested model from other perspectives by using fixed point approached derive its existence and uniqueness of solution. Further the stability of the concerned solution in Hyers-Ulam sense is also investigated. Further to derive the approximate solution in the form of series to the considered model, we use integral transform of Laplace coupled with Adomian decomposition method. The concerned technique is powerful tool to find semi-analytical solutions to many nonlinear problems. Finally, we demonstrate the results of approximate solutions through graphs by using Matlab.

This manuscript introduces the square-mean doubly weighted pseudo almost automorphy and also square-mean doubly weighted pseudo almost automorphy in the sense of Stepanov ( S l 2 ) over time scales. We derive results for a general stochastic dynamic system on time scales which can model a stochastic cellular neural network with time shifting delays on time scales. The coefficients are considered to be doubly weighted Stepanov-like pseudo almost automorphic functions in square-mean sense which is more general than weighted pseudo almost automorphic functions. We present several new and key results such as composition theorem for such functions on time scale. These results play a crucial role in order to study qualitative properties of nonlinear differential equations. Furthermore, we study the existence of a unique solution of stochastic delay cellular neural network on time scales. These results improve and extend the previous works in this direction. At the end, a numerical example is given to illustrate the analytical findings.

In this paper, we study P-type, PIα-type, and D-type iterative learning control for fractional impulsive evolution equations in Banach spaces. We present triple convergence results for open-loop iterative learning schemes in the sense of λ-norm through rigorous analysis. The proposed iterative learning control schemes are effective to fractional hybrid infinite-dimensional distributed parameter systems. Finally, an example is given to illustrate our theoretical results. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we investigate the approximate controllability of fractional stochastic differential inclusions with nonlocal conditions. In particular, we obtain a new set of sufficient conditions for the approximate controllability of nonlinear fractional stochastic differential inclusions under the assumption that the corresponding linear system is approximately controllable. In addition, we establish the approximate controllability results for the fractional stochastic control system with infinite delay. The results are obtained with the help of the fixed-point theorem for multivalued operators and fractional calculus. Also, two examples are provided to illustrate the obtained theory.

We prove existence and uniqueness of mild solutions to Sobolev type fractional nonlocal dynamic equations in Banach spaces. The Sobolev nonlocal condition is considered in terms of a Riemann-Liouville fractional derivative. A Lagrange optimal control problem is considered, and existence of a multi-integral solution obtained. Main tools include fractional calculus, semigroup theory, fractional power of operators, a singular version of Gronwall's inequality, and Leray-Schauder fixed point theorem. An example illustrating the theory is given., This is a preprint of a paper whose final and definite form will appear in Fract. Calc. Appl. Anal. Paper submitted 10/April/2014; accepted for publication 21/Sept/2014

We introduce the optimality question to the relaxation in multiple control problems described by Sobolev type nonlinear fractional differential equations with nonlocal control conditions in Banach spaces. Moreover, we consider the minimization problem of multi-integral functionals, with integrands that are not convex in the controls, of control systems with mixed nonconvex constraints on the controls. We prove, under appropriate conditions, that the relaxation problem admits optimal solutions. Furthermore, we show that those optimal solutions are in fact limits of minimizing sequences of systems with respect to the trajectory, multi-controls, and the functional in suitable topologies., This is a preprint of a paper whose final and definite form will be published in Journal of Optimization Theory and Applications, ISSN 0022-3239 (print), ISSN 1573-2878 (electronic). Submitted: 26-Dec-2014; Revised: 14-Apr-2015; Accepted: 19-Apr-2015

This paper investigates the existence and uniqueness of mild solutions for a class of Sobolev type fractional nonlocal abstract evolution equations in Banach spaces. We use fractional calculus, semigroup theory, a singular version of Gronwall inequality and Leray-Schauder fixed point theorem for the main results. A new kind of Sobolev type appears in terms of two linear operators is introduced. To extend previous works in the field, an existence result of optimal multi-control pairs governed by the presented system is proved. Finally, an example is also given to illustrate the obtained theory.

We study a class of fractional stochastic dynamic control systems of Sobolev type in Hilbert spaces. We use fixed point technique, fractional calculus, stochastic analysis, and methods adopted directly from deterministic control problems for the main results. A new set of sufficient conditions for approximate controllability is formulated and proved. An example is also given to provide the obtained theory.

