Hilbert manifolds with corners of finite codimension and the theory of optimal control

This survey presents a version of Palais-Smale theory for Hilbert manifolds that is convenient for investigation of optimal control problems associated with smooth control systems of constant rank. Results obtained with A. A. Agrachev concerning the finite-dimensional case — the analog of Morse theory for (finite-dimensional) manifolds with corners are given. Simple applications of the theory are discussed. A necessary condition for global controllability of systems of constant rank is obtained, as well as a dual result on the multiplicity of solutions for the corresponding optimization problems.Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 28, pp. 96–171, 1990.     

Model of the soil and computational complex for the analysis of underground explosions

In underground explosions executed in the interests of ejection, downcomer funnel or bulging hillock formation, the soil properties influence not only the quantitative parameters substantially, but also the qualitative pattern of the explosion. Thus, under the same conditions of charge embedding and power, a downcomer funnel or bulging hillock can be formed depending on the properties of the rock. The majority of explosions are performed in hard rock. Hence, the model of the soil should be suitable to describe its fundamental properties. A model of rocky soil is presented in this paper, the scheme for a numerical computation of the problem is described, and results of certain computations are presented.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 153–160, May–June, 1979.     

Chronological series and the Cauchy-Kowalevski theorem

We consider the Cauchy problem for a system of partial differential equations. We prove an existence theorem for a solution of this problem which is analytic in the spatial variable under the assumption of measurability and local integrability of the right side with respect to time only. The solution is represented in the form of a chronological series.Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 12, pp. 165–189, 1981.