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## Archive of Issues

Russia Nizhni Novgorod
Year
2015
Volume
25
Issue
2
Pages
230-243
 Section Mathematics Title On the totally global solvability of a controlled Hammerstein type equation with a varied linear operator Author(-s) Chernov A.V.ab Affiliations Nizhni Novgorod State Technical Universitya, Nizhni Novgorod State Universityb Abstract Let $n,m,\ell,s\in\mathbb{N}$ be given numbers, $\Pi\subset\mathbb{R}^n$ be a measurable bounded set, $\mathcal{X}, \mathcal{Z}, \mathcal{U}$ be Banach ideal spaces of functions measurable on the set $\Pi$, $\mathcal{D}\subset\mathcal{U}^{s}$ be a convex set, $\mathcal{A}$ be some class of linear bounded operators $A:\mathcal{Z}^{m} \to\mathcal{X}^{\ell}$. We study the controlled Hammerstein type functional operator equation as follows $$x(t)=\theta(t)+ A\Bigl[ f(.,x(.),u(.)) \Bigr](t), \quad t\in \Pi , \quad x\in\mathcal{X}^{\ell}, \qquad \qquad (1)$$ where $\{ u,\theta,A\}\in \mathcal{D}\times \mathcal{X}^{\ell}\times \mathcal{A}$ is the set of controlled parameters; $f(t,x,v): \Pi\times\mathbb{R}^{\ell}\times\mathbb{R}^{s}\to\mathbb{R}^{m}$ is a given function measurable with respect to $t\in\Pi$, continuous with respect to $\{x,v\}\in\mathbb{R}^\ell\times\mathbb{R}^s$ and satisfying to certain natural hypotheses. Eq. $(1)$ is a convenient form of representation of the broad class of controlled distributed systems. For the equation under study we prove a theorem concerning sufficient conditions of global solvability for all $u\in\mathcal{D}$, $A\in\mathcal{A}$ and $\theta$ from a pointwise bounded set. For the original equation we define some majorant and minorant inequalities obtaining them from Eq. $(1)$ with the help of upper and lower estimates of the right-hand side. The theorem is proved providing global solvability of the majorant and minorant inequalities. As an application of obtained general results we prove a theorem concerning the total (with respect to the whole set of admissible controls) global solvability of the mixed boundary value problem for a system of hyperbolic equations of the first order with controlled higher coefficients. Keywords totally global solvability, functional operator equation of the Hammerstein type, pointwise estimate of solutions, system of hyperbolic equations of the first order with controlled higher coefficients UDC 517.957, 517.988, 517.977.56 MSC 47J05, 47J35, 47N10 DOI 10.20537/vm150207 Received 29 March 2015 Language Russian Citation 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. References Kalantarov V.K., Ladyzhenskaya O.A. On the appearance of collapses for quasilinear equations of the parabolic and hyperbolic types, Zap. Nauch. Sem. LOMI, 1977, vol. 69, pp. 77-102 (in Russian). Lions J.-L. Upravlenie singulyarnymi raspredelennymi sistemami (Control of singular distributed systems), Moscow: Nauka, 1987, 368 p. Sumin V.I. The features of gradient methods for distributed optimal control problems, USSR Comput. Math. Math. Phys., 1990, vol. 30, no. 1, pp. 1-15. Sumin V.I. Funktsional'nye vol'terrovy uravneniya v teorii optimal'nogo upravleniya raspredelennymi sistemami. Chast' I. Vol'terrovy uravneniya i upravlyaemye nachal'no-kraevye zadachi (Functional Volterra equations in the theory of optimal control of distributed systems. Part I. Volterra equations and controlled initial boundary value problems), Nizhni Novgorod: Nizhni Novgorod State University, 1992, 110 p. Sveshnikov A.G., Al'shin A.B., Korpusov M.O. Nelineinyi funktsional'nyi analiz i ego prilozheniya k uravneniyam v chastnykh proizvodnykh (Nonlinear functional analysis and its applications to partial differential equations), Moscow: Nauchnyi mir, 2008, 400 p. Filippov A.F. Differentsial'nye uravneniya s razryvnoi pravoi chast'yu (Differential equations with discontinuous right-hand side), Moscow: Nauka, 1985, 224 p. Alekseev V.M., Tikhomirov V.M., Fomin S.V. Optimal'noe upravlenie (Optimal control), Moscow: Nauka, 1979, 432 p. 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. Optimization of controlled generalized Volterra systems, Cand. Sci. (Phys.