×
Zamyshlyaeva, A. A.; Tsyplenkova, O. N.

Optimal control of solutions to the initial-final problem for the model of linear waves in a plasma. (English) Zbl 1459.35087

Vestn. Yuzhno-Ural. Gos. Univ., Ser. Mat., Mekh., Fiz. 11, No. 4, 26-31 (2019).
Summary: The optimal control problem for a Sobolev type equation of higher order with a relatively polynomially bounded operator pencil is investigated in the paper. The results are applied to the study of the optimal control of solutions to the initial-final problem for the model of linear waves in plasma. The first results on the investigation of equation that describes the linear ion-acoustic waves in an unmagnetized plasma and on the study of some properties of these waves were obtained by Yu. D. Pletner. The initial-final conditions posed for the fourth-order Sobolev type equation are the generalization of the conditions in the Cauchy problem that is unsolvable at the arbitrary initial values. The research is based on the phase space method developed by G. A. Sviridiuk and the theory of relatively polynomially bounded operator pencil developed by A. A. Zamyshlyaeva. The article considers an equation that describes ion-acoustic waves in a plasma in an external magnetic field.

MSC:

35G16 Initial-boundary value problems for linear higher-order PDEs
35Q93 PDEs in connection with control and optimization

Keywords:

Sobolev-type equations of higher-order; relatively polynomially bounded operator pencil; optimal control problem; initial-final conditions
Cite Review PDF
Full Text: DOI MNR

References:

[1] A. G. Sveshnikov, A. B. Al’shin, M. O. Korpusov, Yu. D. Pletner, Linear and Nonlinear the Sobolev Type Equations, Fizmatlit, M., 2007, 736 pp. (in Russ.) · Zbl 1179.35007
[2] A. A. Zamyshlyaeva, “The Higher-Order Sobolev-Type Models”, Bulletin of the South Ural State University, Series “Mathematical Modelling, Programming & Computer Software”, 7:2 (2014), 5-28 (in Russ.) · Zbl 1335.35149 · doi:10.14529/mmp140201
[3] M.A. Sagadeeva, S.A. Zagrebina, N.A. Manakova, “Optimal Control of Solutions of a Multipoint Initial-Finish Problem for Non-Autonomous Evolutionary Sobolev Type Equation”, Evolution Equations and Control Theory, 8:3 (2019), 473-488 · Zbl 1426.49007 · doi:10.3934/eect.2019023
[4] N.A. Manakova, E.A. Bogatyreva, “Mathematical Model of the Start Control of Electric Field Potential in Conducting Medium without Dispersion Considering Relaxation”, 2016 2nd International Conference on Industrial Engineering, Applications and Manufacturing (ICIEAM) (Chelyabinsk, 2016), 1-5 · doi:10.1109/ICIEAM.2016.7911711
[5] A. A. Zamyshlyaeva, O. N. Tsyplenkova, “Optimal Control of Solutions to the Showalter-Sidorov Problem in a Model of Linear Waves in Plazma”, Journal of Computational and Engineering Mathematics, 5:4 (2018), 46-57 · Zbl 1499.49097 · doi:10.14529/jcem180404
[6] E. A. Bogatyreva, N. A. Manakova, “Numerical Simulation of the Process of Nonequilibrium Counterflow Capillary Imbiition”, Computational mathematics and mathematical physics, 56:1 (2016), 132-139 · Zbl 1383.76332 · doi:10.1134/S0965542516010085
[7] A. V. Keller, A. A. Ebel, “Parallelization of Numerical Algorithm for Optimum Dynamic Measurement Problem Solution”, Proc. 2nd International Ural Conference on Measurements (UralCon) (Chelyabinsk, 2017), 372-377 · doi:10.1109/URALCON.2017.8120738
[8] G. A. Zakirova, “Inverse Spectral Problems and Mathematical Models of Continuum Mechanics”, Bulletin of the South Ural State University, Series “Mathematical Modelling, Programming & Computer Software”, 12:2 (2019), 5-24 · Zbl 07130401 · doi:10.14529/mmp190201
[9] D. E. Shafranov, O. G. Kitaeva, “The Barenblatt-Zheltov-Kochina Model with the Showalter-Sidorov Condition and Additive “white noise” in Spaces of Differential Forms on Riemannian Manifolds without Boundary”, Global and Stochastic Analysis, 5:2 (2018), 139-152
[10] G.A. Sviridyuk, V.E. Fedorov, Linear Sobolev Type Equations and Degenerate Semigroups of Operators, VSP, Utrecht-Boston-Köln-Tokyo, 2003, 216 pp. · Zbl 1102.47061 · doi:10.1515/9783110915501
[11] N. A. Manakova, G. A. Svirigyuk, “An Optimal Control of the Solutions of the Initial-Final Problem for Linear Sobolev Type Equations with Strongly Relatively p-Radial Operator”, Semigroup of operators theory and applications, 2015, 213-224 · Zbl 1341.47053 · doi:10.1007/978-3-319-12145-1_13
[12] G. A. Sviridyuk, A. A. Efremov, “Optimal Control of Sobolev Type Linear Equations with Relatively p-Sectorial Operators”, Differential Equations, 31:11 (1995), 1882-1890 · Zbl 0868.49005
[13] A. A. Zamyshlyaeva, O. N. Tsyplenkova, E. V. Bychkov, “Optimal Control of Solutions to the Showalter-Sidorov Problem for the Sobolev type Equation of Higher Order”, Proc. 2nd International Conference on Industrial Engineering, Applications and Manufacturing (ICIEAM) (Chelyabinsk, 2016), 1-4 · Zbl 1329.76044 · doi:10.1109/ICIEAM.2016.7911725
[14] A. Favini, S. A. Zagrebina, G. A. Sviridyuk, “Multipoint Initial-Final Value Problems for Dynamical Sobolev-type Equations in the Space of Noises”, Electronic Journal of Differential Equations, 2018 (2018), 128 · Zbl 1434.60139
[15] A. A. Zamyshlyaeva, Linear Sobolev Type Equations of Higher Order, Publishing center of SUSU, Chelyabinsk, 2012, 88 pp. (in Russ.) · Zbl 1320.35004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.