arXiv subjects

Türker Özsarı

Publications and source records attributed to Türker Özsarı.

22 records · Page 2Linked to original sources

arXiv2017

Finite-parameter feedback control for stabilizing the complex Ginzburg-Landau equation

In this paper, we prove the exponential stabilization of solutions for complex Ginzburg-Landau equations using finite-parameter feedback control algorithms, which employ finitely many volume elements, Fourier modes or nodal observables (controllers). We also propose a feedback control for steering solutions of the Ginzburg-Landau equation to a desired solution of the non-controlled system. In this latter problem, the feedback controller also involves the measurement of the solution to the non-controlled system.

math.OC↗

arXiv2017

Complex Ginzburg-Landau equations with dynamic boundary conditions

The initial-dynamic boundary value problem (idbvp) for the complex Ginzburg-Landau equation (CGLE) on bounded domains of $\mathbb{R}^N$ is studied by converting the given mathematical model into a Wentzell initial-boundary value problem (ibvp). First, the corresponding linear homogeneous idbvp is considered. Secondly, the forced linear idbvp with both interior and boundary forcings is studied. Then, the nonlinear idbvp with Lipschitz nonlinearity in the interior and monotone nonlinearity on the boundary is analyzed. The local well-posedness of the idbvp for the CGLE with power type nonlinearities is obtained via a contraction mapping argument. Global well-posedness for strong solutions is shown. Global existence and uniqueness of weak solutions are proven. Smoothing effect of the corresponding evolution operator is proved. This helps to get better well-posedness results than the known results on idbvp for nonlinear Schrödinger equations (NLS). An interesting result of this paper is proving that solutions of NLS subject to dynamic boundary conditions can be obtained as inviscid limits of the solutions of the CGLE subject to same type of boundary conditions. Finally, long time behaviour of solutions is characterized and exponential decay rates are obtained at the energy level by using control theoretic tools.

math.AP↗

arXiv2015

Qualitative properties of solutions for nonlinear Schrödinger equations with nonlinear boundary conditions on the half-line

In this paper, we study the interaction between a nonlinear focusing Robin type boundary source, a nonlinear defocusing interior source, and a weak damping term for nonlinear Schrödinger equations posed on the infinite half line. We construct solutions with negative initial energy satisfying a certain set of conditions which blow-up in finite time in the $H^1$-sense. We obtain a sufficient condition relating the powers of nonlinearities present in the model which allows construction of blow-up solutions. In addition to the blow-up property, we also discuss the stabilization property and the critical exponent for this model.

math.AP↗

arXiv2015

Nonlinear Schrödinger equation on the half-line with nonlinear boundary condition

In this paper, we study the initial boundary value problem for nonlinear Schrödinger equations on the half-line with nonlinear boundary conditions of type $u_x(0,t)+λ|u(0,t)|^ru(0,t)=0,$ $λ\in\mathbb{R}-\{0\}$, $r> 0$. We discuss the local well-posedness when the initial data $u_0=u(x,0)$ belongs to an $L^2$-based inhomogeneous Sobolev space $H^s(\mathbb{R}_+)$ with $s\in \left(\frac{1}{2},\frac{7}{2}\right)-\{\frac{3}{2}\}$. We deal with the nonlinear boundary condition by first studying the linear Schrödinger equation with a time-dependent inhomogeneous Neumann boundary condition $u_x(0,t)=h(t)$ where $h\in H^{\frac{2s-1}{4}}(0,T)$. This latter problem is studied by adapting the method of Bona-Sun-Zhang \cite{BonaSunZhang2015} to the case of inhomogeneous Neumann boundary conditions.

math.AP↗