On differentiation of functional in problem on parametric coefficient optimization in semilinear global electric circuit equation

A. V. Chernov

Ufa mathematical journal, Tome 13 (2021) no. 3, pp. 152-173

Résumé

For the problem on parametric optimization with respect to an integral criterion of the coefficient and the right-hand side of the semilinear global electric circuit equation, we obtain formulae for the first partial derivatives of the cost functional with respect to control parameters. The problem on reconstructing unknown parameters of the equation by the observed data of local sensors can be represented in such form. In the paper we generalize a similar result obtained earlier by the author for the case of linear global electric circuit equation. But it is commonly believed by experts that the right hand side treated as the volumetric density of external currents of the equation depends on the gradient, with respect to the collection of space variables, of the unknown electric potential function. Because of this, it is necessary to study the case of a semilinear equation. We use the conditions of preserving global solvability of the semilinear global electric circuit equation and the estimates for the increment of the solutions, which we have obtained formerly. The mathematical novelty of presented research is due to the fact that, unlike the earlier studied linear case, now the right hand side depends nonlinearly on the state, which, in its turn, depends on the controlled parameters. Such more complicated nonlinear dependence of the state on the control parameters requirs, in particular, the development of a special technique to estimate the additional terms arising in the increment of solutions of the controlled equation.

Détail

Keywords: controlled coefficient and right hand side, parametric optimization, semilinear differential global electric circuit equation.

Bibliographie

[1] A. G. Sveshnikov, A. B. Alshin, M. O. Korpusov, Nonlinear functional analysis and its applications to partial differential equations, Nauchnyi mir, M., 2008 (in Russian)

[2] A. A. Zhidkov, A. V. Kalinin, “Some issues of mathematical and numerical modelling of a global eletric circuit in atmosphere”, Vestnik Nizhegorodskogo Univ. im N.I. Lobachevskogo, 6:1 (2009), 150–158 (in Russian)

[3] A. V. Kalinin, N. N. Slyunyaev, “Initial-boundary value problems for the equations of the global atmospheric electric circuit”, J. Math. Anal. Appl., 450:1 (2017), 112–136 | DOI | Zbl

[4] E. A. Mareev, S. V. Anisimov, “Geophysical studies of the global electric circuit”, Izv. Phys. Solid Earth, 44:10 (2008), 760–769 | DOI

[5] E. A. Mareev, “Global electric circuit research: achievements and prospects”, Physics Uspekhi, 53:5 (2010), 504–511 | DOI | DOI

[6] J. Jansky, V. P. Pasko, “Effects of conductivity perturbations in time-dependent global electric circuit model”, J. Geophys. Res.: Space Phys., 120:12 (2015), 10654–10668 | DOI

[7] V. V. Denisenko, M. J. Rycroft, R. G. Harrison, “Mathematical simulation of the ionospheric electric field as a part of the global electric circuit”, Surv. Geophys., 40:1 (2019), 1–35 | DOI

[8] V. N. Morozov, Mathematical modelling of atmosphere-electric processes taking into account influence of aerosol particles and radioactive substances, Ross. Gosud. Gidrometeorol. Univ., St. Petersburg, 2011 (in Russian)

[9] A. J.G. Baumgaertner, J. P. Thayer, R. R. Neely, G. Lucas, “Toward a comprehensive global electric circuit model: Atmospheric conductivity and its variability in CESM1(WACCM) model simulations”, J. Geophys. Res.: Atmospheres, 118:16 (2013), 9221–9232 | DOI

[10] S. S. Davydenko, E. A. Mareev, T. C. Marshall, M. Stolzenburg, “On the calculation of electric fields and currents of mesoscale convective systems”, J. Geophys. Res., 109:D11 (2004), 1–10

[11] G. M. Lucas, A. J.G. Baumgaertner, J. P. Thayer, “A global electric circuit model within a community climate model”, J. Geophys. Res.: Atmospheres, 120:23 (2015), 12054–12066 | DOI

[12] V. Bayona, N. Flyer, G. M. Lucas, A. J.G. Baumgaertner, “A 3-D RBF-FD solver for modeling the atmospheric global electric circuit with topography (GEC-RBFFD v1.0)”, Geoscientific Model Development, 8:10 (2015), 3007–3020 | DOI

[13] N. A. Denisova, A. V. Kalinin, “Influence of the choice of boundary conditions on the distribution of the electric field in models of the global electric circuit”, Radiophys. Quant. Electronics, 61:10 (2019), 741–751 | DOI

[14] A. V. Chernov, “Differentiation of a functional in the problem of parametric coefficient optimization in the global electric circuit equation”, Comput. Math. Math. Phys., 56:9 (2016), 1565–1579 | DOI | Zbl

[15] A. V. Chernov, “Differentiation of the functional in a parametric optimization problem for the higher coefficient of an elliptic equation”, Diff. Equat., 51:4 (2015), 548–557 | DOI | Zbl

[16] A. V. Chernov, “Differentiation of the functional in a parametric optimization problem for a coefficient of a semilinear elliptic equation”, Diff. Equat., 53:4 (2017), 551–562 | DOI | Zbl

[17] A. V. Chernov, “On the uniqueness of solution to the inverse problem of determination parameters in the senior coefficient and the righthand side of an elliptic equation”, Dalnevostochnyi Matem. Zhurn., 16:1 (2016), 96–110 (in Russian) | Zbl

[18] A. V. Chernov, “On uniqueness of solutions to inverse problem of atmospheric electricity”, Vestnik Ross. Univ. Matem., 25:129 (2020), 85–99 (in Russian)

[19] A. V. Chernov, “Preservation of the solvability of a semilinear global electric circuit equation”, Comput. Math. Math. Phys., 58:12 (2018), 2095–2111 | DOI | Zbl

[20] A. V. Chernov, “Total preservation of the solvability of the semilinear global electric circuit equation”, Differ. Equats., 54:8 (2018), 1073–1082 | DOI | Zbl

[21] F. V. Lubyshev, A. R. Manapova, “Difference approximations of optimization problems for semilinear elliptic equations in a convex domain with controls in the coefficients multiplying the highest derivatives”, Comput. Math. Math. Phys., 53:1 (2013), 8–33 | DOI | Zbl

[22] O. A. Ladyzhenskaya, N. N. Ural'tseva, Linear and quasilinear elliptic equations, Academic Press, New York, 1968 | Zbl

[23] H. Gajewski, K. Gröger, K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Akademie-Verlag, Berlin, 1974 | Zbl

[24] D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin, 1983 | Zbl

[25] K. Yosida, Functional analysis, Springer-Verlag, Berlin, 1965 | Zbl

[26] A. V. Chernov, “On total preservation of global solvability of differential operator equation”, Prikl. Matem. Vorp. Upravl., 2 (2017), 94–111 (in Russian)

[27] A. V. Chernov, “Local convexity conditions for reachability tubes of distributed control systems”, Russ. Math., 58:11 (2014), 60–73 | DOI | Zbl

[28] M. A. Krasnosel'skii, Topological methods in the theory of nonlinear integral equations, Pergamon Press, Oxford, 1964 | Zbl