Global dynamics and control of a nonlinear body equation with strong structural damping

A nonlinear hinged extensible elastic body equation with strong structural damping and Balakrishnan-Taylor damping of full exponent is studied as a general model for large space structures of higher dimensions. In this paper, the absorbing sets and flat inertial manifold are obtained for this nonlinear body equation. The control spillover problem associated with the stabilization of this equation is resolved by constructing a linear finite dimensional feedback, control based on the existence of inertial manifolds of the uncontrolled equation. Moreover, the results obtained are robust with respect to the uncertainty in structural parameters. Supported by the National Natural Science Foundation of China (No. 19701023)     

Inverse coefficient problems for nonlinear parabolic differential equations

This paper is devoted to a class of inverse problems for a nonlinear parabolic differential equation. The unknown coefficient of the equation depends on the gradient of the solution and belongs to a set of admissible coefficients. It is proved that the convergence of solutions for the corresponding direct problems continuously depends on the coefficient convergence. Based on this result the existence of a quasisolution of the inverse problem is obtained in the appropriate class of admissible coefficients.     

A certain approach to the problem of regularization on the basis of the perturbation of linear operators

We consider a method for constructing the solutions of a linear Fredholm operator equation that is regularized by means of a special perturbation of the equation by a linear operator.Translated from Matematicheskii Zametki, Vol. 20, No. 5, pp. 747–752, November, 1976.The author is grateful to M. A. Krasnosel'skill for discussing the results of this article.     

Tetrahedron equation and spin integrable models on a cubic lattice

This paper is a survey of results on the three-dimensional generalization of the Yang-Baxter equation obtained since the pioneer works by Zamolodchikov (1979) up to our articles in 1995. The integrability condition for statistical spin models on a simple cubic lattice (tetrahedron equation) is discussed and different versions of this equation are considered together with their symmetrical properties. The solution of the tetrahedron equation corresponding to the Bazhanov-Baxter model is considered in detail. The review contains an updated list of solutions for this equation. Generalization to inhomogeneous spin models with two types of Boltzmann weights forming a chessboard-type lattice is considered. The paper was written at the request of the Editorial Board. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 2, pp. 179–213, February, 1997.     

Width of parametric resonance regions for equations on a torus

We present new proofs of the theorem on the width of the forbidden regions for the Hill equation with a small potential and the theorem on the width of the parametric resonance regions for a first-order differential equation on a torus. These results are special cases of the theorem proved in this paper by the normal form method.Translated fromMatematicheskie Zametki, Vol. 64, No. 1, pp. 129–135, July, 1998.The author wishes to thank M. F. Kondrat'eva, V. V. Sidorenko, and the referee for useful remarks.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-01411.     

On the dynamics of deformable ferromagnets

The paper begins with a fairly detailed presentation of a general mathematical model for the dynamics of ferromagnetic bodies undergoing arbitrarily large deformations. Next, a mathematical study is presented of the evolution problem for the magnetization field in a «soft» ferromagnetic body which is «mechanically at rest». No matter how special and simple this problem within the frame-work of the full theory, the governing equation is interesting: it is identical to the dynamic version of the harmonic-map equation usually referred to in the mathematical literature as the Gilbert form of the Landau-Lifshitz equation. Motivated by recent nonuniqueness results for the dynamic harmonicmap equation, we give a new proof of global existence of weak solutions to the Gilbert-Landau-Lifshitz equation.     

Local integral manifolds for a nonautonomous parabolic equation

The paper deals with a quasilinear nonautonomous parabolic equation. A finite-dimensional Lipschitz integral manifold is constructed in a neighborhood of an equilibrium point of the limiting autonomous equation. It is proved that the integral manifold attracts exponentially all the solutions of the initial nonautonomous equation, passing through this neighborhood. Bibliography: 13 titles. Dedicated to Olga A. Oleinik This work was supported in part by the Russian Foundation for Basic Research and by the International Science Foundation. Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 19, pp. 000-000, 0000.     

New results on periodic solutions for a kind of Rayleigh equation

The paper deals with the existence of periodic solutions for a kind of non-autonomous time-delay Rayleigh equation. With the continuation theorem of the coincidence degree and a priori estimates, some new results on the existence of periodic solutions for this kind of Rayleigh equation are established. This work is supported by Science and Technology Plan of Hunan Province, China under Grant No. 2008FJ3143 and the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of Hunan Province.     

On Singular Sets of Local Solutions to p-Laplace Equations

The author proves that the on the singular set of a local solution to existence of an optimal control problem. right-hand term of a p-Laplace equation is zero the equation. Such a result is used to study the     

Mixed problem for a nonlinear ultraparabolic equation that generalizes the diffusion equation with inertia

We consider a mixed problem for a nonlinear ultraparabolic equation that is a nonlinear generalization of the diffusion equation with inertia and the special cases of which are the Fokker-Planck equation and the Kolmogorov equation. Conditions for the existence and uniqueness of a solution of this problem are established. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 9, pp. 1192–1210, September, 2006.     

