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Evolution Equations & Control Theory

2015 , Volume 4 , Issue 2

Special issue dedicated to Professor Abdelhaq El Jai on the occasion of his retirement

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Preface
Thomas Seidman, Samira El Yacoubi and Abdessamad Tridane
2015, 4(2): i-iv doi: 10.3934/eect.2015.4.2i +[Abstract](211) +[PDF](141.4KB)
Abstract:
This special issue of the Journal of Evolution Equations and Control Theory (EECT) is dedicated to Professor Abdelhaq El Jai on the occasion of his retirement and in celebration of his significant achievements in the field of control and distributed parameter systems theory, both in his own research and in his leadership in the development of a Moroccan research community in DPS.

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Global dynamics on a circular domain of a diffusion predator-prey model with modified Leslie-Gower and Beddington-DeAngelis functional type
Walid Abid, Radouane Yafia, M.A. Aziz-Alaoui, Habib Bouhafa and Azgal Abichou
2015, 4(2): 115-129 doi: 10.3934/eect.2015.4.115 +[Abstract](269) +[PDF](849.6KB)
Abstract:
In this paper, we present a predator-prey model with modified Leslie-Gower and Beddington-DeAngelis functional response. The model is governed by a two dimensional reaction diffusion system defined on a disc domain. The conditions of boundedness, existence of a positively invariant and attracting set are proved. Sufficient conditions of local and global stability of the positive steady state are established. In the end, we carry out some numerical simulations in order to illustrate our theoretical results and to interpret how biological processes affect spatio-temporal pattern formation.
On observers and compensators for infinite dimensional semilinear systems
Mohammed Elarbi Achhab
2015, 4(2): 131-142 doi: 10.3934/eect.2015.4.131 +[Abstract](208) +[PDF](518.2KB)
Abstract:
This paper investigates two important questions for a class of partially observed infinite-dimensional semilinear systems.The first one is the design of an exponential Luenberger-like observer for this class of systems. Then, the stabilization problem around a desired equilibrium profile of the system is solved, yielding a compensator based on the Luenberger-like observer. Finally, the main result is applied to a nonisothermal chemical plug flow reactor model. The approach is illustrated by numerical simulations. The paper also gives a short overview of selected works considering the same questions for linear distributed parameter systems.
Input/state invariant LQ-optimal control: Application to competitive coexistence in a chemostat
Charlotte Beauthier, Joseph J. Winkin and Denis Dochain
2015, 4(2): 143-158 doi: 10.3934/eect.2015.4.143 +[Abstract](194) +[PDF](441.4KB)
Abstract:
This paper is devoted to the application of the input/state-invariant linear quadratic (LQ) problem in order to solve the problem of coexistence of species in competition in a chemostat. The methodology that is used has for objective to guarantee the local positive input/state-invariance of the nonlinear system which describes the chemostat model by ensuring the input/state-invariance of its linear approximation around an equilibrium. This is achieved by applying an appropriate LQ-optimal control to the system, following two different approaches, namely a receding horizon method and an inverse problem approach.
Constrained controllability for lumped linear systems
Larbi Berrahmoune
2015, 4(2): 159-175 doi: 10.3934/eect.2015.4.159 +[Abstract](208) +[PDF](423.6KB)
Abstract:
We consider linear lumped control systems of the form $y'(t)=Ay(t)+Bu(t)$ where $A \in \mathbb{R}^{m\times m}$, $B \in \mathbb{R}^{m\times p}$. Taking into account eventual control constraint (such as saturation), we study the problem of controllability by using a general variational approach. The results are applied to the following saturation constraints on the control $u(t)=(u_{1}(t), ..., u_{p}(t))$: (i) the quadratic one specified by $\underset{j=1}{\overset{p}\sum}\left|u_{j}(t)\right|^{2} \leq 1$ for all $0\leq t\leq T$ and (ii) the polyhedral one characterized by $\underset{1 \leq j \leq p}{\max}\left|u_{j}(t)\right| \leq 1$ for all $0\leq t\leq T$.
Flux reconstruction for hyperbolic systems: Sensors and simulations
Adil Khazari and Ali Boutoulout
2015, 4(2): 177-192 doi: 10.3934/eect.2015.4.177 +[Abstract](173) +[PDF](734.8KB)
Abstract:
This paper aims to establish necessary conditions for sensors structure (number and location) in order to obtain regional boundary gradient observability for hyperbolic system. The obtained results are applied to a two-dimensional diffusion process considering various types of sensors. Also, we will explore an approach that can reconstruct the gradient on a part $\Gamma$ of the boundary $\partial\Omega$ of the evolution domain $\Omega$. The simulations illustrate the established results and lead to some conjectures.
Modeling plant nutrient uptake: Mathematical analysis and optimal control
Loïc Louison, Abdennebi Omrane, Harry Ozier-Lafontaine and Delphine Picart
2015, 4(2): 193-203 doi: 10.3934/eect.2015.4.193 +[Abstract](680) +[PDF](358.1KB)
Abstract:
The article studies the nutrient transfer mechanism and its control for mixed cropping systems. It presents a mathematical analysis and optimal control of the absorbed nutrient concentration, governed by a transport-diffusion equation in a bounded domain near the root system, satisfying to the Michaelis-Menten uptake law.
    The existence, uniqueness and positivity of a solution (the absorbed concentration) is proved. We also show that for a given plant we can determine the optimal amount of required nutrients for its growth. The characterization of the optimal control leading to the desired concentration at the root surface is obtained. Finally, some numerical simulations to evaluate the theoretical results are proposed.
Finite rank distributed control for the resistive diffusion equation using damping assignment
Ngoc Minh Trang Vu and Laurent Lefèvre
2015, 4(2): 205-220 doi: 10.3934/eect.2015.4.205 +[Abstract](183) +[PDF](848.0KB)
Abstract:
A first extension of the IDA-PBC control synthesis to infinite dimensional port Hamiltonian systems is investigated, using the same idea as for the finite dimensional case, that is transform the original model into a closed loop target Hamiltonian model using feedback control. To achieve this goal both finite rank distributed control and boundary control are used. The proposed class of considered port Hamiltonian distributed parameters systems is first defined. Then the matching equation is derived for this class before considering the particular case of damping assignment on the resistive diffusion example, for the radial diffusion of the poloidal magnetic flux in tokamak reactors.
Optimal bounded controls problem for bilinear systems
El Hassan Zerrik and Nihale El Boukhari
2015, 4(2): 221-232 doi: 10.3934/eect.2015.4.221 +[Abstract](176) +[PDF](380.0KB)
Abstract:
The aim of this paper is to study the optimal control problem for finite dimensional bilinear systems with bounded controls. We characterize an optimal control that minimizes a quadratic cost functional using Pontryagin's minimum principle, we derive sufficient conditions of uniqueness from the fixed point theorem, and we develop an algorithm that allows to compute the optimal control and the associated states. Our approach is applied to a cancer treatment by chemotherapy in order to determine the optimal dose of a killing agent.
Relating systems properties of the wave and the Schrödinger equation
Hans Zwart, Yann Le Gorrec and Bernhard Maschke
2015, 4(2): 233-240 doi: 10.3934/eect.2015.4.233 +[Abstract](265) +[PDF](344.1KB)
Abstract:
In this article we show that systems properties of the systems governed by the second order differential equation $\frac{d^{2}w}{dt^{2}}=-A_{0}w$ and the first order differential equation $\frac{dz}{dt}=iA_{0}z$ are related. This can be used to show that, for instance, exact observability of the $N$-dimensional wave equation implies the similar property for the $N$-dimensional Schrödinger equation.

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