On a Multiperson Pursuit Problem with Integral Constraints on the Controls of the Players

We consider a linear multiperson differential game with integral constraints on the control of the players. The pursuit is assumed terminated if the solution of at least one of the equations describing the game reaches the origin at some instant of time.In the case of one pursuer, we obtain a necessary and sufficient condition for terminating pursuit from all points of space. In the case of many pursuers, we obtain a sufficient condition for terminating pursuit from all points of space.     

A differential game of evasion with delays

A game of evasion (or contact avoidance problem) described by a system of differential equations containing delays in state and control is considered. There is given a sufficient condition for the existence of a strategy which ensures the contact avoidance. A method of constructing such a strategy is presented.     

The Central Limit Problem for Random Vectors with Symmetries

Motivated by the central limit problem for convex bodies, we study normal approximation of linear functionals of high-dimensional random vectors with various types of symmetries. In particular, we obtain results for distributions which are coordinatewise symmetric, uniform in a regular simplex, or spherically symmetric. Our proofs are based on Stein’s method of exchangeable pairs; as far as we know, this approach has not previously been used in convex geometry. The spherically symmetric case is treated by a variation of Stein’s method which is adapted for continuous symmetries. This work was done while at Stanford University.     

On a differential game in a nondamped distributed system

We study a differential game of approach in a system whose dynamics is described by a Sobolev‐type second‐order operator differential equation in Hilbert spaces. Solutions of the equation are represented by cosine and sine operator functions. To obtain solvability conditions of the game problem, we use support functionals of two sets, which defined by the behaviors of pursuer and evader. The results are applied to the investigation of conflicted‐controlled dynamics of bending waves in a rod.     

On certain properties of a nonlinear eigenvalue problem for Hamiltonian systems of ordinary differential equations

Properties of the eigenvalues are examined in a nonlinear self-adjoint eigenvalue problem for linear Hamiltonian systems of ordinary differential equations. In particular, it is proved that, under certain assumptions, every eigenvalue is isolated and there exists an eigenvalue with any prescribed index.     

The first variation and Pontryagin’s maximum principle in optimal control for partial differential equations

A modification of the classical needle variation, namely, the so-called two-parameter variation of controls is proposed. The first variation of a functional is understood as a repeated limit. It is shown that the modified needle variation can be effectively used to derive necessary optimality conditions for a rather wide class of optimal control problems involving partial differential equations with weak solutions. Specifically, the two-parameter variation is used to obtain necessary optimality conditions in the form of a maximum principle for the optimal control of divergent hyperbolic equations.     

Existence and uniqueness of a Nash equilibrium feedback for a simple nonzero-sum differential game

Existence and uniqueness of a Nash equilibrium feedback is established for a simple class nonzero-sum differential games on the line.     

On the operator of exterior derivation on the Riemannian manifolds with cylindrical ends

We formulate some conditions for the normal and compact solvability of the operator of exterior derivation on the cylindrical manifolds equipped with some Riemannian metrics. Some analogous results were obtained in the particular case of warped cylinders [1].     

On estimates for integral solutions of linear inequalities

Recently, Bombieri and Vaaler obtained an interesting adelic formulation of the first and the second theorems of Minkowski in the Geometry of Numbers and derived an effective formulation of the well-known “Siegel’s lemma” on the size of integral solutions of linear equations. In a similar context involving linearinequalities, this paper is concerned with an analogue of a theorem of Khintchine on integral solutions for inequalities arising from systems of linear forms and also with an analogue of a Kronecker-type theorem with regard to euclidean frames of integral vectors. The proof of the former theorem invokes Bombieri-Vaaler’s adelic formulation of Minkowski’s theorem.     

Conditions of Existence and Uniqueness of Solutions of the Multipoint Boundary Value Problem for a System of Generalized Ordinary Differential Equations

Effective sufficient conditions are established for the solvability and unique solvability of the boundary value problem where is a matrix-function with bounded variation components, is a vector-function belonging to the Carathéodory class corresponding to are continuous functionals (in general nonlinear) defined on the set of all vector-functions of bounded variation.     

