Mathematics / Matematik

Permanent URI for this collectionhttps://hdl.handle.net/11147/8

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Access Right: Closed AccessAuthor: search.filters.author.Özsarı, Türker

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  • Article
    Citation - WoS: 7
    Citation - Scopus: 7
    Output Feedback Stabilization of the Linearized Korteweg-De Vries Equation With Right Endpoint Controllers
    (Elsevier Ltd., 2019) Batal, Ahmet; Özsarı, Türker
    In this paper, we prove the output feedback stabilization for the linearized Korteweg-de Vries (KdV) equation posed on a finite domain in the case the full state of the system cannot be measured. We assume that there is a sensor at the left end point of the domain capable of measuring the first and second order boundary traces of the solution. This allows us to design a suitable observer system whose states can be used for constructing boundary feedbacks acting at the right endpoint so that both the observer and the original plant become exponentially stable. Stabilization of the original system is proved in the L-2-sense, while the convergence of the observer system to the original plant is also proved in higher order Sobolev norms. The standard backstepping approach used to construct a left endpoint controller fails and presents mathematical challenges when building right endpoint controllers due to the overdetermined nature of the related kernel models. In order to deal with this difficulty we use the method of Ozsan and Batal, (2019) which is based on using modified target systems involving extra trace terms. In addition, we show that the number of controllers and boundary measurements can be reduced to one, with the cost of a slightly lower exponential rate of decay. We provide numerical simulations illustrating the efficacy of our controllers. (C) 2019 Elsevier Ltd. All rights reserved.
  • Article
    Citation - WoS: 6
    Citation - Scopus: 6
    Exponential Stability for the Nonlinear Schrodinger Equation With Locally Distributed Damping
    (Taylor and Francis Ltd., 2020) Cavalcanti, Marcelo M.; Correa, Wellington J.; Özsarı, Türker; Sepulveda, Mauricio; Vejar-Aseme, Rodrigo
    In this paper, we study the defocusing nonlinear Schrodinger equation with a locally distributed damping on a smooth bounded domain as well as on the whole space and on an exterior domain. We first construct approximate solutions using the theory of monotone operators. We show that approximate solutions decay exponentially fast in the L-2-sense by using the multiplier technique and a unique continuation property. Then, we prove the global existence as well as the L-2-decay of solutions for the original model by passing to the limit and using a weak lower semicontinuity argument, respectively. The distinctive feature of the paper is the monotonicity approach, which makes the analysis independent from the commonly used Strichartz estimates and allows us to work without artificial smoothing terms inserted into the main equation. We in addition implement a precise and efficient algorithm for studying the exponential decay established in the first part of the paper numerically. Our simulations illustrate the efficacy of the proposed control design.