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)
Optimal control of the viscous Camassa-Holm equation. (English) Zbl 1154.49300
Summary: This paper studies the problem of optimal control of the viscous Camassa-Holm equation. The existence and uniqueness of weak solution to the viscous Camassa-Holm equation are proved in a short interval. According to variational method, optimal control theories and distributed parameter system control theories, we can deduce that the norm of solution is related to the control item and initial value in the special Hilbert space. The optimal control of the viscous Camassa-Holm equation under boundary condition is given and the existence of optimal solution to the viscous Camassa-Holm equation is proved.

49J20Optimal control problems with PDE (existence)
35K57Reaction-diffusion equations
35B25Singular perturbations (PDE)
76D05Navier-Stokes equations (fluid dynamics)
Full Text: DOI
[1] Holm, Darryl D.; Staley, Martin F.: Wave structures and nonlinear balances in a family of 1+1 evolutionary pdes. SIAM journal of applied dynamical system 2, No. 3, 323-380 (2003) · Zbl 1088.76531
[2] Tian, Lixin; Fang, Guochang; Gui, Guilong: Well-posedness and blowup for an integrable shallow water wave equation with strong dispersive term. International journal of nonlinear science 1, No. 1, 3-13 (2006)
[3] Yu, Liqin; Tian, Lixin; Wang, Xuedi: The bifurcation and peakon for Degasperis--Procesi equation. Chaos, soliton & fractals 30, No. 4, 956-966 (2007) · Zbl 1142.35589
[4] Tian, Lixin; Sun, Lu: Singular solitons of generalized Camassa--Holm models. Chaos, soliton & fractals 32, No. 2, 780-799 (2006) · Zbl 1139.35090
[5] Ding, Danping; Tian, Lixin: The study on solution of Camassa--Holm equation with weak dissipation. Communications on pure and applied analysis 5, No. 3, 483-493 (2006) · Zbl 1140.35306
[6] Tian, Lixin; Gui, Guilong; Liu, Yue: On the well-posedness problem and scattering problem for DGH equation. Communications in mathematical physics 257, No. 3, 667-701 (2005) · Zbl 1080.76016
[7] Foias, C.; Holm, D. D.; Titi, E. S.: The three dimensional viscous Camassa--Holm equation and their relation to the Navier--Stokes equation and turbulence theory. Journal of dynamics and differential equations 14, 1-36 (2002) · Zbl 0995.35051
[8] Chen, S.; Foias, C.; Holm, D. D.; Olson, E. J.; Titi, E. S.; Wynne, S.: The Camassa--Holm equations as a closure model for turbulent channel and pipe flows. Physical review letters 81, 5338-5341 (1998) · Zbl 1042.76525
[9] Chen, S.; Foias, C.; Holm, D. D.; Olson, E. J.; Titi, E. S.; Wynne, S.: A connection between Camassa--Holm equations and turbulent flows in channels and pipes. Physics of fluids 11, 2343-2353 (1999) · Zbl 1147.76357
[10] Chen, S.; Foias, C.; Holm, D. D.; Olson, E. J.; Titi, E. S.; Wynne, S.: The Camassa--Holm equations and turbulence. Physica D 133, 49-65 (1999) · Zbl 1194.76069
[11] Ding, Danping; Tian, Lixin: The study of solution of dissipative Camassa--Holm equation on total space. International journal of nonlinear science 1, No. 1, 37-42 (2006)
[12] Camassa, R.; Holm, D.: An integrable shallow water equation with peaked solitions. Physical review letters 71, 1661-1664 (1993) · Zbl 0972.35521
[13] Constantin, A.; Escher, J.: Global existence and blow-up for a shallow water equation. Annalidella scuola normale superiore di Pisa 26, 303-328 (1998) · Zbl 0918.35005
[14] Li, Y.; Olver, P.: Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation. Journal of differential equations 162, 27-63 (2000) · Zbl 0958.35119
[15] Constantin, A.: Global existence of solutions and breaking waves for a shallow water equation: A geometric approach. Université de Grenoble annales de 1’Institut Fourier 50, 321-362 (2000) · Zbl 0944.35062
