On the analogue of Wintner's theorem for a controlled elliptic equation

A. V. Chernov

Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 46 (2015) no. 2, pp. 228-235.

Résumé

For a homogeneous Dirichlet problem associated with a controlled semilinear partial differential elliptic equation of the second order, referred as a stationary diffusion–reaction equation, we state analogue of the classical Wintner's theorem concerning the solvability of the Cauchy problem for an ordinary differential equation.

Détail

Keywords: controlled semilinear elliptic equation, total preservation of solvability.

Bibliographie

[1] Hartman Ph., Ordinary differential equations, John Wiley Sons, New York, 1964, xiv+612 pp. | MR | Zbl

[2] Ladyzhenskaya O. A., Ural'tseva N. N., Linear and quasilinear elliptic equations, Academic Press, New York–London, 1968, xviii+495 pp. | MR | Zbl

[3] Gilbarg D., Trudinger N. S., Elliptic partial differential equations of second order, Springer, Berlin, 1983, xiii+513 pp. | MR | Zbl

[4] Chernov A. V., “A majorant criterion for the total preservation of global solvability of controlled functional operator equation”, Russian Mathematics, 55:3 (2011), 85–95 | DOI | MR

[5] Kalantarov V. K., Ladyzhenskaya O. A., “On the appearance of collapses for quasilinear equations of the parabolic and hyperbolic types”, Zap. Nauch. Sem. LOMI, 69, 1977, 77–102 (in Russian) | MR | Zbl

[6] Sumin V. I., “The features of gradient methods for distributed optimal control problems”, USSR Comput. Math. Math. Phys., 30:1 (1990), 1–15 | Zbl

[7] Sumin V. I., Functional Volterra equations in the theory of optimal control of distributed systems, v. I, Volterra equations and controlled initial boundary value problems, NNSU, Nizhni Novgorod, 1992, 110 pp.

[8] Petrosyan L. A., Zenkevich N. A., Semina E. A., Game Theory, Vysshaya Shkola, M., 1998, 304 pp.

[9] Casas E., “Boundary control of semilinear elliptic equations with pointwise state constraints”, SIAM J. Control Optim., 31 (1993), 993–1006 | DOI | MR | Zbl

[10] Tröltzsch F., Optimal control of partial differential equations: theory, methods and applications, Graduate Studies in Mathematics, 112, American mathematical society, Providence, R.I., 2010, xv+399 pp. | MR | Zbl

[11] Chernov A. V., “A majorant-minorant criterion for the total preservation of global solvability of a functional operator equation”, Russian Mathematics, 56:3 (2012), 55–65 | DOI | MR | Zbl

[12] Chernov A. V., “On a generalization of the method of monotone operators”, Differential Equations, 49:4 (2013), 517–527 | DOI | MR | Zbl

[13] Chernov A. V., “On the convergence of the conditional gradient method in distributed optimization problems”, Comput. Math. Math. Phys., 51:9 (2011), 1510–1523 | DOI | MR | Zbl

[14] Chernov A. V., “Smooth finite-dimensional approximations of distributed optimization problems via control discretization”, Comput. Math. Math. Phys., 53:12 (2013), 1839–1852 | DOI | DOI | MR | Zbl

[15] Chernov A. V., “On the smoothness of an approximated optimization problem for a Goursat–Darboux system on a varied domain”, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 20, no. 1, 2014, 305–321 (in Russian) | MR

[16] Chernov A. V., “Sufficient conditions for the controllability of nonlinear distributed systems”, Comput. Math. Math. Phys., 52:8 (2012), 1115–1127 | DOI | MR | Zbl

[17] Chernov A. V., “On Volterra functional operator games on a given set”, Automation and Remote Control, 75:4 (2014), 787–803 | DOI | MR | Zbl

[18] Chernov A. V., “On $\varepsilon$-equilibrium in noncooperative functional operator $n$-person games”, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 19, no. 1, 2013, 316–328 (in Russian) | MR

[19] Chernov A. V., “On existence of $\varepsilon$-equilibrium in noncooperative $n$-person games associated with elliptic partial differential equations”, Matematicheskaya teoriya igr i prilozheniya, 6:1 (2014), 91–115 (in Russian) | Zbl

[20] Vorob'ev A. Kh., Diffusion problems in chemical kinetics, Moscow State University, M., 2003, 98 pp.

[21] Lubyshev F. V., Manapova A. R., “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 | DOI | MR | Zbl

[22] Vakhitov I. S., “Inverse identification problem for unknown coefficient in the diffusion–reaction equation”, Dal'nevost. Mat. Zh., 10:2 (2010), 93–105 (in Russian)