Mathematics / Matematik

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

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  • Article
    Citation - WoS: 11
    Citation - Scopus: 10
    Pseudo-Backstepping and Its Application To the Control of Korteweg-De Vries Equation From the Right Endpoint on a Finite Domain
    (Society for Industrial and Applied Mathematics Publications, 2019) Özsarı, Türker; Batal, Ahmet
    In this paper, we design Dirichlet-Neumann boundary feedback controllers for the Korteweg-de Vries equation that act at the right endpoint of the domain. The length of the domain is allowed to be critical. Constructing backstepping controllers that act at the right endpoint of the domain is more challenging than its left endpoint counterpart. The standard application of the backstepping method fails, because corresponding kernel models become overdetermined. In order to deal with this difficulty, we introduce the pseudo-backstepping method, which uses a pseudo-kernel that satisfies all but one desirable boundary condition. Moreover, various norms of the pseudo-kernel can be controlled through a parameter in one of its boundary conditions. We prove that the boundary controllers constructed via this pseudo-kernel still exponentially stabilize the system with the cost of a low exponential rate of decay. We show that a single Dirichlet controller is sufficient for exponential stabilization with a slower rate of decay. We also consider a second order feedback law acting at the right Dirichlet boundary condition. We show that this approach works if the main equation includes only the third order term, while the same problem remains open if the main equation involves the first order and/or the nonlinear terms. At the end of the paper, we give numerical simulations to illustrate the main result.
  • Article
    Citation - WoS: 7
    Citation - Scopus: 7
    Finite-Parameter Feedback Control for Stabilizing the Complex Ginzburg–landau Equation
    (Elsevier Ltd., 2017) Kalantarova, Jamila; Özsarı, Türker
    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.