[16] Constantin, A.; Mckean, H. P.: A shallow water equation on the circle. Communications on pure and applied mathematics 52, 949-982 (1999) · Zbl 0940.35177
[17] Hakkaev, Sevdzhan; Kirchev, Kiril: On the well-posedness and stability of peakons for a generalized Camassa--Holm equation. International journal of nonlinear science 1, No. 3, 139-148 (2006) · Zbl 1076.35098
[18] Ding, Danping; Tian, Lixin: The attractor in dissipative Camassa--Holm equation. Acta mathematic applicate sinica 27, No. 3, 536-545 (2004) · Zbl 1080.35007
[19] Ito, K.; Ravindranss, S. S.: A reduced-basis method for control problems governed by PDES. Control and estimation of distributed parameter systems. International series of numerical mathematics 126, 153-168 (1998)
[20] Atweil, J. A.; King, B. B.: Proper orthogonal decomposition for reduced basis feedback controller for parabolic equation. Mathematical and modelling 33, 1-19 (2001)
[21] Huchang-Bing; Roger, Temam: Robust control the kuramto--sivashing equation. Dynamics of continuous. Discrete and impulsive systems series B 8, 315-338 (2001)
[22] S. Volkwein, Mesh-independence of an augmented Lagrangian--SQP method in Hilbert spaces and control problems for burgers equation, Ph.D. Thesis, Department of mathematics, Technical University of Berlin, October 1997
[23] Hinze, M.; Volkwein, S.: Analysis of instantaneous control for the Burgers equation. Nonlinear analysis 50, No. 1, 1-26 (2002) · Zbl 1022.49001
[24] Volkwein, S.: Distributed control problems for the Burgers equation. Computational optimization and applications 16, 57-81 (2000) · Zbl 0974.49020
[25] Zhu, Min; Tian, Lixin; Zhao, Zhifeng: Optimal control of KdV--Burgers equation. Journal of JiangSu university 25, No. 3, 235-238 (2004) · Zbl 1073.49004
[26] Zhao, Zhifeng; Tian, Lixin: Optimal control of sufficient nonlinear KdV--Burgers equation. Journal of JiangSu university 26, No. 2, 140-143 (2005) · Zbl 1130.49022
[27] Armaou, A.; Christofides, P. D.: Feedback control of kuramto--sivashing equation. Physics D 137, 49-61 (2000) · Zbl 0952.93060
[28] Zhao, Zhifeng: Optimal control of kuramto--sivashing equation. International journal of nonlinear science 1, No. 1, 54-58 (2006)
[29] Oksendal, Bernt: Optimal control of stochastic partial differential equations. Stochastic analysis and applications 23, No. 1, 165-179 (2005)
[30] Lagnese, J. E.; Leugering, G.: Time-domain decomposition of optimal control problems for the wave equation. System and control letters 48, 229-242 (2003) · Zbl 1134.49313
[31] Ghattas, Omar; Bark, Jai-Hyeong: Optimal control of two- and three-dimensional in compressible Navier--Stokes flows. Journal of computational physics 136, 231-244 (1997) · Zbl 0893.76067
[32] Ji, Guangcao; Martin, Clyde: Optimal boundary control of the heat equation with target function at terminal time. Applied mathematics and computation 127, 335-345 (2002) · Zbl 1040.49037
[33] Camassa, R.; Holm, D. D.: An integrable shallow water equation with peaked solitons. Physical review letters 71, No. 11, 1661-1664 (1993) · Zbl 0972.35521
[34] Liu, Zhengrong; Qian, Tifei: Peakons of the Camassa--Holm equation. Applied mathematical modeling 26, 473-480 (2002) · Zbl 1018.35061
[35] Wouk, A.: A course of applied functional analysis. (1979) · Zbl 0407.46001
[36] Temam, R.: Navier--Stokes equations, studies in mathematics and its applications. (1979) · Zbl 0426.35003
[37] Dautray, R.; Lions, J. L.: Mathematical analysis and numerical methods for science and technology. Volume 5: evolution problems I. (1992) · Zbl 0755.35001
[38] Kreyszig, E.: Introductory functional analysis with applications. (1978) · Zbl 0368.46014