Stabilization Of Two Types Of Fractional-order Of Evolution Equations

Fractional calculus is the generalization of integral calculus.At present,the theory of integral calculus is quite mature,and boundary control of classical reac-tion diffusion equation has been widely researched in various fields engineering and science.Recent years,the fractional differential equations that are abstracted from some practical problems have attracted much attention in mathematics.This paper mainly introduces the boundary control for two types of fractional-order of evolution equations.The boundary feedback stabilization of an unstable time fractional-order wave equation is discussed by the backstepping method.We firstly solve the kernel function in the forward and inverse transformation;then,we obtain the polynomial stability of the closed-loop system.Then,we consider the boundary control problem for a fractional-order heat equation with spatial memory.Via constructing appropriate transition system,we obtain the existence of kernels and avoid the difficulty in solving the kernels in directly backstepping transforma-tion.Meanwhile,we prove the invertibility of two steps backstepping transformation and obtain the rapid L2 Mittag-Leffler stability of closed-loop system.