Kernel well-posedness and computation by power series in backstepping output feedback for radially-dependent reaction–diffusion PDEs on multidimensional balls

Abstract

Recently, the problem of boundary stabilization and estimation 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. Some early success has been achieved under simplifying conditions, such as radially-varying reaction coefficients under revolution symmetry, on a disk or a sphere. These particular cases notwithstanding, the problem remains open. The main issue is that the equations become singular in the radius; when applying the backstepping method, the same type of singularity appears in the kernel equations. Traditionally, well-posedness of these equations has been proved by transforming them into integral equations and then applying the method of successive approximations. In this case, with the resulting integral equation becoming singular, successive approximations do not easily apply. This paper takes a different route and directly addresses the kernel equations via a power series approach (in the spirit of the method of Frobenius for ordinary differential equations), finding in the process the required conditions for the radially-varying reaction (namely, analyticity and evenness) and showing the existence and convergence of the series solution. This approach provides a direct numerical method that can be readily applied, despite singularities, to both control and observer boundary design problems.