zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Robust stabilization the Korteweg-de Vries-Burgers equation by boundary control. (English) Zbl 1183.76660
Summary: The problem of robust global stabilization by nonlinear boundary feedback control for the Korteweg-de Vries-Burgers equation on the domain [0,1] is considered. The main purpose of this paper is to derive nonlinear robust boundary control laws which make the system robustly globally asymptotically stable in spite of uncertainty in the system parameters. Furthermore, we show that the proposed boundary controllers guarantee $L_{2}$-robust exponential stability, $L_{\infty }$-robust asymptotic stability and boundedness in terms of both $L_{2}$ and $L_{\infty }$.

76B75Flow control and optimization
76B25Solitary waves (inviscid fluids)
93C20Control systems governed by PDE
Full Text: DOI
[1] Gao, P., Zhao, Y.: Boundary stabilization for the general Korteweg--de Vries--Burgers equation. Acta Anal. Funct. Appl. 5(2), 110--118 (2003) · Zbl 1033.35098
[2] Balogh, A., Krstic, M.: Boundary control of the Korteweg--de Vries--Burgers equation: further results on stabilization and well-posedness, with numerical demonstration. IEEE Trans. Autom. Control 45(9), 1739--1745 (2000) · Zbl 0990.93049 · doi:10.1109/9.880639
[3] Barbu, P.: Analysis and Control of Nonlinear Infinite Dimensional Systems. Academic, Boston (1993) · Zbl 0776.49005
[4] Crandall, M.G., Liggett, T.: Generation of semi-groups of nonlinear transformations in general Banach spaces. Am. J. Math. 93, 265--298 (1971) · Zbl 0226.47038 · doi:10.2307/2373376
[5] Liu, W.J., Krstic, M.: Global boundary stabilization of the Korteweg--de Vries--Burgers equation. Comput. Appl. Math. 21, 315--354 (2002) · Zbl 1125.35404
[6] Russell, D.L., Zhang, B.Y.: Exact controllability and stabilizability of the Korteweg--de Vries equation. Trans. Am. Math. Soc. 348(9), 3643--3672 (1996) · Zbl 0862.93035 · doi:10.1090/S0002-9947-96-01672-8
[7] Sakthivel, R., Ito, H.: Nonlinear robust boundary control of the Kuramoto--Sivashinsky equation. IMA J. Math. Control Inf. 24, 47--55 (2007) · Zbl 1137.93043 · doi:10.1093/imamci/dnl009
[8] Smaoui, N., Al-Jamal, R.H.: Boundary control of the generalized Korteweg--de Vries--Burgers equation. Nonlinear Dyn. 51(3), 439--446 (2008) · Zbl 1170.93018 · doi:10.1007/s11071-007-9222-5
[9] Smaoui, N., Al-Jamal, R.A.: Nonlinear boundary control for the dynamics of the generalized Korteweg--de Vries--Burgers equation. Kuwait J. Sci. Eng. 34, 57--76 (2007) · Zbl 1207.93083
[10] Smaoui, N.: Nonlinear boundary control of the generalized Burgers equation. Nonlinear Dyn. 37(1), 75--86 (2004) · Zbl 1078.76026 · doi:10.1023/B:NODY.0000040023.92220.09
[11] Tian, L.X., Zhao, Z.F., Wang, J.F.: Boundary control of MKDV-Burgers equation. Appl. Math. Mech. 27, 109--116 (2006) · Zbl 1144.76013 · doi:10.1007/s10483-006-0114-z
[12] Zhang, B.Y.: Boundary Stabilization of the Korteweg--de Vries equation. Int. Ser. Numer. Math. 118, 371--389 (1994) · Zbl 0811.35133