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Invited researcher Andrey Alexandrovich Agrachev
Contract number
14.B25.31.0029
Time span of the project
2013-2015

As of 30.01.2020

16
Number of staff members
32
scientific publications
14
Objects of intellectual property
General information

Name of the project: Geometric control theory and analysis on metric structures

Strategy for Scientific and Technological Development Priority Level: а


Goals and objectives

Research directions:

- Problems of metric analysis, geometric control theory and differential geometry on Riemann structures and nonholonomic structures and aspects of their applications to optimal control theory

- Optimization over infinite time intervals and long-term behavior of dissipative systems, research of anisotropic diffusion controlled by hypoelliptic equations.

- Usage of methods of geometric control theory to solving applied problems: control of configuration of moving bodies, image recovery and quantum systems control.

Project objective: Designing new powerful geometrical and analytical means for solving complex problems in geometric control theory and analysis on metric structures, as well as applications of achieved results to problems in adjacent domains of pure mathematics and applied fields of studies.


The practical value of the study

  • Our researchers have computed and thoroughly studied curvature operators for multi-dimensional contact sub-Riemannian manifolds.
  • We have found formulas to compute squares of mappings-graphs on five-dimensional sub-Lorentz structures with two «negative» directions of different orders.
  • Universal methods have been proposed to search for geodesic normal coordinates on Lee groups with left-invariant (sub)-Riemannian metrics.
  • We have investigated the problem of energy functional minimization on a new class of mappings compared to classes researched earlier.
  • Our researchers have proven solvability of the boundary value problem for stationary systems of Navier-Stokes equations in in unlimited (external) regions with inhomogeneous boundary data, in absence of traditional suggestions concerning minuteness of flows of liquid through components of the boundary. For the axis-symmetrical case we have solved the problem that remained open for over 80 years since the famous 1933 publication by Jean Leray.
  • An original method has been developed for lifting vector fields that allows to both achieve new results and significantly simplify proving existing ones.
  • We have obtained description of isomorphic composition operators of Sobolev classes on Carnot–Carathéodory spaces.
  • We have found necessary and sufficient conditions at which homomorphism of regions in the Euclidean space generate a bounded embedding operator for Sobolev–Orlich spaces determined by a special class of N functions.
  • Our researchers have studied the structure of extremals in the problem of search for a light cone in the sub-Lorentzian Geometry over an Engel group.
  • We have found the main geometrical objects for all the Lee groups that locally isometrically cover the SO_2(2,1) Lee groups with a left-invariant (sub)-Riemannian metric that is invariant under right shifts on elements of the SO(2) subgroup. Results were expressed in terms of matrix elements of corresponding Lee groups.
  • Our researchers are currently investing properties of measurable mappings on a Carnot group that induce isomorphisms of the Sobolev spaces according to the rule of variable substitution. We have proven that such mapping can be re-determined on a set of null measure in such a way that it would be either quasi-conformal (when the summability indicator matches the Hausdorff measure of the group) or лquasi-isometric (when the summability indicator does not match the Hausdorff measure of the group).
  • We have found the formula of square for spatially similar surfaces-graphs on two-stage four-dimensional sub-Lorentzian structures, as well as obtained descriptions of basic properties of maximal surfaces including in terms of sub-Lorentzian mean curvature
  • We have proven that the conditions of (q1,1)- and (1, q2)-quasi-metricity of the function of distance is sufficient for existence of 1-quasi-metric that is bi-Lipschitz-equivalent to it.
  • We have proven the existence of (q1, q2)-quasi-metrics for which there is no 1-quasi-metric that are Lipschitz-equivalent to them from which, in particular, follows another proof of the known result of W. Schroeder.
  • Our researchers have proven the theorem of regularization of (q1, q2)-quasi-metrics. 
  • We have investigated axioms of separability on lim-weak symmetrical (q1, q2)-quasi-metrical spaces.
  • Our researchers have determined the formula of square for geometric mappings of Carnot groups. The main tool of research is the notion of polynomial sub-Riemannian differentiability introduced by the author.
  • We have given an exhaustive description of boundary values of conform mappings of flat finitely-connected regions in terms of conform modules of pairs of boundary components of the reviewed region in case when its connectivity is less or equal than 3.

Education and career development:

  • Two scientific schools and 22 scientific seminars have been conducted.
  • We have conducted additional training courses for 39 young scientists from the Novosibirsk State University, the Altai State University, the Gorno-Altaisk State University, the Novosibirsk State Technical University, the V. A. Steklov Mathematical Institute of the Russian Academy of Sciences, V. M. Matrosov Institute for System Dynamics and Control Theory of the Siberian Branch of the Russian Academy of Sciences (Russia), the Eindhoven University of Technology (the Netherlands) and the Laboratory of Analysis, Topology, Probabilities (France).
  • 8 candidate dissertations and one doctoral dissertation have been defended.
  • The Laboratory has developed the course «Optimal control theory» for fourth-year students of the Faculty of Mathematics and Mechanics of the Novosibirsk State University (Russia).

Organizational and structural changes: We have created the Regional Mathematical Center at the Novosibirsk State University (Russia).

Collaborations:

  • International School for Advanced Studies (Italy): organizing internships for young scientists
  • Ecole Polytechnique (France): joint research, organizing internships for young scientists
  • V. A. Steklov Mathematical Institute of the Russian Academy of Sciences (Russia), A. K. Ailamazian Institute of Software Systems of the Russian Academy of Sciences (Russia), Gorno-Altaisk State University (Russia): collaborative scientific events

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Journal of Dynamical and Control Systems. 2014. V. 20, № 1. P. 123-148.
Selivanova S. Metric Geometry of Nonregular Weighted Carnot–Carathéodory Spaces
Korobkov M.V., Pileckas K., Russo R.// Ann. of Math., 2015, Vol. 181, No. 2, 769–807.
Solution of Leray's problem for stationary Navier-Stokes equations in plane and axially symmetric spatial domains
Bourgain J., Korobkov M.V., Kristensen J. // Periodica Mathematica Hungarica, 2016, vol. 72, no. 2, pp. 252-257.
On the Morse--Sard property and level sets of $W^{n,1}$ Sobolev functions on $\mathbb R^n$ // Journal fur die reine und angewandte Mathematik (Crelles Journal), 2015, Vol.2015, No. 700, 93–112.
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