# Preservation of the Solvability of a Semilinear Global Electric Circuit Equation

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### Abstract

For an initial–boundary value problem associated with a controlled semilinear differential equation of a global electric circuit, a uniqueness theorem and the preservation of global solvability under variations in the controls involved in the higher coefficient and the right-hand side are proved.

## Keywords:

semilinear differential equation of global electric circuit solution uniqueness stable existence of global solutions control of higher coefficient and right-hand side## Notes

### ACKNOWLEDGMENTS

I am sincerely grateful to A.V. Kalinin, who acquainted me with the global electric circuit problem and related mathematical issues, and to V.I. Sumin for helpful discussions of the present material.

## REFERENCES

- 1.A. A. Zhidkov and A. V. Kalinin, “Several problems in mathematical and numerical modeling of global electric circuit in the atmosphere,” Vestn. Nizhegorod. Univ. im. N.I. Lobachevskogo, No. 6(1), 150–158 (2009).Google Scholar
- 2.A. V. Kalinin and N. N. Slyunyaev, “Initial-boundary value problems for the equations of the global atmospheric electric circuit,” J. Math. Anal. Appl.
**450**, 112–136 (2017).MathSciNetCrossRefzbMATHGoogle Scholar - 3.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), 1565–1579 (2016).MathSciNetCrossRefzbMATHGoogle Scholar - 4.A. V. Chernov, “Differentiation of the functional in a parametric optimization problem for the higher coefficient of an elliptic equation,” Differ. Equations
**51**(4), 548–557 (2015).MathSciNetCrossRefzbMATHGoogle Scholar - 5.A. V. Chernov, “On the uniqueness of solution to the inverse problem of determination of parameters in the senior coefficient and the right-hand side of an elliptic equation,” Dal’nevost. Mat. Zh.
**16**(1), 96–110 (2016).MathSciNetzbMATHGoogle Scholar - 6.V. I. Sumin, Candidate’s Dissertation in Mathematics and Physics (Gorky State Univ., Gorky, 1975).Google Scholar
- 7.V. I. Sumin, “On the stability of existence of a global solution to the Dirichlet boundary value problem for a controlled parabolic equation,” Differ. Uravn.
**22**(9), 1587–1595 (1986).Google Scholar - 8.V. I. Sumin, “Volterra functional operator equations in the theory of optimal control of distributed systems,” Dokl. Akad. Nauk SSSR
**305**(5), 1056–1059 (1989).MathSciNetGoogle Scholar - 9.V. I. Sumin, “The features of gradient methods for distributed optimal-control problems,” USSR Comput. Math. Math. Phys.
**30**(1), 1–15 (1990).CrossRefzbMATHGoogle Scholar - 10.V. I. Sumin, “On sufficient conditions for the stability of existence of global solutions to controlled boundary value problems,” Differ. Uravn.
**26**(12), 2097–2109 (1990).Google Scholar - 11.V. I. Sumin, “Volterra functional operator equations and stability of existence of global solutions to boundary value problems,” Ukr. Mat. Zh.
**43**(4), 555–561 (1991).MathSciNetGoogle Scholar - 12.V. I. Sumin,
*Volterra Functional Equations in the Theory of Optimal Control of Distributed Systems*, Part 1:*Volterra Equations and Controlled Initial–Boundary Value Problems*(Nizhegorod. Gos. Univ., Novgorod, 1992) [in Russian].Google Scholar - 13.V. I. Sumin and A. V. Chernov, “Conditions for the stability of existence of global solutions to the Cauchy problem for a hyperbolic equation,” Vestn. Nizhegorod. Gos. Univ. Mat. Model. Optim. Upr., 94–103 (1997).Google Scholar
- 14.V. I. Sumin, “Controlled Volterra functional equations in Lebesgue spaces,” Vestn. Nizhegorod. Gos. Univ. Mat. Model. Optim. Upr., No. 2(19), 138–151 (1998).Google Scholar
- 15.V. I. Sumin, Doctoral Dissertation in Physics and Mathematics (Nizhny Novgorod State Univ., Nizhny Novgorod, 1998).Google Scholar
- 16.A. V. Chernov, “On conditions for the stability of existence of global solutions to the controlled Goursat problem for a hyperbolic equation,” Vestn. Nizhegorod. Gos. Univ. Mat. Model. Optim. Upr., No. 1(20), 154–162 (1999).Google Scholar
- 17.I. V. Lisachenko and V. I. Sumin, “Nonlinear Goursat–Darboux control problem: conditions for the preservation of global solvability,” Differ. Equations
**47**(6), 863–876 (2011).MathSciNetCrossRefzbMATHGoogle Scholar - 18.V. I. Sumin, “Stability of existence of global solutions for controlled boundary value problems and Volterra functional equations,” Vestn. Nizhegorod. Gos. Univ. Mat., No. 1, 91–107 (2003).Google Scholar
- 19.V. I. Sumin and A. V. Chernov, “Volterra functional operator equations in the theory of optimization of distributed systems,”
*Proceedings of International Conference on Dynamics of Control Systems and Processes Dedicated to Academician N.N. Krasovskii on the Occasion of His 90th Birthday, Yekaterinburg, Russia, September 15–20, 2014*(Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, Yekaterinburg, 2015), pp. 293–300.Google Scholar - 20.V. I. Sumin and A. V. Chernov, “Volterra operator equations in Banach spaces: Stability of existence of global solutions,” Available from VINITI, No. 1198-V00 (Nizhny Novgorod State Univ., Nizhny Novgorod, 2000) [in Russian].Google Scholar
- 21.A. V. Chernov, Candidate’s Dissertation in Physics and Mathematics (Nizhny Novgorod State Univ., Nizhny Novgorod, 2000).Google Scholar
- 22.V. I. Sumin and A. V. Chernov, “On sufficient conditions of existence stability of global solutions of Volterra operator equations,” Vestn. Nizhegorod. Gos. Univ. Mat. Model. Optim. Upr., No. 1(26), 39–49 (2003).Google Scholar
- 23.A. V. Chernov, “A majorant-minorant criterion for the total preservation of global solvability of a functional operator equation,” Russ. Math.
**56**(3), 55–65 (2012).CrossRefzbMATHGoogle Scholar - 24.A. V. Chernov, “On the totally global solvability of a controlled Hammerstein-type equation with a varied linear operator,” Vestn. Udmurt. Univ. Mat. Mekh. Komp’yut. Nauki
**25**(2), 230–243 (2015).CrossRefzbMATHGoogle Scholar - 25.A. V. Chernov, “On a majorant–minorant criterion for the total preservation of global solvability of distributed controlled systems,” Differ. Equations
**52**(1), 111–121 (2016).MathSciNetCrossRefzbMATHGoogle Scholar - 26.O. A. Ladyzhenskaya and N. N. Uraltseva,
*Linear and Quasilinear Elliptic Equations*(Nauka, Moscow, 1973; Academic, New York, 1987).Google Scholar - 27.H. Gajewski, K. Gröger, and K. Zacharias,
*Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen*(Akademie, Berlin, 1974).zbMATHGoogle Scholar - 28.A. G. Sveshnikov, A. B. Al’shin, and M. O. Korpusov,
*Nonlinear Functional Analysis and Its Applications to Partial Differential Equations*(Nauchnyi Mir, Moscow, 2008) [in Russian].Google Scholar

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