Uniform Exponential Stability for a Schrödinger Equation and Its Semidiscrete Approximation

In this article, we investigate the uniform exponential stability of a semidiscrete scheme for a Schrödinger equation under boundary stabilizing feedback control in the natural state space $L^{2}(0,1)$. This study is significant since a time domain energy multiplier that allows proving the exponential stability of this continuous Schrödinger system has not yet found, thus leading to a major mathematical challenge to the uniform exponential stability of the corresponding semidiscretization systems, which is an open problem for a long time. Although the powerful frequency domain energy multiplier approach has been used in proving exponential stability for partial differential equations (PDEs) since 1980s, its use to the uniform exponential stability of the semidiscrete scheme for PDEs has not been reported yet. The difficulty associated with the uniformity is that due to the parameter of the step size, it involves infinitely many matrices in different state spaces that need to be considered simultaneously. Based on the Huang–Prüss frequency domain criterion for uniform exponential stability of a family of $C_{0}$-semigroups in Hilbert spaces, we solve this problem for the first time by proving the uniform boundedness for all the resolvents of these matrices on the imaginary axis. The proof almost exactly follows the procedure for the exponential stability of the continuous counterpart, highlighting the advantage of this discretization method.