Stability Analysis Of Time-fractional Schr(?)dinger Differential Equations With Delay

In recent years,due to the nonlocal properties of fractional derivative and its memory and genetic effects,many practical phenomena,such as quantum mechanics,anomalous diffusion and neural networks,can be described by fractional differential equations.The theory and application of fractional differential equations are widely concerned by scholars.As a basic equation in quantum mechanics,the fractional model of Schr(?)dinger equation provides a better mathematical tool for quantum mechanics.However,due to the transmission process of matter,energy or information,delay is an inevitable inherent phenomenon.Generally speaking,delay may cause the transient response of the system,resulting in the complex behavior of the solution.Therefore,the stability of time-fractional Schr(?)dinger equations with delay is investigated in this paper.In the second chapter,we investigate the asymptotic stability of time-fractional Schr(?)dinger delay equations with Dirichlet boundary conditions.As a generalization of this model,the abstract delay evolution equation is considered at first.Through the spectral decomposition of Laplace operator,the equation is decomposed into an infinite dimensional fractional delay ordinary differential equation.By using the Laplace transform method and root locus technique,a necessary and sufficient condition for the stability of the equation is given,which is a delay-independent coefficient-type criterion expressed by the coefficients and fractional exponents.Based on this conclusion,the stability conditions for linear and nonlinear time-fractional Schr(?)dinger delay equations,time-space fractional Schr(?)dinger delay equations and time-fractional fourth-order delay equations under Dirichlet boundary conditions are given.Numerical examples are given to illustrate the effectiveness and applicability of the theoretical resultsIn the third chapter,the asymptotic stability of time-fractional Schr(?)dinger delay equations with Neumann boundary conditions is studied.The solutions can not be stable because of the spectral structure of Laplace operator under Neumann boundary conditions.Therefore,by designing a state-dependent linear feedback controller,the sufficient and necessary condition of stabilization is obtained.Based on this conclusion,a state-dependent linear feedback controller for time-space fractional Schr(?)dinger delay equations is designed.Numerical examples are given to illustrate the effectiveness and feasibility of the theoretical results.