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Mat. Sb., 2001, Volume 192, Number 4, Pages 115–160 (Mi msb560)  

This article is cited in 35 scientific papers (total in 35 papers)

Stabilizability of a quasi-linear parabolic equation by means of a boundary control with feedback

A. V. Fursikov

M. V. Lomonosov Moscow State University

Abstract: The problem of stabilizability from the boundary $\partial\Omega$ for a parabolic equation given in a bounded domain $\Omega\in\mathbb R^n$, consists in choosing a boundary condition (a control) such that the solution of the resulting mixed boundary-value problem tends as $t\to\infty$ to a given steady-state solution at a prescribed rate $\exp(-\sigma_0t)$. Furthermore, it is required that the control be with feedback, that is, that it react to unpredictable fluctuations of the system by suppressing the results of their action on the stabilizable solution. A new mathematical formulation of the concept of feedback is presented and then used in solving the problem of stabilizability of linear as well as quasi-linear parabolic equations by means of a control with feedback defined on part of the boundary.

DOI: https://doi.org/10.4213/sm560

Full text: PDF file (519 kB)
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English version:
Sbornik: Mathematics, 2001, 192:4, 593–639

Bibliographic databases:

UDC: 517.977.1
MSC: Primary 35K15, 93D15, 93B52, 35K20; Secondary 35K55, 93B05, 35B37, 47A52, 49N35
Received: 31.08.2000

Citation: A. V. Fursikov, “Stabilizability of a quasi-linear parabolic equation by means of a boundary control with feedback”, Mat. Sb., 192:4 (2001), 115–160; Sb. Math., 192:4 (2001), 593–639