-Math.) Dissertation, Gorkii, 1975, 158 p (in Russian). Morozov S.F., Sumin V.I. Optimization of nonlinear transport processes, Sov. Math., Dokl., 1979, vol. 20, pp. 802-806. Morozov S.F., Sumin V.I. Optimization of the non-linear systems of transport theory, USSR Comput. Math. Math. Phys., 1979, vol. 19, no. 1, pp. 101-114. Sumin V.I. Volterra functional-operator equations in the theory of optimal control of distributed systems, Sov. Math., Dokl., 1989, vol. 39, no. 2, pp. 374-378. Sumin V.I. Sufficient conditions for stable existence of solutions to global problems in control theory, Differ. Equations, 1990, vol. 26, no. 12, pp. 1579-1590. 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. Functional Volterra equations in the mathematical theory of optimal control of distributed systems, Dr. Sci. (Phys.-Math.) Dissertation, Nizhni Novgorod, 1998, 346 p (in Russian). Sumin V.I., Chernov A.V. Conditions for existence stability of global solutions to controlled Cauchy problem for a hyperbolic equation, Vestn. Nizhegorod. Univ. N.I. Lobachevskogo. Mat. Model. Optim. Upr., 1997, pp. 94-103 (in Russian). Sumin V.I., Chernov A.V. Volterra operator equations in Banach spaces: existence stability of global solutions, Nizhni Novgorod State University, Nizhni Novgorod, 2000, 75 p. Deposited in VINITI 25.04.2000, no. 1198-V00 (in Russian). Chernov A.V. Volterra operator equations and their application in the theory of optimization of hyperbolic systems, Cand. Sci. (Phys.-Math.) Dissertation, Nizhni Novgorod, 2000, 177 p (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. 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. Sufficient conditions for the controllability of nonlinear distributed systems, Comput. Math. Math. Phys., 2012, vol. 52, no. 8, pp. 1115-1127. DOI: 10.1134/S0965542512050053 Chernov A.V. On controllability of nonlinear distributed systems on a set of discretized controls, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2013, no. 1, pp. 83-98 (in Russian). Chernov A.V. On the convergence of the conditional gradient method in distributed optimization problems, Comput. Math. Math. Phys., 2011, vol. 51, no. 9, pp. 1510-1523. DOI: 10.1134/S0965542511090077 Chernov A.V. Smooth finite-dimensional approximations of distributed optimization problems via control discretization, Comput. Math. Math. Phys., 2013, vol. 53, no. 12, pp. 1839-1852. DOI: 10.1134/S096554251312004X 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, 2014, vol. 20, no. 1, pp. 305-321 (in Russian). Chernov A.V. On Volterra functional operator games on a given set, Automation and Remote Control, 2014, vol. 75, no. 4, pp. 787-803. DOI: 10.1134/S0005117914040195 Vainberg M.M. Variational method and method of monotone operators in the theory of nonlinear equations, New York-Toronto: John Wiley & Sons, 1973; Jerusalem-London: Israel Program for Scientific Translations, 1973, xi+356 p. Original Russian text published in Vainberg M.M. Variatsionnyi metod i metod monotonnykh operatorov v teorii nelineinykh uravnenii, Moscow: Nauka, 1972, 416 p. 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 Chernov A.V. A generalization of Bihari's lemma to the case of Volterra operators in Lebesgue spaces, Mathematical Notes, 2013, vol. 94, no. 5, pp. 703-714. DOI: 10.1134/S0001434613110114 Sumin V.I., Chernov A.V. Operators in spaces of measurable functions: the Volterra property and quasinilpotency, Differ. Equations, 1998, vol. 34, no. 10, pp. 1403-1411. Sumin V.I., Chernov A.V. On some indicators of the quasi-nilpotency of functional operators, Russian Mathematics, 2000, vol. 44, no. 2, pp. 75-78. Kantorovich L.V., Akilov G.P. Funktsional'nyi analiz (Functional Analysis), Moscow: Nauka, 1984, 752 p (in Russian). Chernov A.V. On the existence stability of global solutions to a system of hyperbolic equations of the first order under the higher coefficients control, Trudy XXIII konferentsii molodykh uchenykh (Proceedings of XXIII conference of young scientists), Lomonosov Moscow State University, Moscow, 2001, pp. 352-355 (in Russian). Chernov A.V. On necessary optimality conditions in the problem of higher coefficients control in a system of hyperbolic equations of the first order, Matematicheskoe modelirovanie i kraevye zadachi: Trudy II Vserossiiskoi nauchnoi konferentsii (Mathematical modeling and boundary value problems: Proceedings of the Second All-Russian scientific conference), Part 2, Samara State Technical University, Samara, 2005, pp. 259-262 (in Russian). Full text