2018/01/22 by Türker Özsarı, Özsarı, Türker, Ahmet Batal +1
Computer Science · Engineering · Mathematics · #35A01 #35B45 #35Q53 #93C10 #93C20 #93D15 #93D20 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods for differential equations #Optimization and Control (math.OC) #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1801.07206
openalex publication_date 2018/01/22 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
In this paper, we design Dirichlet-Neumann boundary feedback controllers for\nthe Korteweg-de Vries (KdV) equation that act at the right endpoint of the\ndomain. The length of the domain is allowed to be critical. Constructing\nbackstepping controllers that act at the right endpoint of the domain is more\nchallenging than its left endpoint counterpart. The standard application of the\nbackstepping method fails, because corresponding kernel models become\noverdetermined. In order to deal with this difficulty, we introduce the\npseudo-backstepping method, which uses a pseudo-kernel that satisfies all but\none desirable boundary condition. Moreover, various norms of the pseudo-kernel\ncan be controlled through a parameter in one of its boundary conditions. We\nprove that the boundary controllers constructed via this pseudo-kernel still\nexponentially stabilize the system with the cost of a low exponential rate of\ndecay. We show that a single Dirichlet controller is sufficient for exponential\nstabilization with a slower rate of decay. We also consider a second order\nfeedback law acting at the right Dirichlet boundary condition. We show that\nthis approach works if the main equation includes only the third order term,\nwhile the same problem remains open if the main equation involves the first\norder and/or the nonlinear term(s). At the end of the paper, we give numerical\nsimulations to illustrate the main result.\n