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EECT is primarily devoted to papers on analysis and control of infinite dimensional systems with emphasis on applications to PDE's and FDEs. Topics include:

  *  Modeling of physical systems as infinite-dimensional processes
  *  Direct problems such as existence, regularity and well-posedness
  *  Stability, long-time behavior and associated dynamical attractors
  *  Indirect problems such as exact controllability, reachability theory and inverse problems
  *  Optimization - including shape optimization - optimal control, game theory and calculus of variations
  *  Well-posedness, stability and control of coupled systems with an interface. Free boundary problems and problems with moving interface(s)
  *  Applications of the theory to physics, chemistry, engineering, economics, medicine and biology

The journal also welcomes excellent contributions on interesting and challenging ODE systems which arise as simplified models of infinite-dimensional structures.

  • AIMS is a member of COPE. All AIMS journals adhere to the publication ethics and malpractice policies outlined by COPE.
  • Publishes 4 issues a year in March, June, September and December.
  • Publishes online only.
  • Indexed in Science Citation Index-Expanded, Current Contents/Physical, Chemical & Earth Sciences (CC/PC&ES), Web of Science, MathSciNet and Zentralblatt MATH.
  • Archived in Portico and CLOCKSS.
  • EECT is a publication of the American Institute of Mathematical Sciences. All rights reserved.

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Energy decay for the damped wave equation on an unbounded network
Rachid Assel and Mohamed Ghazel
2018, 7(3) : 335-351 doi: 10.3934/eect.2018017 +[Abstract](207) +[HTML](70) +[PDF](431.8KB)
Abstract:

We study the wave equation on an unbounded network of \begin{document} $N, N∈\mathbb{N}^*$ \end{document}, finite strings and a semi-infinite one with a single vertex identified to 0. We consider continuity and dissipation conditions at the vertex and Dirichlet conditions at the extremities of the finite edges. The dissipation is given by a damping constant $α>0$ via the condition \begin{document} $\sum_{j = 0}^N\partial_xu_j(0, t) = α \partial_tu_0(0, t)$ \end{document}. We give a complete spectral description and we use it to study the energy decay of the solution. We prove that for \begin{document} $α\not = N+1$ \end{document} we have an exponential decay of the energy and we give an explicit formula for the decay rate when the finite edges have the same length.

Rate of convergence of inertial gradient dynamics with time-dependent viscous damping coefficient
Hedy Attouch, Alexandre Cabot, Zaki Chbani and Hassan Riahi
2018, 7(3) : 353-371 doi: 10.3934/eect.2018018 +[Abstract](155) +[HTML](66) +[PDF](478.17KB)
Abstract:

In a Hilbert space \begin{document} $\mathcal H$ \end{document}, we study the convergence properties when \begin{document} $t→+∞$ \end{document} of the trajectories of the second-order differential equation

where \begin{document} $\nablaΦ$ \end{document} is the gradient of a convex continuously differentiable function \begin{document} $Φ: \mathcal H→\mathbb R$ \end{document}, and \begin{document} $γ(t)$ \end{document} is a time-dependent positive viscous damping coefficient. This study aims to offer a unifying vision on the subject, and to complement the article by Attouch and Cabot (J. Diff. Equations, 2017). We obtain convergence rates for the values \begin{document} $Φ(x(t))-{\rm{inf}}_\mathcal{H} Φ$ \end{document} and the velocities under general conditions involving only \begin{document} $γ(·)$ \end{document} and its derivatives. In particular, in the case \begin{document} $γ(t) = \frac{α}{t}$ \end{document}, which is directly connected to the Nesterov accelerated gradient method, our approach allows us to cover all the positive values of \begin{document} $α$ \end{document}, including the subcritical case \begin{document} $α<3$ \end{document}. Our approach is based on the introduction of a new class of Lyapunov functions.

