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Normal Equation Generated from Helmholtz System: Nonlocal Stabilization by Starting Control and Properties of Stabilized Solutions

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Recent Developments in Integrable Systems and Related Topics of Mathematical Physics (MP 2016)

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Abstract

We consider the semilinear normal parabolic equation (NPE) corresponding to the 3D Helmholtz system with periodic boundary conditions. First, we recall the main definitions and results associated with the NPE including a result on stabilization to zero of the solution for NPE with arbitrary initial condition by starting control. The main content of the paper is to study properties of stabilized solution of NPE.

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Notes

  1. 1.

    Of course, many other equations including the Helmholtz system possess this property.

  2. 2.

    Note that because of periodic boundary conditions the Stokes system should not contain the pressure term \(\nabla p\).

  3. 3.

    Strictly speaking, one needs to add \(|\widehat{z}(0)|^2\) to the right-hand side of (30). However, we did not do this, since starting from (31) it will be assumed that this Fourier coefficient is zero.

  4. 4.

    Here and below we use for brevity notation \(S(t;y_0)\) instead of \(S(t,\cdot ;y_0)\)

References

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  7. Fursikov, A.V., Shatina, L.S.: Nonlocal stabilization of the normal equation connected with Helmholtz system by starting control. Dis. Cont. Dyn. Syst.-A. 38(3), 1187–1242 (2018)

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Acknowledgments

The work has been fulfilled by RAS program “Theoretical problems of modern mathematics", project “Optimization of numerical algorithms of Mathematical Physics problems". The author was supported in part by RFBI Grants 15-01-03576 and 15-01-08023.

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Authors and Affiliations

  1. Faculty of Mechanics and Mathematics, “Lomonosov” Moscow State University, GSP-1 Leninskie Gory, 119991, Moscow, Russian Federation

    A. V. Fursikov

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  1. A. V. Fursikov
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Correspondence to A. V. Fursikov .

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Editors and Affiliations

  1. Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia

    Victor M. Buchstaber

  2. Centre of Integrable Systems, P.G. Demidov Yaroslavl State University, Yaroslavl, Russia

    Sotiris Konstantinou-Rizos

  3. School of Mathematics, University of Leeds, Leeds, UK

    Alexander V. Mikhailov

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Fursikov, A.V. (2018). Normal Equation Generated from Helmholtz System: Nonlocal Stabilization by Starting Control and Properties of Stabilized Solutions. In: Buchstaber, V., Konstantinou-Rizos, S., Mikhailov, A. (eds) Recent Developments in Integrable Systems and Related Topics of Mathematical Physics. MP 2016. Springer Proceedings in Mathematics & Statistics, vol 273. Springer, Cham. https://doi.org/10.1007/978-3-030-04807-5_11

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