+7 (3412) 91 60 92

## Archive of Issues

Russia Nizhni Novgorod
Year
2018
Volume
28
Issue
4
Pages
531-548
 Section Mathematics Title Majorant sign of the first order for totally global solvability of a controlled functional operator equation Author(-s) Chernov A.V.ab Affiliations Nizhni Novgorod State Technical Universitya, Nizhni Novgorod State Universityb Abstract We consider a nonlinear functional operator equation of the Hammerstein type which is a convenient form of representation for a wide class of controlled distributed parameter systems. For the equation under study we prove a solution uniqueness theorem and a majorant sign for the totally (with respect to a whole set of admissible controls) global solvability subject to Volterra property of the operator component and differentiability with respect to a state variable of the functional component in the right hand side. Moreover, we use hypotheses on the global solvability of the original equation for a fixed admissible control $u=v$ and on the global solvability for some majorant equation with the right hand side depending on maximal deviation of admissible controls from the control $v$. For example we consider the first boundary value problem associated with a parabolic equation of the second order with right hand side $f\bigl( t, x(t),u(t)\bigr)$, $t=\{ t_0,\overline{t}\}\in\Pi=(0,T)\times Q$, $Q\subset\mathbb{R}^n$, where $x$ is a phase variable, $u$ is a control variable. Here, a solution to majorant equation can be represented as a solution to the zero initial-boundary value problem of the same type for analogous equation with the right hand side $bx^{q/2}+a_0x+Z$, where $Z(t)=\max\limits_{u\in\mathcal{V}(t)} |f(t,x[v](t),u)-f(t,x[v](t),v(t))|$, $\mathcal{V}(t)\subset\mathbb{R}^s$ is a set of admissible values for control at $t\in\Pi$, $q>2$, $s\in\mathbb{N}$; $a_0(.)$ and $b\geqslant0$ are parameters defined from $f^\prime_x$. Keywords functional operator equation of the Hammerstein type, totally global solvability, majorant equation, Volterra property UDC 517.957, 517.988, 517.977.56 MSC 47J05, 47J35, 47N10 DOI 10.20537/vm180407 Received 23 May 2018 Language Russian Citation Chernov A.V. Majorant sign of the first order for totally global solvability of a controlled functional operator equation, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2018, vol. 28, issue 4, pp. 531-548. References Kalantarov V.K., Ladyzhenskaya O.A. The occurrence of collapse for quasilinear equations of parabolic and hyperbolic types, Journal of Soviet Mathematics, 1978, vol. 10, issue 1, pp. 53-70. DOI: 10.1007/BF01109723 Sumin V.I. The features of gradient methods for distributed optimal control problems, USSR Computational Mathematics and Mathematical Physics, 1990, vol. 30, no. 1, pp. 1-15. DOI: 10.1016/0041-5553(90)90002-A Sumin V.I. Funktsional'nye vol'terrovy uravneniya v teorii optimal'nogo upravleniya raspredelennymi sistemami. Chast' 1. Vol'terrovy uravneniya i upravlyaemye nachal'no-kraevye zadachi (Functional Volterra equations in the theory of optimal control of distributed systems. Part 1. Volterra equations and controlled initial boundary value problems), Nizhny Novgorod: Nizhny Novgorod State University, 1992, 110 p. Ikehata R. On solutions to some quasilinear hyperbolic equations with nonlinear inhomogeneous terms, Nonlinear Analysis: Theory, Methods & Applications, 1991, vol. 17, no. 2, pp. 181-203. DOI: 10.1016/0362-546X(91)90221-L Chernov A.V. On total preservation of solvability of controlled Hammerstein-type equation with non-isotone and non-majorizable operator, Russian Mathematics, 2017, vol. 61, no. 6, pp. 72-81. DOI: 10.3103/S1066369X1706010X Tröltzsch F. Optimal control of partial differential equations: theory, methods and applications, Providence, R.I.: American