Summary: In this article, a distributed control design is developed via partial differential equation (PDE) backstepping for a class of reaction-diffusion PDEs on \(n\)-ball under revolution symmetry conditions in the presence of radial spatially varying actuator delay. The actuator delay is modeled by a two-dimensional (2D) first-order hyperbolic PDE with spatially varying boundary, resulting in a cascaded transport-diffusion PDE system. Two-step backstepping transformation is introduced for the control design. First, a new Volterra transformation for the \(n\)-ball under revolution symmetry is proposed that allows the kernel function to be solvable. Different from the kernel in the 1D domain, which is expressed by the classical Fourier series, the kernel of the \(n\)-ball under revolution symmetry conditions is based on the Fourier-Bessel series. The resulting Volterra transformation is an implicit one that contains the state of the target system on both sides of the definition. Based on the implicit transformation, we employ the successive approximations approach to derive the second-step transformation, namely, an explicit transformation, and arrive at a stable target system. The inverse transformation is also derived, to prove \(L^2\) exponential stability of the closed-loop system. The stability analysis requires more technical skills that are not needed in solving the spatially varying delay compensation problem when the domain is 1D. Numerical simulation examples of 3D and 4D revolution-symmetrical systems illustrate the effectiveness of the controller.
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