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Evolution Equations & Control Theory

December 2019 , Volume 8 , Issue 4

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Controllability of the semilinear wave equation governed by a multiplicative control
Mohamed Ouzahra
2019, 8(4): 669-686 doi: 10.3934/eect.2019039 +[Abstract](793) +[HTML](251) +[PDF](358.83KB)
Abstract:

In this paper we establish several results on approximate controllability of a semilinear wave equation by making use of a single multiplicative control. These results are then applied to discuss the exact controllability properties for the one dimensional version of the system at hand. The proof relies on linear semigroup theory and the results on the additive controllability of the semilinear wave equation. The approaches are constructive and provide explicit steering controls. Moreover, in the context of undamped wave equation, the exact controllability is established for a time which is uniform for all initial states.

Simultaneous controllability of two vibrating strings with variable coefficients
Jamel Ben Amara and Emna Beldi
2019, 8(4): 687-694 doi: 10.3934/eect.2019032 +[Abstract](641) +[HTML](235) +[PDF](288.11KB)
Abstract:

We study the simultaneous exact controllability of two vibrating strings with variable physical coefficients and controlled from a common endpoint. We give sufficient conditions on the physical coefficients for which the eigenfrequencies of both systems do not coincide and the associated spectral gap is uniformly positive. Under these conditions, we show that these systems are simultaneously exactly controllable.

A penalty decomposition method for nuclear norm minimization with l1 norm fidelity term
Duo Wang, Zheng-Fen Jin and Youlin Shang
2019, 8(4): 695-708 doi: 10.3934/eect.2019034 +[Abstract](730) +[HTML](283) +[PDF](1627.75KB)
Abstract:

It is well known that the nuclear norm minimization problems are widely used in numerous fields such as machine learning, system recognition, and image processing, etc. It has captured considerable attention and taken good progress. Many researchers have made great contributions to the nuclear norm minimization problem with \begin{document}$ l_{2} $\end{document} norm fidelity term, which is just able to deal with those problems with Gaussian noise. In this paper, we propose an efficient penalty decomposition(PD) method to solve the nuclear norm minimization problem with \begin{document}$ l_{1} $\end{document} norm fidelity term. One subproblem admits a closed-form solution due to its special structure, and another can also get a closed-form solution by linearizing its quadratic term. The convergence results of the proposed method are given as well. Finally, the effectiveness and efficiency of the proposed method are verified by some numerical experiments.

Well-posedness and asymptotic behavior of the dissipative Ostrovsky equation
Hongwei Wang and Amin Esfahani
2019, 8(4): 709-735 doi: 10.3934/eect.2019035 +[Abstract](740) +[HTML](245) +[PDF](420.63KB)
Abstract:

In this paper we study the global well-posedness and the large-time behavior of solutions to the initial-value problem for the dissipative Ostrovsky equation. We show that the associated solutions decay faster than the solutions of the dissipative KdV equation.

Discontinuous solutions for the generalized short pulse equation
Giuseppe Maria Coclite and Lorenzo di Ruvo
2019, 8(4): 737-753 doi: 10.3934/eect.2019036 +[Abstract](728) +[HTML](289) +[PDF](345.08KB)
Abstract:

The generalized short pulse equation is a non-slowly-varying envelope approximation model that describes the physics of few-cycle-pulse optical solitons. This is a nonlinear evolution equation. In this paper, we prove the wellposedness of the Cauchy problem associated with this equation within a class of discontinuous solutions.

On some nonlinear problem for the thermoplate equations
Suma'inna, Hirokazu Saito and Yoshihiro Shibata
2019, 8(4): 755-784 doi: 10.3934/eect.2019037 +[Abstract](631) +[HTML](250) +[PDF](456.56KB)
Abstract:

In this paper, we prove the local and global well-posedness of some nonlinear thermoelastic plate equations with Dirichlet boundary conditions. The main tool for proving the local well-posedness is the maximal \begin{document}$ L_p $\end{document}-\begin{document}$ L_q $\end{document} regularity theorem for the linearized equations, and the main tool for proving the global well-posedness is the exponential stability of \begin{document}$ C_0 $\end{document} analytic semigroup associated with linear thermoelastic plate equations with Dirichlet boundary conditions.

A new energy-gap cost functional approach for the exterior Bernoulli free boundary problem
Julius Fergy T. Rabago and Hideyuki Azegami
2019, 8(4): 785-824 doi: 10.3934/eect.2019038 +[Abstract](717) +[HTML](232) +[PDF](1196.47KB)
Abstract:

The solution to a free boundary problem of Bernoulli type, also known as Alt-Caffarelli problem, is studied via shape optimization techniques. In particular, a novel energy-gap cost functional approach with a state constraint consisting of a Robin condition is proposed as a shape optimization reformulation of the problem. Accordingly, the shape derivative of the cost is explicitly determined, and using the gradient information, a Lagrangian-like method is used to formulate an efficient boundary variation algorithm to numerically solve the minimization problem. The second order shape derivative of the cost is also computed, and through its characterization at the solution of the Bernoulli problem, the ill-posedness of the shape optimization formulation is proved. The analysis of the proposed formulation is completed by addressing the existence of optimal solution of the shape optimization problem and is accomplished by proving the continuity of the solution of the state problems with respect to the domain. The feasibility of the newly proposed method and its comparison with the classical energy-gap type cost functional approach is then presented through various numerical results. The numerical exploration issued in the study also includes results from a second-order optimization procedure based on a Newton-type method for resolving such minimization problem. This computational scheme put forward in the paper utilizes the Hessian information at the optimal solution and thus offers a state-of-the-art numerical approach for solving such free boundary problem via shape optimization setting.