Mathematical Society, 2010, xv+399 p. DOI: 10.1090/gsm/112 Turo J. Global solvability of the mixed problem for first order functional partial differential equations, Ann. Polon. Math., 1991, vol. 52, no. 3, pp. 231-238. DOI: 10.4064/ap-52-3-231-238 Carasso C., Hassnaoui E.H. Mathematical analysis of the model arising in study of chemical reactions in a catalytic cracking reactor, Mathematical and Computer Modelling, 1993, vol. 18, no. 2, pp. 93-109. DOI: 10.1016/0895-7177(93)90010-V Kobayashi T., Pecher H., Shibata Y. On a global in time existence theorem of smooth solutions to a nonlinear wave equation with viscosity, Mathematische Annalen, 1993, vol. 296, issue 1, pp. 215-234. DOI: 10.1007/BF01445103 Biler P., Hilhorst D., Nadzieja T. Existence and nonexistence of solutions for a model of gravitational interaction of particles. II, Colloq. Math., 1994, vol. 67, no. 2, pp. 297-308. DOI: 10.4064/cm-67-2-297-308 Hu B., Yin H.-M. Global solutions and quenching to a class of quasilinear parabolic equations, Forum Math., 1994, vol. 6, no. 6, pp. 371-383. DOI: 10.1515/form.1994.6.371 Lu G. Global existence and blow-up for a class of semilinear parabolic systems: a Cauchy problem, Nonlinear Analysis: Theory, Methods & Applications, 1995, vol. 24, no. 8, pp. 1193-1206. DOI: 10.1016/0362-546X(94)00190-S Doppel K., Herfort W., Pflüger K. A nonlinear beam equation arising in the theory of elastic bodies, Z. Anal. Anwend., 1997, vol. 16, no. 4, pp. 945-960. DOI: 10.4171/ZAA/798 Yamazaki T. Scattering for a quasilinear hyperbolic equation of Kirchhoff type, J. Differential Equations, 1998, vol. 143, no. 1, pp. 1-59. DOI: 10.1006/jdeq.1997.3372 Nakamura M., Ozawa T. The Cauchy problem for nonlinear wave equations in the homogeneous Sobolev space, Annales de l’Institut Henri Poincaré. Physique Théorique, 1999, vol. 71, no. 2, pp. 199-215. http://www.numdam.org/item/AIHPA_1999__71_2_199_0/ Catalano F. The nonlinear Klein-Gordon equation with mass decreasing to zero, Advances in Differential Equations, 2002, vol. 7, no. 9, pp. 1025-1044. https://projecteuclid.org/euclid.ade/1367241458 Cavalcanti M.M., Domingos Cavalcanti V.N., Soriano J.A. On existence and asymptotic stability of solutions of the degenerate wave equation with nonlinear boundary conditions, J. Math. Anal. Appl., 2003, vol. 281, no. 1, pp. 108-124. DOI: 10.1016/S0022-247X(02)00558-9 Rozanova-Pierrat A. Qualitative analysis of the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation, Mathematical Models and Methods in Applied Sciences, 2008, vol. 18, no. 5, pp. 781-812. DOI: 10.1142/S0218202508002863 Zhao X. Self-similar solutions to a generalized Davey-Stewartson system, Mathematical and Computer Modelling, 2009, vol. 50, issues 9-10, pp. 1394-1399. DOI: 10.1016/j.mcm.2009.04.023 Tersenov A. The Dirichlet problem for second order semilinear elliptic and parabolic equations, Differ. Equ. Appl., 2009, vol. 1, no. 3, pp. 393-411. DOI: 10.7153/dea-01-22 Guo C., Zhao X., Wei X. Cauchy problem for higher-order Schrödinger equations in anisotropic Sobolev space, Appl. Anal., 2009, vol. 88, no. 9, pp. 1329-1338. DOI: 10.1080/00036810903277127 Korpusov M.O., Ovchinnikov A.V., Sveshnikov A.G. On blow up of generalized Kolmogorov-Petrovskii-Piskunov equation, Nonlinear Analysis: Theory, Methods & Applications, 2009, vol. 71, no. 11, pp. 5724-5732. DOI: 10.1016/j.na.2009.05.002 Blokhin A.M., Tkachev D.L. Asymptotic stability of the stationary solution for a new mathematical model of charge transport in semiconductors, Quarterly of Applied Mathematics, 2012, vol. 70, no. 2, pp. 357-382. DOI: 10.1090/S0033-569X-2012-01251-7 Kharibegashvili S., Jokhadze O. The Cauchy-Darboux problem for wave equations with a nonlinear dissipative term, Mem. Differ. Equ. Math. Phys., 2016, vol. 69, pp. 53-75. Saito H. Global