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Chizhonkov E.V., “Numerical aspects of one stabilization method”, Russian J. Numer. Anal. Math. Modelling, 18:5 (2003), 363–376  crossref  mathscinet  zmath  isi  elib
    2. A. V. Gorshkov, “Stabilization of a semilinear parabolic equation in the exterior of a bounded domain by means of boundary controls”, Sb. Math., 194:10 (2003), 1475–1502  mathnet  crossref  crossref  mathscinet  zmath  isi
    3. A. V. Fursikov, “Stabilizatsiya s granitsy reshenii sistemy Nave-Stoksa: razreshimost i obosnovanie chislennogo modelirovaniya”, Dalnevost. matem. zhurn., 4:1 (2003), 86–100  mathnet  elib
    4. Fursikov A.V., “Feedback stabilization for the 2D Oseen equations: Additional remarks”, Control and Estimation of Distributed Parameter Systems, International Series of Numerical Mathematics, 143, 2003, 169–187  zmath  isi
    5. Chizhonkov E.V., Ivanchikov A.A., “On numerical stabilization of solutions of Stokes and Navier–Stokes equations by the boundary conditions”, Russian J. Numer. Anal. Math. Modelling, 19:6 (2004), 477–494  crossref  mathscinet  zmath  isi  elib
    6. Gorshkov A.V., “Stabilization of linear parabolic equations defined in an exterior of a bounded domain by a boundary control”, J. Dynam. Control Systems, 10:1 (2004), 1–9  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    7. Fursikov A.V., “Stabilization for the 3D Navier–Stokes system by feedback boundary control”, Discrete Contin. Dyn. Syst., 10:1-2 (2004), 289–314  crossref  mathscinet  zmath  isi  elib
    8. Duan Jinqiao, Fursikov A.V., “Feedback stabilization for Oseen fluid equations: A stochastic approach”, J. Math. Fluid Mech., 7:4 (2005), 574–610  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus  scopus
    9. Dymkou V., Rabenstein R., Steffen P., “Discrete simulation of a class of distributed systems using functional analytic methods”, Multidimens. Syst. Signal Process., 17:2-3 (2006), 177–209  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    10. Ivanchikov A.A., “On numerical stabilization of unstable Couette flow by boundary conditions”, Russian J. Numer. Anal. Math. Modelling, 21:6 (2006), 519–537  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    11. M. A. Pribyl, “Spectral analysis of linearized stationary equations of a compressible viscous fluid”, Sb. Math., 198:10 (2007), 1495–1515  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    12. Fursikov A.V., “Analyticity of stable invariant manifolds of 1D-semilinear parabolic equations”, Control Methods in PDE-Dynamical Systems, Contemporary Mathematics Series, 426, 2007, 219–242  crossref  mathscinet  zmath  isi
    13. M. A. Pribyl', “Spectral analysis of linearized stationary equations of viscous compressible fluid in $\mathbb{R}^3$, with periodic boundary conditions”, St. Petersburg Math. J., 20:2 (2009), 267–288  mathnet  crossref  mathscinet  zmath  isi  elib
    14. Dymkou V., Pothérat A., “Spectral methods based on the least dissipative modes for wall bounded MHD flows”, Theoret. Comput. Fluid Dynamics, 23:6 (2009), 535–555  crossref  zmath  adsnasa  isi  elib  scopus  scopus  scopus
    15. A. Yu. Chebotarev, “Ustoichivyi sintez optimalnogo upravleniya v ekstremalnoi zadache dlya ellipticheskogo uravneniya”, Dalnevost. matem. zhurn., 9:1-2 (2009), 204–208  mathnet  elib
    16. Raymond J.-P., Thevenet L., “Boundary Feedback Stabilization of the Two Dimensional Navier–Stokes Equations with Finite Dimensional Controllers”, Discrete and Continuous Dynamical Systems, 27:3 (2010), 1159–1187  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    17. Chebotarev A.Yu., “Stable synthesis of optimal control in stationary extremal problems”, Differ Equ, 46:12 (2010), 1779–1785  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    18. Jean-Pierre Raymond, Henda Fekih, Faker Belgacem, Hassen Arfaoui, “Boundary stabilizability of the linearized viscous Saint-Venant system”, DCDS-B, 15:3 (2011), 491  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    19. Faker Ben Belgacem, Duc Thang Du, Faten Jelassi, “Extended-domain-Lavrentiev's regularization for the Cauchy problem”, Inverse Problems, 27:4 (2011), 045005  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    20. Pribyl M., “Analysis of Spectral Properties of Operators for Linearized Steady-State Equations of a Viscous Compressible Heat-Conducting Fluid”, J Dynam Control Systems, 17:2 (2011), 187–205  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    21. Andrei Fursikov, Alexey V. Gorshkov, “Certain questions of feedback stabilization for Navier–Stokes equations”, EECT, 1:1 (2012), 109  crossref  mathscinet  zmath  scopus  scopus  scopus
    22. Andrei Fursikov, “The simplest semilinear parabolic equation of normal type”, MCRF, 2:2 (2012), 141  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    23. A. V. Gorshkov, “Stabilizing a solution of the 2D Navier-Stokes system in the exterior of a bounded domain by means of a control on the boundary”, Sb. Math., 203:9 (2012), 1244–1268  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    24. Barbu V., “Boundary Stabilization of Equilibrium Solutions to Parabolic Equations”, IEEE Trans. Autom. Control, 58:9 (2013), 2416–2420  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    25. Andrei Fursikov, “Stabilization of the simplest normal parabolic equation”, CPAA, 13:5 (2014), 1815  crossref  mathscinet  zmath  scopus  scopus  scopus
    26. Badra M., Takahashi T., “On the Fattorini Criterion For Approximate Controllability and Stabilizability of Parabolic Systems”, ESAIM-Control OPtim. Calc. Var., 20:3 (2014), 924–956  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    27. Amangaliyeva M.M., Jenaliyev M.T., Imanberdiyev K.B., Ramazanov M.I., “On a Stability of a Solution of the Loaded Heat Equation”, International Conference on Analysis and Applied Mathematics, AIP Conference Proceedings, 1611, eds. Ashyralyev A., Malkowsky E., Amer Inst Physics, 2014, 161–166  crossref  isi  scopus  scopus  scopus
    28. Munteanu I., “Boundary Stabilization of the Stochastic Heat Equation By Proportional Feedbacks”, Automatica, 87 (2018), 152–158  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    29. K. A. Zhukov, A. A. Kornev, A. V. Popov, “Acceleration of transition to stationary state for solutions to a linearized viscous gas dynamics system. I”, Moscow University Mathematics Bulletin, 73:1 (2018), 24–29  mathnet  crossref  isi
    30. K. A. Zhukov, A. A. Kornev, A. V. Popov, “Acceleration of the process of entering stationary mode for molutions of a linearized system of viscous gas dynamics. II”, Moscow University Mathematics Bulletin, 73:3 (2018), 85–89  mathnet  crossref  isi
    31. Jenaliyev M.T., Amangaliyeva M.M., Imanberdiyev K.B., Ramazanov M.I., “On a Stability of a Solution of the Loaded Heat Equation”, Bull. Karaganda Univ-Math., 90:2 (2018), 56–71  isi
    32. K. A. Zhukov, A. A. Kornev, M. A. Lozhnikov, A. V. Popov, “Acceleration of transition to stationary mode for solutions to a system of viscous gas dynamics”, Moscow University Mathematics Bulletin, 74:2 (2019), 55–61  mathnet  crossref  isi
    33. Raymond J.-P., “Stabilizability of Infinite Dimensional Systems By Finite Dimensional Control”, Comput. Methods Appl. Math., 19:2, SI (2019), 267–282  crossref  mathscinet  isi  scopus
    34. Jenaliyev M.T., Imanberdiyev K.B., Kassymbekova A.S., Sharipov K.S., “Spectral Problems Arising in the Stabilization Problem For the Loaded Heat Equation: a Two-Dimensional and Multi-Point Cases”, Eurasian J. Math. Comput. Appl., 7:1 (2019), 23–37  crossref  isi
    35. Munteanu I., “Boundary Stabilisation to Non-Stationary Solutions For Deterministic and Stochastic Parabolic-Type Equations”, Int. J. Control, 92:8 (2019), 1720–1728  crossref  isi
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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