关于多项式的Waring问题（Ⅱ）

设ｆ＿ｌ（ｘ）是首项系数为正的ｌ次整系数多项式，满足条件：不存在整数ｄ，ｑ＞１使得ｆ＿ｌ（ｘ）≡ｄ（ｍｏｄｑ）对所有ｘ成立．记Ｒ＿ｋ（ｎ）为方程的正整数的解数，本文的主要结果是：对于及充分大的ｎ，我们有Ｒ＿ｋ（ｎ）》这是Ｖａｕｇｈａｎ关于Ｗａｒｉｎｇ问题的一个结果对多项式的推广。     

The boundedness for solutions of a certain two-dimensional fractional differential system with delay

In this paper, we study the components-wise upper bounds for solutions of two-dimensional fractional differential system with delay. Prior to the main results, we derive some results on two-dimensional nonlinear integral inequalities, then we investigate upper bounds of solutions basing on the obtained inequalities, finally, an example is given to illustrate the theoretical results.     

Quasilinear Pursuit Differential Game with a Simple Dynamics under Phase Constraints

On a fixed closed time interval we consider a quasilinear pursuit differential game with a convex compact target set under a phase constraint in the form of a convex closed set. We construct a convex compact guaranteed capture set similar to an alternating Pontryagin sum and define the guaranteed piecewise-programmed strategy of the pursuer ensuring the hitting of the target set by the phase vector satisfying the phase constraint in finite time. Under certain conditions, we prove the convergence of the constructed alternating sum in the Hausdorff metric to a convex compact set, which is an analog of the alternating Pontryagin integral for the differential game.     

On the properties of a Cauchy-type problem for an abstract differential equation with fractional derivatives

We study the relationship between the solutions of abstract differential equations with fractional derivatives and their stability with respect to the perturbation by a bounded operator. Besides, we obtain representations for the solution of an inhomogeneous equation and for an equation containing a fractional power of the generator of a cosine operator function.     

Smoothness of generalized solutions of boundary value problems for certain degenerate nonlinear ordinary differential equations

The variational method is applied to the study of a boundary value problem of the first kind for a class of nonlinear ordinary differential equations of order 2r with strong degeneracy at the endpoints of the interval (a, b). An inequality is obtained in which the norm of the solutionU of the problem under study in the sense ofW _p, ^r (a, b) is estimated from above by the norms of the given functions (x) andF(x).Translated fromMatematicheskie Zametki, Vol. 63, No. 6, pp. 882–890, June, 1998.     

On the Unique Solvability of an Inverse Problem for Parabolic Equations under a Final Overdetermination Condition

We consider the unique solvability of the inverse problem of determining the right-hand side of a parabolic equation with the leading coefficient depending on time and space variables under a final overdetermination condition. We obtain two types of conditions that are sufficient for the local solvability of the inverse problem and also prove the so-called Fredholm solvability of the inverse problem under study.     

A numerical method to solve a class of linear integro‐differential equations with weakly singular kernel

In this paper, a collocation method based on the Bessel polynomials is introduced for the approximate solution of a class of linear integro‐differential equations with weakly singular kernel under the mixed conditions. The exact solution can be obtained if the exact solution is polynomial. In other cases, increasing number of nodes, a good approximation can be obtained with applicable errors. In addition, the method is presented with error and stability analysis. Copyright © 2012 John Wiley & Sons, Ltd.     

Systems of matrix linear differential equations of first order

Sufficient conditions for the solvability in quadratures of systems of matrix linear ordinary differential equations of first order with one-sided multiplication by variable matrix coefficients are given in this paper. These conditions are stated in terms of the theory of Lie algebras. We consider matrix equations of higher orders that are equivalent to such systems. An illustrative example is considered. Translated fromMatematicheskie Zametki, Vol. 66, No. 1, pp. 63–75, July, 1999.     

On certain two-sided analogues of Newton’s method for solving nonlinear eigenvalue problems

Iterative algorithms for finding two-sided approximations to the eigenvalues of nonlinear algebraic eigenvalue problems are examined. These algorithms use an efficient numerical procedure for calculating the first and second derivatives of the determinant of the problem. Computational aspects of this procedure as applied to finding all the eigenvalues from a given complex-plane domain in a nonlinear eigenvalue problem are analyzed. The efficiency of the algorithms is demonstrated using some model problems.     

Continuity of optimal control in differential games and certain properties of weakly and strongly convex functions

In the first part of the paper, the equivalence of Lipschitzian differentiability of a function and a set of conditions of weak convexity and weak concavity of this function is proved, as well as sufficient conditions for the continuous dependence of the saddle point on a strongly convex-concave function of a parameter are given. In the second part, it is proved that the value function of a game is smooth and the optimal positional and programmed strategies of the players are continuous in zero-sum nonlinear differential games with strongly convex-concave Lagrangian. Translated fromMatematicheskie Zametki, Vol. 66, No. 6, pp. 816–839 December, 1999.