We establish existence, approximate controllability and optimal control of a class of impulsive non-local non-linear fractional dynamical systems in Banach spaces. We use fractional calculus, sectorial operators and Krasnoselskii fixed point theorems for the main results. Approximate controllability results are discussed with respect to the inhomogeneous non-linear part. Moreover, we prove existence results of optimal pairs of corresponding fractional control systems with a Bolza cost functional., Comment: This is a preprint of a paper whose final and definite form is with 'Miskolc Math. Notes', ISSN 1787-2405 (printed version), ISSN 1787-2413 (electronic version), available at [http://mat76.mat.uni-miskolc.hu/~mnotes]

We introduce a nonlocal control condition and the notion of approximate controllability for fractional order quasilinear control inclusions. Approximate controllability of a fractional control nonlocal delay quasilinear functional differential inclusion in a Hilbert space is studied. The results are obtained by using the fractional power of operators, multi-valued analysis, and Sadovskii's fixed point theorem. Main result gives an appropriate set of sufficient conditions for the considered system to be approximately controllable. As an example, a fractional partial nonlocal control functional differential inclusion is considered., Comment: This is a preprint of a paper whose final and definite form will be published in Applied Mathematics and Computation, ISSN 0096-3003 (see http://www.sciencedirect.com/science/journal/00963003). Submitted 12/Feb/2013; Revised 04/May/2014; Accepted 25/May/2014

We discuss the existence of solutions for a class of some separated boundary differential inclusions of fractional orders2<α<3involving the Caputo derivative. In order to obtain necessary conditions for the existence result, we apply the fixed point technique, fractional calculus, and multivalued analysis.

We introduce a new concept called implicit evolution system to establish the existence results of mild and strong solutions of a class of fractional nonlocal nonlinear integrodifferential system, then we prove the exact null controllability result of a class of fractional evolution nonlocal integrodifferential control system in Banach space. As an application that illustrates the abstract results, two examples are provided.

In this article, sufficient conditions for the existence result of quasilinear multi-delay integro-differential equations of fractional orders with nonlocal impulsive conditions in Banach spaces have been presented using fractional calculus, resolvent operators, and Banach fixed point theorem. As an application that illustrates the abstract results, a nonlocal impulsive quasilinear multi-delay integro-partial differential system of fractional order is given. AMS Subject Classifications. 34K05, 34G20, 26A33, 35A05.

We introduce a new notion called fractional stochastic nonlocal condition, and then we study approximate controllability of class of fractional stochastic nonlin- ear differential equations of Sobolev type in Hilbert spaces. We use Holder's inequality, fixed point technique, fractional calculus, stochastic analysis and methods adopted di- rectly from deterministic control problems for the main results. A new set of sufficient conditions is formulated and proved for the fractional stochastic control system to be approximately controllable. An example is given to illustrate the abstract results.

We investigate nonexistence results of nontrivial solutions of fractional differential inequalities of the form $$\bigl(\mathrm{FS}^{m}_{q}\bigr)\mbox{:}\quad \left \{ \textstyle\begin{array}{l} \mathbf{D}^{q}_{0/t}x_{i}-\Delta_{\mathbb{H}}(\lambda_{i}x_{i}) \geq {|\eta|}^{\alpha_{i+1}} {| x_{i+1} |}^{\beta_{i+1}}, \quad (\eta ,t) \in{\mathbb{H}}^{N}\times\, ]0,+\infty [ , 1 \leq i \leq m, \\ x_{m+1}=x_{1} , \end{array}\displaystyle \right . $$ where $\mathbf{D}^{q}_{0/t}$ is the time-fractional derivative of order $q \in(1,2)$ in the sense of Caputo, $\Delta_{\mathbb{H}}$ is the Laplacian in the $(2N+1)$ -dimensional Heisenberg group ${\mathbb {H}}^{N}$ , ${|\eta|}$ is the distance from η in ${\mathbb {H}}^{N}$ to the origin, $m\geq2$ , $\alpha_{m+1}=\alpha_{1}$ , $\beta _{m+1}=\beta_{1}$ , and $\lambda_{i}\in L^{\infty}({\mathbb{H}}^{N} \times\, ]0,+\infty [ )$ , $1 \leq i \leq m$ . The main results are concerned with $Q \equiv2N + 2$ , less than the critical exponents that depend on q, $\alpha_{i}$ , and $\beta_{i}$ , $1 \leq i \leq m$ . For $q=2$ , we deduce the results given by El Hamidi and Kirane (Abstr. Appl. Anal. 2004(2):155-164, 2004) and El Hamidi and Obeid (J. Math. Anal. Appl. 208(1):77-90, 2003) from the hyperbolic systems. For $m=1$ , we study the scalar case $$(\mathrm{FI}_{q})\mbox{:}\quad \mathbf{D}^{q}_{0/t}x - \Delta_{\mathbb{H}}(\lambda x) \geq {|\eta|}^{\alpha} {| x |}^{\beta}, $$ where $\beta>1$ , α are real parameters. In the last case, for $q=2$ , we return to the approach of Pohozaev and Veron (Manuscr. Math. 102:85-99, 2000) from the hyperbolic inequalities.