Enlarged Controllability of Riemann-Liouville Fractional Differential Equations

We investigate exact enlarged controllability for time fractional diffusion systems of Riemann-Liouville type. The Hilbert uniqueness method is used to prove exact enlarged controllability for both cases of zone and pointwise actuators. A penalization method is given and the minimum energy control is characterized.Comment: This is a preprint of a paper whose final and definite form is with 'Journal of Computational and Nonlinear Dynamics', ISSN 1555-1415, eISSN 1555-1423, CODEN JCNDDM, available at [http://computationalnonlinear.asmedigitalcollection.asme.org]. Submitted 10-Aug-2017; Revised 28-Sept-2017 and 24-Oct-2017; Accepted 05-Nov-201

Enlarged controllability and optimal control of sub-diffusion processes with Caputo fractional derivatives

We investigate the exact enlarged controllability and optimal control of a fractional diffusion equation in Caputo sense. This is done through a new definition of enlarged controllability that allows us to extend available contributions. Moreover, the problem is studied using two approaches: a reverse Hilbert uniqueness method, generalizing the approach introduced by Lions in 1988, and a penalization method, which allow us to characterize the minimum energy control.publishe

The stability and stabilization of infinite dimensional Caputo-time fractional differential linear systems

We investigate the stability and stabilization concepts for infinite dimensional time fractional differential linear systems in Hilbert spaces with Caputo derivatives. Firstly, based on a family of operators generated by strongly continuous semigroups and on a probability density function, we provide sufficient and necessary conditions for the exponential stability of the considered class of systems. Then, by assuming that the system dynamics is symmetric and uniformly elliptic and by using the properties of the Mittag-Leffler function, we provide sufficient conditions that ensure strong stability. Finally, we characterize an explicit feedback control that guarantees the strong stabilization of a controlled Caputo time fractional linear system through a decomposition approach. Some examples are presented that illustrate the effectiveness of our results.publishe

Regional enlarged observability of fractional differential equations with Riemann—Liouville time derivatives

We introduce the concept of regional enlarged observability for fractional evolution differential equations involving Riemann-Liouville derivatives. The Hilbert Uniqueness Method (HUM) is used to reconstruct the initial state between two prescribed functions, in an interested subregion of the whole domain, without the knowledge of the statepublishe

A Decay Estimate for Constrained Semilinear Systems

The paper proposes a constrained feedback control that guarantee weak and strong stabilizability for distributed semilinear systems of the form : dy(t) dt = Ay(t)+ p(t)Ny(t), where A is the infinitesimal generator of a linear C0−semigroup of contractions on a Hilbert space H and N is a (nonlinear) operator from H into its self. A decay rate of the state is estimated. Also the robustness of the considered control is discussed. Applications and simulations are provided

Gradient Controllability for Hyperbolic Systems

This paper deals with the problem of regional gradient controllability of hyperbolic systems. We show how one can reach a desired state gradient given only on a part of the system evolution domain. Also we explore a numerical approach using Hilbert UniquenessMethod (HUM) that leads to an explicit formula of the optimal control. The obtained results are successfully tested through computer simulations leading to some conjectures

Regional Controllability of Riemann–Liouville Time-Fractional Semilinear Evolution Equations

In this paper, we discuss the exact regional controllability of fractional evolution equations involving Riemann–Liouville fractional derivative of order q∈0,1. The result is obtained with the help of the theory of fractional calculus, semigroup theory, and Banach fixed-point theorem under several assumptions on the corresponding linear system and the nonlinear term. Finally, some numerical simulations are given to illustrate the obtained result

Regional Constrained Observability for Distributed Hyperbolic Systems

The aim of this paper is to develop the question of the regional constrained observability for distributed hyperbolic system evolving in spatial domain W. It consists in the reconstruction of the initial conditions, in a subregion w of W, knowing that the initial position is between two prescribed functions in w and also the initial speed is between two others functions also prescribed in w. We give some definitions and proprieties concerning this concept and then we describe two approaches for solving this problem. The first is based on subdifferential technics and the second uses the Lagrangian multiplier method. This last approach leads to an algorithm for the reconstruction of the initial conditions. The obtained results are illustrated by numerical simulations which lead to some conjectures