On quasioptimum selection of the regularization parameter in M. M. Lavrent'ev's method

Moscow. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 34, No. 4, pp. 117–126, July–August, 1993.     

On nonconvex optimization with integral constraints

We consider the problem of minimizing f(y)dm with y dm=c,c fixed. The functionf is assumed to be continuous, but need not be convex. For this problem, we give necessary and sufficient conditions for the existence of solutions. We also give conditions under which uniqueness in a certain sense holds, and we show a relation which holds between the minimizers of two different problems and the corresponding values of the constraintsc.This research was supported by FINEP-Brazil, Grant Nos. 62.24-0416-00 and 4.2.82.0719-00.     

Spectral asymptotics for problems with integral constraints

The eigenvalue problem for differential operators of arbitrary order with integral constraints is considered. The asymptotics of the eigenvalues is obtained. The results are applied to finding the asymptotics of the probability of small deviations for some detrended processes of nth order.     

On linear differential games with integral constraints

The paper is devoted to quasilinear conflict-controlled processes of general form with a cylindrical terminal set. A specific feature is that, instead of a dynamical system, we start with representation of a solution in a form that allows one to include an additive term with the initial data and a control unit. This makes it possible to consider a broad spectrum of dynamic processes in a unified scheme. Our study is based on the method of resolving functions. We obtain sufficient conditions for the solvability of the pursuit problem at a certain guaranteed time in the class of strategies that use information on the behavior of the opponent in the past, as well as in the class of stroboscopic strategies. We also find conditions under which information on the prehistory of the evader does not matter. The guaranteed times of various schemes of the resolving function method are compared with the guaranteed time of Pontryagin’s first direct method. The qualitative results are illustrated by an example of a game problem with simple motions and incomplete sweeping for special control domains in the Pontryagin condition.     

Quasiconformal extremals of smooth functionals and of the energy integral on Riemann surfaces

Krasnodar. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 29, No. 3, pp. 163–174, May–June, 1988.     

Variational problems with obstacles and integral constraints

Various problems in mathematics and physics can be formulated in terms of a variational problem with obstacles and integral constraints, e.g. finding a surface of minimal area with prescribed volume in a bounded region.We are concerned with the regularity of solutions of variational problems: We show that the minima of a variational integral under all Sobolewfunctions with prescribed boundary values, lying between two obstacles, and fulfilling some integral constraints, are bounded and Hölder-continuous. We do not assume any differentiability or convexity of the integrand, but only a Caratheodory and a growth condition.This research has been supported by the Sonderforschungsbereich 72 of the Deutsche Forschungsgemeinschaft.     

A maximum principle for an optimal control problem with integral constraints

In this paper, necessary corditions are obtained for an optimal control problem whose state variables are given in terms of integral equations. The conditions are obtained separately for Volterra equations and Fredholm equations. The main result for each case is the maximum principle and multiplier rule. For the Volterra equations, transversality conditions are obtained.     

On inner dilatations of the mappings with unbounded characteristic

For the mappings f:D ? D￠,  D,  D￠ ì \mathbbRⁿ f:D \to D',\,\,D,\,\,D' \subset {\mathbb{R}^n} , n ≥ 2, satisfying certain geometric conditions in the fixed domain D, we have proved estimates of the form K _I (x, f) ≤ Q(x) almost everywhere, where K _I (x, f) is the inner dilatation of f at a point x, and Q(x) is a fixed real-valued function responsible for the “control” over a distortion of the families of curves in D at a mapping f.     

Extremal problems in a class of mappings with bounded integral characteristics

We consider mappings with bounded integral characteristics. We construct extremal mappings of plane rings realizing the minimum of these characteristics.