A boundary function equation and its numerical solution

We consider the asymptotic solution of the plasma-sheath integro-differential equation, which is singularly perturbed due to the presence of a small coefficient multiplying the highest order (second) derivative. The asymptotic solution is obtained by the boundary function method. A second-order differential equation is derived describing the behavior of the zeroth-order boundary functions. A numerical algorithm for this equation is discussed. __________ Translated from Prikladnaya Matematika i Informatika, No. 24, pp. 24–34, 2006.     

Direct solution to problems of hydraulic jump in horizontal triangular channels

The specific force equation has many applications in open channel flow problems. Quantifying of the hydraulic jump phenomenon is an important application of this equation. This equation has a direct solution only for the rectangular channels. The trial and error procedure as well as the graphical solution are the existing methods of solving hydraulic jump equations. No direct solutions are available in technical literature for sequent depth ratios in horizontal triangular channels because it is presumed that the governing equation is a quintic equation. In the present study, considering physical concepts this quintic equation has been reduced to a quartic equation. In the next step, this quartic equation has been converted to a resolvent cubic equation and two quadratic equations. This research shows these steps clearly to reach an acceptable physical analytic solution for sequent depth ratios in horizontal triangular channels.     

A note on an open problem about the first Painlevé equation

In this note, we apply numerical analysis to the first equation, find the conditions for it to have oscillating solutions and therefore solve an open problem posed by Peter A. Clarkson.     

Using the Renyi entropy to describe quantum dissipative systems in statistical mechanics

We develop a formalism for describing quantum dissipative systems in statistical mechanics based on the quantum Renyi entropy. We derive the quantum Renyi distribution from the principle of maximum quantum Renyi entropy and differentiate this distribution (the temperature density matrix) with respect to the inverse temperature to obtain the Bloch equation. We then use the Feynman path integral with a modified Mensky functional to obtain a Lindblad-type equation. From this equation using projection operators, we derive the integro-differential equation for the reduced temperature statistical operator, an analogue of the Zwanzig equation in statistical mechanics, and find its formal solution in the form of a series in the class of summable functions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 3, pp. 444–453, September, 2008.     

Splitting of potential curves in the two-center quantum-mechanical problem for the Dirac equation

An asymptotic (over a large internuclear distance) theory is presented for the two-center quantum-mechanical problem described by the Dirac equation. The asymptotic equation of the two-center wave function of a Dirac electron is constructed for a system consisting of an arbitrary ion and an atom. Using this function, we calculate the asymptotic leading term of the potential curve splitting in the relativistic two-center problem.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 109, No. 2, pp. 232–249, November, 1996.     

Classification of integrable Vlasov-type equations

The classification of integrable Vlasov-type equations reduces to a functional equation for a generating function. We find a general solution of this functional equation in terms of hypergeometric functions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 2, pp. 249–260, February, 2008.     

Optimization for nonlinear hyperbolic equations without the uniqueness theorem for a solution of the boundary-value problem

We consider the optimal control problem for a system governed by a nonlinear hyperbolic equation without any constraints on the parameter of nonlinearity. No uniqueness theorem is established for a solution to this problem. The control-state mapping of this system is not Gateaux differentiable. We study an approximate solution of the optimal control problem by means of the penalty method.     

A self-adaptive trust region method for the extended linear complementarity problems

By using some NCP functions, we reformulate the extended linear complementarity problem as a nonsmooth equation. Then we propose a self-adaptive trust region algorithm for solving this nonsmooth equation. The novelty of this method is that the trust region radius is controlled by the objective function value which can be adjusted automatically according to the algorithm. The global convergence is obtained under mild conditions and the local superlinear convergence rate is also established under strict complementarity conditions. This work is supported by National Natural Science Foundation of China (No. 10671126) and Shanghai Leading Academic Discipline Project (S30501).     

Junctions of singularly degenerating domains with different limit dimensions. II

This paper is aimed at finding asymptotic formulas for solutions to the mixed boundary problem for the Poisson equation in a domain obtained by joining singularly degenerating domains. In this paper, which is the second part of the work (the first part was published in No. 18), the main attention is given to three-dimensional problems in which a thin plate or a periodic family of thin rods is joined to a massive body (the distances between the rods are comparable with the diameters of their cross-sections). The distinctive feature of such problems is that an integral equation arises as one of the limit problems. Bibliography: 48 titles. Translated from Trudy Seminara imeni l. G. Petrovskogo, No. 20, pp. 155–195, 1997.     

Solving an optimal control problem for parabolic equations

We consider optimal boundary control of a distributed-parameter system. The system state is described by two parabolic equations of second order, where the coefficients of one equation depend on the gradient of the solution of the second equation. An existence and uniqueness theorem is proved for the optimal control in this problem and the necessary conditions of optimality are derived.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 59, pp. 90–98, 1986.