Existence and continuous-discrete asymptotic behaviour for Tikhonov-like dynamical equilibrium systems
Aicha Balhag, Zaki Chbani and Hassan Riahi
2018, 7(3) : 373-401 doi: 10.3934/eect.2018019 +[Abstract](153) +[HTML](69) +[PDF](574.93KB)
Abstract:

We consider the regularized Tikhonov-like dynamical equilibrium problem: find \begin{document} $u: [0, +∞ [\to\mathcal H$ \end{document} such that for a.e. \begin{document} $t \ge 0$ \end{document} and every \begin{document} $y∈K$ \end{document}, \begin{document} $\langle \dot{u}(t), y-u(t)\rangle +F(u(t), y)+\varepsilon(t) \langle u(t), y-u(t)\rangle \ge 0$ \end{document}, where \begin{document} $F:K×K \to \mathbb{R}$ \end{document} is a monotone bifunction, \begin{document} $K$ \end{document} is a closed convex set in Hilbert space \begin{document} $\mathcal H$ \end{document} and the control function \begin{document} $\varepsilon(t)$ \end{document} is assumed to tend to 0 as \begin{document} $t \to +∞$ \end{document}. We first establish that the corresponding Cauchy problem admits a unique absolutely continuous solution. Under the hypothesis that \begin{document} $\int_{0}^{+∞} \varepsilon (t) dt <∞$ \end{document}, we obtain weak ergodic convergence of \begin{document} $u(t)$ \end{document} to \begin{document} $x∈K$ \end{document} solution of the following equilibrium problem \begin{document} $F(x, y) \ge 0, \;\forall y∈K$ \end{document}. If in addition the bifunction is assumed demipositive, we show weak convergence of \begin{document} $u(t)$ \end{document} to the same solution. By using a slow control \begin{document} $\int_{0}^{+∞} \varepsilon (t) dt = ∞$ \end{document} and assuming that the bifunction \begin{document} $F$ \end{document} is 3-monotone, we show that the term \begin{document} $\varepsilon (t)u(t)$ \end{document} asymptotically acts as a Tikhonov regularization, which forces all the trajectories to converge strongly towards the element of minimal norm of the closed convex set of equilibrium points of \begin{document} $F$ \end{document}. Also, in the case where \begin{document} $\varepsilon $ \end{document} has a slow control property and \begin{document} $\int_{0}^{+∞}\vert \dot{\varepsilon} (t) \vert dt < +∞ $ \end{document}, we show that the strong convergence property of \begin{document} $u(t)$ \end{document} is satisfied. As applications, we propose a dynamical system to solve saddle-point problem and a neural dynamical model to handle a convex programming problem. In the last section, we propose two Tikhonov regularization methods for the proximal algorithm. We firstly use the prox-penalization algorithm \begin{document} $(ProxPA)$ \end{document} by iteration \begin{document} $ x_{n+1} = J^{F_n}_{λ_n}(x_n)$ \end{document} where \begin{document} $F_n(x, y) = F(x, y)+\varepsilon_n \langle x, y-x\rangle$ \end{document}, and \begin{document} $\varepsilon_n$ \end{document} is the Liapunov parameter; afterwards, we propose the descent-proximal (forward-backward) algorithm \begin{document} $(DProxA)$ \end{document}: \begin{document} $x_{n+1} = J^F_{λ_n} ((1 - λ_n\varepsilon_n)x_n)$ \end{document}. We provide low conditions that guarantee a strong convergence of these algorithms to least norm element of the set of equilibrium points.

Exact boundary controllability for the Boussinesq equation with variable coefficients
Jamel Ben Amara and Hedi Bouzidi
2018, 7(3) : 403-415 doi: 10.3934/eect.2018020 +[Abstract](167) +[HTML](57) +[PDF](486.31KB)
Abstract:

In this paper we study the exact boundary controllability for the following Boussinesq equation with variable physical parameters:

where \begin{document}$l>0$\end{document}, the coefficients \begin{document}$ρ(x)>0, \sigma (x)>0 $\end{document}, \begin{document}$q(x)≥0$\end{document} in \begin{document}$\left[ {0,l} \right]$\end{document} and \begin{document}$u$\end{document} is the control acting at the end \begin{document}$x=l$\end{document}. We prove that the linearized problem is exactly controllable in any time \begin{document}$T>0$\end{document}. Our approach is essentially based on a detailed spectral analysis together with the moment method. Furthermore, we establish the local exact controllability for the nonlinear problem by fixed point argument. This problem has been studied by Crépeau [Diff. Integ. Equat., 2002] in the case of constant coefficients \begin{document}$ρ\equiv\sigma \equiv q\equiv1$\end{document}.

