圆盘上奇异反应扩散方程的边界控制
Boundary control of a singular reaction-diffusion equation on a disk
摘要：
近年来，N球上不稳定线性常系数反应扩散方程的边界稳定化问题已被用反推方法解决.然而，这一结果的空间变化系数的扩展是远远不够的。这项工作涉及径向变化的反应系数旋转对称条件下的磁盘（2-D的情况下）。在这些条件下，方程在半径上变得奇异。当应用反演方法时，反演核方程也会出现同样类型的奇异性。传统上，核方程的适定性是通过将其转化为积分方程，然后应用逐次逼近的方法来证明的。在这种情况下，得到的积分方程是奇异的。一个连续的近似级数仍然可以公式化，但它的收敛性是具有挑战性的显示，由于奇异性。这个问题是通过一个相当非标准的证明来解决的，这个证明使用了加泰罗尼亚数的性质，加泰罗尼亚数是一个在组合数学中经常使用的著名序列。
英文摘要：
Recently, the problem of boundary stabilization for unstable linear constant-coefficient reaction-diffusion equation on N-balls has been solved by means of the backstepping method. However, the extension of this result to spatially-varying coefficients is far from trivial. 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 used in combinatorial mathematics.
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