TY - JOUR
T1 - On the control issues for higher-order nonlinear dispersive equations on the circle
AU - Capistrano–Filho, Roberto de A.
AU - Kwak, Chulkwang
AU - Vielma Leal, Francisco J.
N1 - Funding Information:
We greatly appreciate the referees’ careful reading and helpful suggestions. R. de A. Capistrano–Filho was supported by CNPq, Brazil Grant Numbers 307808/2021-1 and 408181/2018-4, CAPES, Brazil Grant Numbers 88881.311964/2018-01 and 88881.520205/2020-01, MATHAMSUD, Brazil grant 21-MATH-03 and Propesqi (UFPE, Brazil). C. Kwak was partially supported by the Ewha Womans University, South Korea Research Grant of 2020 and the National Research Foundation of Korea (NRF), South Korea grant funded by the Korea government (MSIT) (No. 2020R1F1A1A0106876811). F. Vielma Leal was partially supported by FAPESP/Brazil grant 2020/14226-4. This work was carried out while the second author was visiting the Federal University of Pernambuco. He thanks the institution for their hospitality.
Funding Information:
We greatly appreciate the referees’ careful reading and helpful suggestions. R. de A. Capistrano–Filho was supported by CNPq, Brazil Grant Numbers 307808/2021-1 and 408181/2018-4 , CAPES, Brazil Grant Numbers 88881.311964/2018-01 and 88881.520205/2020-01 , MATHAMSUD, Brazil grant 21-MATH-03 and Propesqi ( UFPE, Brazil ). C. Kwak was partially supported by the Ewha Womans University, South Korea Research Grant of 2020 and the National Research Foundation of Korea (NRF), South Korea grant funded by the Korea government (MSIT) (No. 2020R1F1A1A0106876811 ). F. Vielma Leal was partially supported by FAPESP/Brazil grant 2020/14226-4 . This work was carried out while the second author was visiting the Federal University of Pernambuco. He thanks the institution for their hospitality.
Publisher Copyright:
© 2022 Elsevier Ltd
PY - 2022/12
Y1 - 2022/12
N2 - The local and global control results for a general higher-order KdV-type operator posed on the unit circle are presented. Using spectral analysis, we are able to prove local results, that is, the equation is locally controllable and exponentially stable. To extend the local results to the global one we captured the smoothing properties of the Bourgain spaces, the so-called propagation of regularity, which are proved with a new perspective. These propagation, together with the Strichartz estimates, are the key to extending the local control properties to the global one, precisely, higher-order KdV-type equations are globally controllable and exponentially stabilizable in the Sobolev space Hs(T) for any s≥0. Our results recover previous results in the literature for the KdV and Kawahara equations and extend, for a general higher-order operator of KdV-type, the Strichartz estimates as well as the propagation results, which are the main novelties of this work.
AB - The local and global control results for a general higher-order KdV-type operator posed on the unit circle are presented. Using spectral analysis, we are able to prove local results, that is, the equation is locally controllable and exponentially stable. To extend the local results to the global one we captured the smoothing properties of the Bourgain spaces, the so-called propagation of regularity, which are proved with a new perspective. These propagation, together with the Strichartz estimates, are the key to extending the local control properties to the global one, precisely, higher-order KdV-type equations are globally controllable and exponentially stabilizable in the Sobolev space Hs(T) for any s≥0. Our results recover previous results in the literature for the KdV and Kawahara equations and extend, for a general higher-order operator of KdV-type, the Strichartz estimates as well as the propagation results, which are the main novelties of this work.
KW - Bourgain spaces
KW - Control problems
KW - KdV-type equation
KW - Propagation of regularity/compactness
UR - http://www.scopus.com/inward/record.url?scp=85134221667&partnerID=8YFLogxK
U2 - 10.1016/j.nonrwa.2022.103695
DO - 10.1016/j.nonrwa.2022.103695
M3 - Article
AN - SCOPUS:85134221667
SN - 1468-1218
VL - 68
JO - Nonlinear Analysis: Real World Applications
JF - Nonlinear Analysis: Real World Applications
M1 - 103695
ER -