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In this paper, a delay compensation design method based on PDE backstepping is developed for a two-dimensional reaction-diffusion partial differential equation (PDE) with bilateral input delays. The PDE is defined in a rectangular domain, and the bilateral control is imposed on a pair of opposite sides of the rectangle. To represent the delayed bilateral inputs, we introduce two 2-D transport PDEs that form a cascade system with the original PDE. A novel set of backstepping transformations is proposed for delay compensator design, including one Volterra integral transformation and two affine Volterra integral transformations. Unlike the kernel equation for 1-D PDE systems with delayed boundary input, the resulting kernel equations for the 2-D system have singular initial conditions governed by the Dirac Delta function. Consequently, the kernel solutions are written as a double trigonometric series with singularities. To address the challenge of stability analysis posed by the singularities, we prove a set of inequalities by using the Cauchy-Schwarz inequality, the 2-D Fourier series, and the Parseval's theorem. A numerical simulation illustrates the effectiveness of the proposed delay-compensation method., Comment: 11 pages, 3 figures(including 8 sub-figures)

This paper proposes a delay-compensated boundary controller for an unstable reaction-diffusion partial differential equation (PDE) with uncertain input delay. The delay is stochastic and modeled by a finite-state Markov process. We utilize the PDE backstepping method for control design. Compared to the case of constant delay, a group of transport PDEs is applied to represent the stochastic properties of the delay which brings challenges to the stability analysis of the closed-loop system. An extra first-order PDE is introduced to construct a new target system. Since the transformation is invertible, we can establish the equivalence between the target system and the original closed-loop system in terms of well-posedness and stability. We first prove the well-posedness of the system by the semigroup theory and then establish the exponential stability of the closed-loop system with respect to the $L^2$-norm of the state and the $H^1$-norm of the infinite-dimensional representation of the actuator state through the Cauchy-Schwarz inequality and Fourier series. A numerical simulation illustrates the effectiveness of the controller., Comment: 10 pages, 3 figures, the control design for the reaction-diffusion system with stochastic delay

Majorana fermions have been intensively studied in recent years for their importance to both fundamental science and potential applications in topological quantum computing1,2. Majorana fermions are predicted to exist in a vortex core of superconducting topological insulators3. However, they are extremely difficult to be distinguished experimentally from other quasiparticle states for the tiny energy difference between Majorana fermions and these states, which is beyond the energy resolution of most available techniques. Here, we overcome the problem by systematically investigating the spatial profile of the Majorana mode and the bound quasiparticle states within a vortex in Bi2Te3/NbSe2. While the zero bias peak in local conductance splits right off the vortex center in conventional superconductors, it splits off at a finite distance ~20nm away from the vortex center in Bi2Te3/NbSe2, primarily due to the Majorana fermion zero mode. While the Majorana mode is destroyed by reducing the distance between vortices, the zero bias peak splits as a conventional superconductor again. This work provides strong evidences of Majorana fermions and also suggests a possible route to manipulating them., Comment: 16 pages, 4 figures