Local controllability and stability of the periodic fifth-order KdV equation with a nonlinear dispersive term

In this paper, we study the local exact controllability and the exponential stability of the fifth-order KdV equation partial derivative(t)u - partial derivative(5)(x)u + c(1)u partial derivative(3)(x)u + c(2)partial derivative(x)u partial derivative(2)(x)u + c(3)u partial derivative(x)u = F, c(1) not equal 0, (0.1) posed on a periodic domain T = R/(2 pi Z). We prove that there are infinite source terms F-k where k is an element of [3,4] such that (0.1) is locally exact control and locally exponential stable in H-s (T) with s >= 3/4. Moreover, as k = 4, (0.1) is locally exact control and locally exponential stable in L-2(T). Our results relax the range of the index s in recent work by Flores and Smith (2019) [10], where they have established the local exact controllability and the local exponential stability of (0.1) in H-s(T) with s > 2. (C) 2020 Elsevier Inc. All rights reserved.