Regional gradient controllability of ultra-slow diffusions involving the Hadamard-Caputo time fractional derivative

This paper investigates the regional gradient controllability for ultra-slow diffusion processes governed by the time fractional diffusion systems with a Hadamard-Caputo time fractional derivative. Some necessary and sufficient conditions on regional gradient exact and approximate controllability are first given and proved in detail. Secondly, we propose an approach on how to calculate the minimum number of $\omega-$strategic actuators. Moreover, the existence, uniqueness and the concrete form of the optimal controller for the system under consideration are presented by employing the Hilbert Uniqueness Method (HUM) among all the admissible ones. Finally, we illustrate our results by an interesting example.Comment: 16 page

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

Boundary Controllability of Riemann-Liouville Fractional Semilinear Equations

We study the boundary regional controllability of a class of Riemann-Liouville fractional semilinear sub-diffusion systems with boundary Neumann conditions. The result is obtained by using semi-group theory, the fractional Hilbert uniqueness method, and Schauder's fixed point theorem. Conditions on the order of the derivative, internal region, and on the nonlinear part are obtained. Furthermore, we present appropriate sufficient conditions for the considered fractional system to be regionally controllable and, therefore, boundary regionally controllable. An example of a population density system with diffusion is given to illustrate the obtained theoretical results. Numerical simulations show that the proposed method provides satisfying results regarding two cases of the control operator.Comment: This is a preprint version of the paper published open access in 'Commun. Nonlinear Sci. Numer. Simul.' [https://doi.org/10.1016/j.cnsns.2023.107814]. Submitted 26/Jul/2022; Revised 08/Dec/2022 and 16/Oct/2023; Accepted for publication 30/Dec/2023; Available online 03/Jan/202

Subdiffusive Source Sensing by a Regional Detection Method.

Motivated by the fact that the danger may increase if the source of pollution problem remains unknown, in this paper, we study the source sensing problem for subdiffusion processes governed by time fractional diffusion systems based on a limited number of sensor measurements. For this, we first give some preliminary notions such as source, detection and regional spy sensors, etc. Secondly, we investigate the characterizations of regional strategic sensors and regional spy sensors. A regional detection approach on how to solve the source sensing problem of the considered system is then presented by using the Hilbert uniqueness method (HUM). This is to identify the unknown source only in a subregion of the whole domain, which is easier to be implemented and could save a lot of energy resources. Numerical examples are finally included to test our results

Regional controllability and minimum energy control of delayed caputo fractional-order linear systems

We study the regional controllability problem for delayed fractional control systems through the use of the standard Caputo derivative. First, we recall several fundamental results and introduce the family of fractional-order systems under consideration. Afterward, we formulate the notion of regional controllability for fractional systems with control delays and give some of their important properties. Our main method consists of defining an attainable set, which allows us to prove exact and weak controllability. Moreover, the main results include not only those of controllability but also a powerful Hilbert uniqueness method, which allows us to solve the minimum energy optimal control problem. More precisely, an explicit control is obtained that drives the system from an initial given state to a desired regional state with minimum energy. Two examples are given to illustrate the obtained theoretical results.This research was funded by The Portuguese Foundation for Science and Technology (FCT—Fundação para a Ciência e a Tecnologia), grant number UIDB/04106/2020 (CIDMA).publishe

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

Boundary Regional Controllability of Semilinear Systems Involving Caputo Time Fractional Derivatives

We study boundary regional controllability problems for a class of semilinear fractional systems. Sufficient conditions for regional boundary controllability are proved by assuming that the associated linear system is approximately regionally boundary controllable. The main result is obtained by using fractional powers of an operator and the fixed point technique under the approximate controllability of the corresponding linear system in a suitable subregion of the space domain. An algorithm is also proposed and some numerical simulations performed to illustrate the effectiveness of the obtained theoretical results.Comment: This is a preprint of a paper whose final and definite form is published in 'Libertas Mathematica (new series)', Volume 43 (2023), No. 1 [http://system.lm-ns.org/index.php/lm-ns/article/view/1488

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

Regional Gradient Observability for Fractional Differential Equations with Caputo Time-Fractional Derivatives

We investigate the regional gradient observability of fractional sub-diffusion equations involving the Caputo derivative. The problem consists of describing a method to find and recover the initial gradient vector in the desired region, which is contained in the spacial domain. After giving necessary notions and definitions, we prove some useful characterizations for exact and approximate regional gradient observability. An example of a fractional system that is not (globally) gradient observable but it is regionally gradient observable is given, showing the importance of regional analysis. Our characterization of the notion of regional gradient observability is given for two types of strategic sensors. The recovery of the initial gradient is carried out using an expansion of the Hilbert Uniqueness Method. Two illustrative examples are given to show the application of the developed approach. The numerical simulations confirm that the proposed algorithm is effective in terms of the reconstruction error.Comment: This is a 22 pages preprint of a paper whose final and definite form is published in 'Int. J. Dyn. Control' (ISSN 2195-268X). Submitted 11/July/2022; Revised 07/Nov/22; and Accepted 26/Dec/202