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

• Published in: Journal of mathematical analysis and applications Vol. 494; no. 1; p. 124635
• Main Authors: Gu, Jiawen, Zhou, Deqin
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.02.2021
• Subjects: Exact controllability; Exponential stability; Fifth-order KdV equation; Fifth-order KdV equation; Exponential stability; Exact controllability
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2020.124635

In this paper, we study the local exact controllability and the exponential stability of the fifth-order KdV equation(0.1)∂tu−∂x5u+c1u∂x3u+c2∂xu∂x2u+c3u∂xu=F,c1≠0, posed on a periodic domain T=R/(2πZ). We prove that there are infinite source terms Fk where k∈[3,4] such that (0.1) is locally exact control and locally exponential stable in Hs(T) with s≥34. Moreover, as k=4, (0.1) is locally exact control and locally exponential stable in L2(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 Hs(T) with s>2.