solvability of the Navier-Stokes equations with a free surface in the maximal $L_p$-$L_q$ regularity class, Journal of Differential Equations, 2018, vol. 264, no. 3, pp. 1475-1520. DOI: 10.1016/j.jde.2017.09.045 Sumin V.I. Stability problem for the existence of global solutions to boundary value control problems and Volterra functional equations, Vestn. Nizhegorod. Univ. N.I. Lobachevskogo. Mat., 2003, no. 1, pp. 91-107 (in Russian). Sumin V.I., Chernov A.V. Volterra functional operator equations in the theory of distributed systems optimization, Dinamika sistem i protsessy upravleniya: Trudy Mezhdunarodnoi konferentsii (System dynamic and control processes: Proceedings of International Conference Dedicated to the 90th Anniversary of Academician N.N. Krasovskii), Ekaterinburg: Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 2015, pp. 293-300 (in Russian). https://elibrary.ru/item.asp?id=23795754 Chernov A.V. On the totally global solvability of a controlled Hammerstein type equation with a varied linear operator, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2015, vol. 25, issue 2, pp. 230-243 (in Russian). DOI: 10.20537/vm150207 Chernov A.V. A majorant criterion for the total preservation of global solvability of controlled functional operator equation, Russian Mathematics, 2011, vol. 55, no. 3, pp. 85-95. DOI: 10.3103/S1066369X11030108 Chernov A.V. A majorant-minorant criterion for the total preservation of global solvability of a functional operator equation, Russian Mathematics, 2012, vol. 56, no. 3, pp. 55-65. DOI: 10.3103/S1066369X12030085 Hartman Ph. Ordinary differential equations, New York: John Wiley & Sons, 1964, xiv+612 p. Chernov A.V. A majorant criterion of the first order for the total preservation of global solvability of a controlled Hammerstein type equation, Vestn. Tambov. Univ. Ser. Estestv. Tekh. Nauki, 2015, vol. 20, issue 5, pp. 1526-1529 (in Russian). Sumin V.I. On functional Volterra equations, Russian Mathematics, 1995, vol. 39, no. 9, pp. 65-75. Sumin V.I. Controlled functional Volterra equations in Lebesgue spaces, Vestn. Nizhegorod. Univ. N.I. Lobachevskogo. Mat. Model. Optim. Upr., 1998, no. 2 (19), pp. 138-151 (in Russian). Sumin V.I., Chernov A.V. On sufficient conditions of existence stability of global solutions of Volterra operator equations, Vestn. Nizhegorod. Univ. N.I. Lobachevskogo. Mat. Model. Optim. Upr., 2003, no. 1 (26), pp. 39-49 (in Russian). Chernov A.V. Pointwise estimation of the difference of the solutions of a controlled functional operator equation in Lebesgue spaces, Mathematical Notes, 2010, vol. 88, no. 2, pp. 262-274. DOI: 10.1134/S0001434610070242 Chernov A.V. On total preservation of global solvability of functional operator equations, Vestn. Nizhegorod. Univ. N.I. Lobachevskogo, 2009, no. 3, pp. 130-137 (in Russian). Mordukhovich B.Sh. Metody approksimatsii v zadachakh optimizatsii i upravleniya (Approximation methods in problems of optimization and control), Moscow: Nauka, 1988, 360 p. Krasnosel'skii M.A., Zabreiko P.P., Pustyl'nik E.I., Sobolevskii P.E. Integral'nye operatory v prostranstvakh summiruemykh funktsii (Integral operators in spaces of summable functions), Moscow: Nauka, 1966, 500 p. Ladyzhenskaya O.A., Solonnikov V.A., Ural'tseva N.N. Lineinye i kvazilineinye uravneniya parabolicheskogo tipa (Linear and quasilinear equations of parabolic type), Moscow: Nauka, 1967, 736 p. Chernov A.V. On nonnegativity of the solution to the first boundary value problem for a parabolic equation, Vestn. Nizhegorod. Univ. N.I. Lobachevskogo, 2012, no. 5 (1), pp. 167-170 (in Russian). Skrypnik I.V. Metody issledovaniya nelineinykh ellipticheskikh granichnykh zadach (Methods for the investigation of nonlinear elliptic boundary value problems), Moscow: Nauka, 1990, 448 p. Full text