doi: 10.3934/mcrf.2021040
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Stable invariant manifolds with application to control problems

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics, Russia, 119991, Moscow, GSP-1, 1 Leninskiye Gory, Main Building, Russia

* Corresponding author: Alexey Gorshkov

Received  May 2019 Revised  August 2020 Early access September 2021

Fund Project: This article was recruited by Andrii Mironchenko

In this article we develop the theory of stable invariant manifolds for evolution equations with application to control problem. We will construct invariant subspaces for linear equations which can be extended to the non-linear equations in the neighbourhood of the equilibrium with help of perturbation theory. Here will be considered both cases of the discrete and continuous spectrum of the generator associated with resolving semi-group. The example of global invariant manifold will be presented for Burgers equation.

Citation: Alexey Gorshkov. Stable invariant manifolds with application to control problems. Mathematical Control and Related Fields, doi: 10.3934/mcrf.2021040
References:
[1]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Nauka, Moscow English transl., 1983; Stud. Math. Appl., vol. 25, North-Holland, Amsterdam 1992.

[2]

A. V. Fursikov, Stabilizability of a quasi-linear parabolic equation by means of a boundary control with feedback, Sb. Math., 192 (2001), 593-639.  doi: 10.1070/sm2001v192n04ABEH000560.

[3]

A. V. Fursikov, Stabilizability of two-dimensional Navier-Stokes equations with help of a boundary feedback control, J. Math. Fluid Mech., 3 (2001), 259-301.  doi: 10.1007/PL00000972.

[4]

A. V. Fursikov and A. V. Gorshkov, Certain questions of feedback stabilization for Navier-Stokes equations, Evol. Equ. Control Theory, 1 (2012), 109-140.  doi: 10.3934/eect.2012.1.109.

[5]

I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Math. Monogr., vol. 18, Amer. Math. Soc., Providence, RI 1969.

[6]

A. V. Gorshkov, Invariant manifolds for the Burgers equation defined on a semiaxis, Comput. Math. Math. Phys., 58 (2018), 90-101.  doi: 10.1134/S0965542518010062.

[7]

A. V. Gorshkov, Boundary stabilization of Stokes system in exterior domains, J. Math. Fluid Mech., 18 (2016), 679-697.  doi: 10.1007/s00021-016-0258-5.

[8]

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, Sbornik: Mathematics, 203 (2012), 1244-1268.  doi: 10.1070/SM2012v203n09ABEH004263.

[9]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., vol. 840, Springer-Verlag, Berlin-New York 1981.

[10]

T. Kato, Perturbation Theory of Linear Operators, Principles of Mathematical Sciences, Springer-Verlag, 1966.

[11]

M. V. Keldysh, On the completeness of the eigenfunctions of some classes of non-selfadjoint linear operators, Russian Math. Surveys, 26 (1971), 15-41. 

[12]

A. S. Markus, Introduction to the Spectral Theory of Polynomial Operator Pencils, Transl. Math. Monogr., vol. 71, Amer. Math. Soc., Providence, RI 1988. doi: 10.1090/mmono/071.

[13]

A. A. Shkalikov, On the basis property of root vectors of a perturbed self-adjoint operator, Proc. Steklov Inst. Math., 269 (2010), 284-298.  doi: 10.1134/S0081543810020240.

[14]

C. E. Wayne, Invariant manifolds for parabolic partial differential equations on unbounded domains, Arch. Rational Mech. Anal., 138 (1997), 279-306. doi: 10.1007/s002050050042.

[15]

K. Yosida, Functional Analysis, 6th ed., Springer-Verlag, Berlin-New York, 1980.

show all references

References:
[1]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Nauka, Moscow English transl., 1983; Stud. Math. Appl., vol. 25, North-Holland, Amsterdam 1992.

[2]

A. V. Fursikov, Stabilizability of a quasi-linear parabolic equation by means of a boundary control with feedback, Sb. Math., 192 (2001), 593-639.  doi: 10.1070/sm2001v192n04ABEH000560.

[3]

A. V. Fursikov, Stabilizability of two-dimensional Navier-Stokes equations with help of a boundary feedback control, J. Math. Fluid Mech., 3 (2001), 259-301.  doi: 10.1007/PL00000972.

[4]

A. V. Fursikov and A. V. Gorshkov, Certain questions of feedback stabilization for Navier-Stokes equations, Evol. Equ. Control Theory, 1 (2012), 109-140.  doi: 10.3934/eect.2012.1.109.

[5]

I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Math. Monogr., vol. 18, Amer. Math. Soc., Providence, RI 1969.

[6]

A. V. Gorshkov, Invariant manifolds for the Burgers equation defined on a semiaxis, Comput. Math. Math. Phys., 58 (2018), 90-101.  doi: 10.1134/S0965542518010062.

[7]

A. V. Gorshkov, Boundary stabilization of Stokes system in exterior domains, J. Math. Fluid Mech., 18 (2016), 679-697.  doi: 10.1007/s00021-016-0258-5.

[8]

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, Sbornik: Mathematics, 203 (2012), 1244-1268.  doi: 10.1070/SM2012v203n09ABEH004263.

[9]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., vol. 840, Springer-Verlag, Berlin-New York 1981.

[10]

T. Kato, Perturbation Theory of Linear Operators, Principles of Mathematical Sciences, Springer-Verlag, 1966.

[11]

M. V. Keldysh, On the completeness of the eigenfunctions of some classes of non-selfadjoint linear operators, Russian Math. Surveys, 26 (1971), 15-41. 

[12]

A. S. Markus, Introduction to the Spectral Theory of Polynomial Operator Pencils, Transl. Math. Monogr., vol. 71, Amer. Math. Soc., Providence, RI 1988. doi: 10.1090/mmono/071.

[13]

A. A. Shkalikov, On the basis property of root vectors of a perturbed self-adjoint operator, Proc. Steklov Inst. Math., 269 (2010), 284-298.  doi: 10.1134/S0081543810020240.

[14]

C. E. Wayne, Invariant manifolds for parabolic partial differential equations on unbounded domains, Arch. Rational Mech. Anal., 138 (1997), 279-306. doi: 10.1007/s002050050042.

[15]

K. Yosida, Functional Analysis, 6th ed., Springer-Verlag, Berlin-New York, 1980.

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