文章基本信息

标题：Boundary control of a singular reaction-diffusion equation on a disk
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作者：Rafael Vazquez ; Miroslav Krstic
期刊名称：IFAC PapersOnLine
印刷版ISSN：2405-8963
出版年度：2016
卷号：49
期号：8
页码：74-79
DOI：10.1016/j.ifacol.2016.07.421
语种：English
出版社：Elsevier
摘要：AbstractRecently, the problem of boundary stabilization for unstable linear constant-coefficient reaction-diffusion equation on n-balls (in particular, disks and spheres) has been solved by means of the backstepping method. However, the extension of this result to spatially-varying coefficients is far from trivial. As a first step, this work deals with radially-varying reaction coefficients under revolution symmetry conditions on a disk (the 2-D case). Under these conditions, the equations become singular in the radius. When applying the backstepping method, the same type of singularity appears in the backstepping kernel equations. Traditionally, well-posedness of the kernel equations is proved by transforming them into integral equations and then applying the method of successive approximations. In this case, the resulting integral equation is singular. A successive approximation series can still be formulated, however its convergence is challenging to show due to the singularities. The problem is solved by a rather non-standard proof that uses the properties of the Catalan numbers, a well-known sequence frequently appearing in combinatorial mathematics.