Control problems and invariant subspaces for sabra shell model of turbulence
Tania Biswas and Sheetal Dharmatti
2018, 7(3) : 417-445 doi: 10.3934/eect.2018021 +[Abstract](88) +[HTML](50) +[PDF](455.62KB)
Abstract:

Shell models of turbulence are representation of turbulence equations in Fourier domain. Various shell models and their existence theory along with numerical simulations have been studied earlier. One of the most suitable shell model of turbulence is so called sabra shell model. The existence, uniqueness and regularity property of this model are extensively studied in [11]. We follow the same functional setup given in [11] and study control problems related to it. We associate two cost functionals: one ensures minimizing turbulence in the system and the other addresses the need of taking the flow near a priori known state. We derive optimal controls in terms of the solution of adjoint equations for corresponding linearized problems. Another interesting problem studied in this work is to establish feedback controllers which would preserve prescribed physical constraints. Since fluid equations have certain fundamental invariants, we would like to preserve these quantities via a control in the feedback form. We utilize the theory of nonlinear semi groups and represent the feedback control as a multi-valued feedback term which lies in the normal cone of the convex constraint space, under certain assumptions. Moreover, one of the most interesting result of this work is that we can design a feedback control with only finitely many modes, which is able to preserve the flow in the neighborhood of the constraint set.

Null controllability of the incompressible Stokes equations in a 2-D channel using normal boundary control
Shirshendu Chowdhury, Debanjana Mitra and Michael Renardy
2018, 7(3) : 447-463 doi: 10.3934/eect.2018022 +[Abstract](80) +[HTML](49) +[PDF](408.6KB)
Abstract:

In this paper, we consider the Stokes equations in a two-dimensional channel with periodic conditions in the direction of the channel. We establish null controllability of this system using a boundary control which acts on the normal component of the velocity only. We show null controllability of the system, subject to a constraint of zero average, by proving an observability inequality with the help of a Müntz-Szász Theorem.

Carleman estimates for forward and backward stochastic fourth order Schrödinger equations and their applications
Peng Gao
2018, 7(3) : 465-499 doi: 10.3934/eect.2018023 +[Abstract](100) +[HTML](47) +[PDF](538.97KB)
Abstract:

In this paper, we establish the Carleman estimates for forward and backward stochastic fourth order Schrödinger equations, on basis of which, we can obtain the observability, unique continuation property and the exact controllability for the forward and backward stochastic fourth order Schrödinger equations.

Optimal control of second order delay-discrete and delay-differential inclusions with state constraints
Elimhan N. Mahmudov
2018, 7(3) : 501-529 doi: 10.3934/eect.2018024 +[Abstract](117) +[HTML](52) +[PDF](532.41KB)
Abstract:

The present paper studies a new class of problems of optimal control theory with state constraints and second order delay-discrete (DSIs) and delay-differential inclusions (DFIs). The basic approach to solving this problem is based on the discretization method. Thus under the "regularity condition the necessary and sufficient conditions of optimality for problems with second order delay-discrete and delay-approximate DSIs are investigated. Then by using discrete approximations as a vehicle, in the forms of Euler-Lagrange and Hamiltonian type inclusions the sufficient conditions of optimality for delay-DFIs, including the peculiar transversality ones, are proved. Here our main idea is the use of equivalence relations for subdifferentials of Hamiltonian functions and locally adjoint mappings (LAMs), which allow us to make a bridge between the basic optimality conditions of second order delay-DSIs and delay-discrete-approximate problems. In particular, applications of these results to the second order semilinear optimal control problem are illustrated as well as the optimality conditions for non-delayed problems are derived.