Optimal energy decay rates for some wave equations with double damping terms
Ryo Ikehata and Shingo Kitazaki
2019, 8(4): 825-846 doi: 10.3934/eect.2019040 +[Abstract](747) +[HTML](267) +[PDF](378.58KB)
Abstract:

We consider the Cauchy problem in \begin{document}$ {\bf R}^{n} $\end{document} for some wave equations with double damping terms, that is, one is the frictional damping \begin{document}$ u_{t}(t, x) $\end{document} and the other is very strong structural damping expressed as \begin{document}$ (-\Delta)^{\theta}u_{t}(t, x) $\end{document} with \begin{document}$ \theta > 1 $\end{document}. We will derive optimal decay rates of the total energy and the \begin{document}$ L^{2} $\end{document}-norm of solutions as \begin{document}$ t \to \infty $\end{document}. These results can be obtained in the case when the initial data have a sufficient high regularity in order to guarantee that the corresponding high frequency parts of such energy and \begin{document}$ L^{2} $\end{document}-norm of solutions are remainder terms. A strategy to get such results comes from a method recently developed by the first author [11].

Exponential stability for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping
Adriana Flores de Almeida, Marcelo Moreira Cavalcanti and Janaina Pedroso Zanchetta
2019, 8(4): 847-865 doi: 10.3934/eect.2019041 +[Abstract](748) +[HTML](269) +[PDF](439.35KB)
Abstract:

The following coupled damped Klein-Gordon-Schrödinger equations are considered

where \begin{document}$ \Omega $\end{document} is a bounded domain of \begin{document}$ \mathbb{R}^2 $\end{document}, with smooth boundary \begin{document}$ \Gamma $\end{document} and \begin{document}$ \omega $\end{document} is a neighbourhood of \begin{document}$ \partial \Omega $\end{document} satisfying the geometric control condition. Here \begin{document}$ \chi_{\omega} $\end{document} represents the characteristic function of \begin{document}$ \omega $\end{document}. Assuming that \begin{document}$ a, b\in L^{\infty}(\Omega) $\end{document} are nonnegative functions such that \begin{document}$ a(x) \geq a_0 >0 $\end{document} in \begin{document}$ \omega $\end{document} and \begin{document}$ b(x) \geq b_{0} > 0 $\end{document} in \begin{document}$ \omega $\end{document}, the exponential decay rate is proved for every regular solution of the above system. Our result generalizes substantially the previous ones given by Cavalcanti et. al in the reference [9] and [1].

Existence and extinction in finite time for Stratonovich gradient noise porous media equations
Mattia Turra
2019, 8(4): 867-882 doi: 10.3934/eect.2019042 +[Abstract](690) +[HTML](218) +[PDF](369.31KB)
Abstract:

We study existence and uniqueness of distributional solutions to the stochastic partial differential equation \begin{document}$ dX - \bigl( \nu \Delta X + \Delta \psi (X) \bigr) dt = \sum_{i = 1}^N \langle b_i, \nabla X \rangle \circ d\beta_i $\end{document} in \begin{document}$ ]0,T[ \times \mathcal{O} $\end{document}, with \begin{document}$ X(0) = x(\xi) $\end{document} in \begin{document}$ \mathcal{O} $\end{document} and \begin{document}$ X = 0 $\end{document} on \begin{document}$ ]0,T[ \times \partial \mathcal{O} $\end{document}. Moreover, we prove extinction in finite time of the solutions in the special case of fast diffusion model and of self-organized criticality model.

Sliding mode control of the Hodgkin–Huxley mathematical model
Cecilia Cavaterra, Denis Enăchescu and Gabriela Marinoschi
2019, 8(4): 883-902 doi: 10.3934/eect.2019043 +[Abstract](726) +[HTML](252) +[PDF](612.91KB)
Abstract:

In this paper we deal with a feedback control design for the action potential of a neuronal membrane in relation with the non-linear dynamics of the Hodgkin-Huxley mathematical model. More exactly, by using an external current as a control expressed by a relay graph in the equation of the potential, we aim at forcing it to reach a certain manifold in finite time and to slide on it after that. From the mathematical point of view we solve a system involving a parabolic differential inclusion and three nonlinear differential equations via an approximating technique and a fixed point result. The existence of the sliding mode and the determination of the time at which the potential reaches the prescribed manifold are proved by a maximum principle argument. Numerical simulations are presented.

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