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Статья
2016

On the structure of a solution set of controlled initial-boundary value problems


A. V. ChernovA. V. Chernov
Русская математика
https://doi.org/10.3103/S1066369X16020109
Abstract / Full Text

For a controlled nonlinear functional-operator equation of the Hammerstein type describing a wide class of controlled initial-boundary value problems, we obtain simple sufficient conditions for the convexity, pointwise boundedness and precompactness of the set of solutions (the reachability tube) in the Lebesgue space. As for boundedness and precompactness, we mean certain conditions of the majorant but not Volterra type requirements which give also the total (with respect to the whole set of admissible controls) preservation of solvability of mentioned equation. As some examples of reduction of a controlled initial-boundary (boundary) value problem to the equation under investigation, and verification of the proposed hypotheses for this equation, we consider the first initial-boundary value problem associated with a semilinear parabolic equation of the second order in a rather general form, and also the Dirichlet problem associated with a semilinear elliptic equation of the second order.

Author information
  • Nizhny Novgorod State University, pr. Gagarina 23, Nizhny Novgorod, 603950, RussiaA. V. Chernov
References
  1. Chernov, A. V. On the Convexity of Reachability Sets of Controlled Initial-Boundary Value Problem, Differential Equations 50, No. 5, 700–710 (2014).
  2. Polyak, B. Convexity of the Reachable Set of Nonlinear Systems Under L 2 Bounded Controls, Dynamics of continuous, discrete and impulsive systems. Ser. A: Math. Anal. 11, No. 2–3, 255–268 (2004).
  3. Reißig G. Convexity of Reachable Sets of Nonlinear Ordinary Differential Equations, Automation and Remote Control 68, No. 9, 1527–1543 (2007).
  4. Cannarsa, P., Sinestrari, C. Convexity Properties of the Minimum Time Function, Calc. Var. Partial Diff. Equat. 3, No. 3, 273–298 (1995).
  5. Chernov, A. V. A Majorant Criterion for the Total Preservation of Global Solvability of Controlled Functional Operator Equation, RussianMathematics (Iz. VUZ) 55, No. 3, 85–95 (2011).
  6. Chernov, A. V. A Majorant–Minorant Criterion for the Total Preservation of Global Solvability of a Functional Operator Equation, RussianMathematics (Iz. VUZ) 56, No. 3, 55–65 (2012).
  7. Chernov, A. V. On Volterra Type Generalization of Monotonization Method for Nonlinear Functional Operator Oquations, Vestn. Udmurtsk. Univ. Mat. Mekh., 2, 84–99 (2012). [in Russian].
  8. Chernov, A. V. On a Generalization of the Method of Monotone Operators, Differential Equations 49, No. 4, 517–527 (2013).
  9. Potapov, D. K. Control Problems for Equations With a Spectral Parameter and a Discontinuous Operator Under Perturbations, Zh. Sib. Fed. Univ. Math. Phys. 5, No. 2, 239–245 (2012). [in Russian].
  10. Vainberg, M. M. Variational Method and the Monotone Operator Method in the Theory of Nonlinear Equations (Nauka, Moscow, 1972) [in Russian].
  11. Mordukhovich, B. S. Approximation Methods in Problems of Optimization and Control (Nauka, Moscow, 1988) [in Russian].
  12. Krasnosel’skii, M. A. Topological Methods in the Theory of Nonlinear Integral Equations, (GITTL, Moscow, 1956; Pergamon Press, Oxford, 1964).
  13. Kantorovich, L. V., Akilov, G. P. Functional Analysis (Nauka, Moscow, 1984) [in Russian].
  14. Ladyzhenskaya, O. A, Solonnikov, V. A., and Ural’tseva, N. N. Linear and Quasilinear Equations of Parabolic Type (Nauka, Moscow, 1967) [in Russian].
  15. Chernov, A. V. On Nonnegativity of the Solution to the First Boundary-Value Problem for a Parabolic Equation, Vestnik of Lobachevsky University of Nizhny Novgorod, No. 5 (1), 167–170 (2012). [in Russian].
  16. Ladyzhenskaya, O. A, Ural’tseva, N. N. Linear and Quasilinear Elliptic Equations (Nauka, Moscow, 1973) [in Russian].
  17. Vorob’yov, A. Kh. Diffusion Problems in Chemical Kinetics (Moscow State University, Moscow, 2003) [in Russian].
  18. Tröltzsch, F. Optimal Control of Partial Differential Equations: Theory, Methods and Applications, Graduate Stud. Math. 112 (AmericanMath. Soc., Providence, R. I., 2010).
  19. 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, Computational Mathematics and Mathematical Physics 53, No. 1, 8–33 (2013).
  20. Vakhitov, I. S. Inverse Problem of Identification of the Diffusion Coefficient in Diffision–Reaction Equation, Dal’nevost. Mat. Zh. 10, No. 2, 93–105 (2010). [in Russian].
  21. Karchevskii, M. M., Pavlova, M. F. Equations of mathematical Physics. Additional Chapters (Kazan State University, Kazan, 2012) [in Russian].
  22. Gilbarg, D., Trudinger, N. Elliptic Partial Differential Equations of Second Order (Springer, Berlin, Heidelberg, New York, 1983; Nauka, Moscow, 1989).