Controllability for fractional evolution inclusions without compactness
Yong Zhou, V. Vijayakumar and R. Murugesu
2015, 4(4) : 507-524 doi: 10.3934/eect.2015.4.507 +[Abstract](738) +[PDF](424.8KB) Cited By(52)
Hyperbolic Navier-Stokes equations I: Local well-posedness
Reinhard Racke and Jürgen Saal
2012, 1(1) : 195-215 doi: 10.3934/eect.2012.1.195 +[Abstract](635) +[PDF](426.3KB) Cited By(19)
On well-posedness of incompressible two-phase flows with phase transitions: The case of equal densities
Jan Prüss, Yoshihiro Shibata, Senjo Shimizu and Gieri Simonett
2012, 1(1) : 171-194 doi: 10.3934/eect.2012.1.171 +[Abstract](655) +[PDF](459.7KB) Cited By(15)
Feedback control of nonlinear dissipative systems by finite determining parameters - A reaction-diffusion paradigm
Abderrahim Azouani and Edriss S. Titi
2014, 3(4) : 579-594 doi: 10.3934/eect.2014.3.579 +[Abstract](507) +[PDF](418.2KB) Cited By(14)
On Kelvin-Voigt model and its generalizations
Miroslav Bulíček, Josef Málek and K. R. Rajagopal
2012, 1(1) : 17-42 doi: 10.3934/eect.2012.1.17 +[Abstract](972) +[PDF](596.6KB) Cited By(13)
The effect of the wave speeds and the frictional damping terms on the decay rate of the Bresse system
Abdelaziz Soufyane and Belkacem Said-Houari
2014, 3(4) : 713-738 doi: 10.3934/eect.2014.3.713 +[Abstract](544) +[PDF](498.1KB) Cited By(13)
Fluid-structure interaction with and without internal dissipation of the structure: A contrast study in stability
George Avalos and Roberto Triggiani
2013, 2(4) : 563-598 doi: 10.3934/eect.2013.2.563 +[Abstract](546) +[PDF](827.5KB) Cited By(12)
Rational decay rates for a PDE heat--structure interaction: A frequency domain approach
George Avalos and Roberto Triggiani
2013, 2(2) : 233-253 doi: 10.3934/eect.2013.2.233 +[Abstract](596) +[PDF](610.3KB) Cited By(11)
Regularity and stability of a wave equation with a strong damping and dynamic boundary conditions
Nicolas Fourrier and Irena Lasiecka
2013, 2(4) : 631-667 doi: 10.3934/eect.2013.2.631 +[Abstract](807) +[PDF](803.7KB) Cited By(9)
On the Cauchy problem for the Schrödinger-Hartree equation
Binhua Feng and Xiangxia Yuan
2015, 4(4) : 431-445 doi: 10.3934/eect.2015.4.431 +[Abstract](475) +[PDF](441.7KB) Cited By(8)
Martingale solutions for stochastic Navier-Stokes equations driven by Lévy noise
Kumarasamy Sakthivel and Sivaguru S. Sritharan
2012, 1(2) : 355-392 doi: 10.3934/eect.2012.1.355 +[Abstract](552) +[PDF](583.6KB) PDF Downloads(268)
The controllability of a thermoelastic plate problem revisited
Moncef Aouadi and Taoufik Moulahi
2018, 7(1) : 1-31 doi: 10.3934/eect.2018001 +[Abstract](904) +[HTML](389) +[PDF](612.96KB) PDF Downloads(210)
Stability problem for the age-dependent predator-prey model
Antoni Leon Dawidowicz and Anna Poskrobko
2018, 7(1) : 79-93 doi: 10.3934/eect.2018005 +[Abstract](890) +[HTML](369) +[PDF](421.72KB) PDF Downloads(134)
On state-dependent sweeping process in Banach spaces
Dalila Azzam-Laouir and Fatiha Selamnia
2018, 7(2) : 183-196 doi: 10.3934/eect.2018009 +[Abstract](326) +[HTML](172) +[PDF](393.79KB) PDF Downloads(107)
Inverse observability inequalities for integrodifferential equations in square domains
Paola Loreti and Daniela Sforza
2018, 7(1) : 61-77 doi: 10.3934/eect.2018004 +[Abstract](541) +[HTML](323) +[PDF](435.73KB) PDF Downloads(104)
Heat-viscoelastic plate interaction: Analyticity, spectral analysis, exponential decay
Roberto Triggiani and Jing Zhang
2018, 7(1) : 153-182 doi: 10.3934/eect.2018008 +[Abstract](629) +[HTML](394) +[PDF](626.99KB) PDF Downloads(104)
Energy decay for the damped wave equation on an unbounded network
Rachid Assel and Mohamed Ghazel
2018, 7(3) : 335-351 doi: 10.3934/eect.2018017 +[Abstract](207) +[HTML](70) +[PDF](431.8KB) PDF Downloads(100)
Self-similar solutions to nonlinear Dirac equations and an application to nonuniqueness
Hyungjin Huh
2018, 7(1) : 53-60 doi: 10.3934/eect.2018003 +[Abstract](682) +[HTML](339) +[PDF](323.9KB) PDF Downloads(90)
The recovery of a parabolic equation from measurements at a single point
Amin Boumenir, Vu Kim Tuan and Nguyen Hoang
2018, 7(2) : 197-216 doi: 10.3934/eect.2018010 +[Abstract](339) +[HTML](167) +[PDF](5897.52KB) PDF Downloads(84)
Exact boundary controllability for the Boussinesq equation with variable coefficients
Jamel Ben Amara and Hedi Bouzidi
2018, 7(3) : 403-415 doi: 10.3934/eect.2018020 +[Abstract](167) +[HTML](57) +[PDF](486.31KB) PDF Downloads(78)

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