I algebraic geometry

We cooperate mainly with:

• 
  University of Warsaw: Krzysztof Barański, Anna Zdunik,
  
• 
  Warsaw University of Technology: Janina Kotus, Grzegorz Świątek
  
• 
  University of Warwick, that coordinated the network CODY.
  

X. GEOMETRIC METHODS IN ODE'S AND CONTROL THEORY
Team Leader: prof. Bronislaw Jakubczyk
Members:

• 
  prof. Stanislaw Janeczko (also in team IX)
  
• 
  Pawel Nurowski (visiting associate professor)
  
• 
  dr hab. Piotr Mormul
  
• 
  prof. Witold Respondek
  
• 
  dr Wojciech Krynski
  
• 
  dr Gabriel Pietrzkowski
  
• 
  dr Marek Grochowski
  

Outside member of the team:

• 
  prof. Witold Respondek (INSA-Rouen)
  

Research mission and objectives:

Ordinary Differential Equations (ODEs) and Control Theory (CT) provide adequate language and tools for stating and solving many problems in Mechanics, Mathematcal Physics, Engineering and other applied fields. When nonlinear problems are considered, the use of differential-geometric methods is natural. Such methods play an increasing role in understanding and solving some of the basic problems in these fields.

We intend to combine and extend the existing results in geometric theories of both fields (ODEs and CT) and in Symplectic Geometry in order to create more effective tools for solving some of the problems appearing in applications. In particular, we want to extend the notion of curvature so that it can be used for the analysis of control systems and optimality of extremals.

Invariants non-holonomic distributions and of mechanical control systems are to be analysed and exploited. Intrinsic geometries hidden in the geometric structures defined by ODEs are hoped to be revealed.

Research Programme and future research plans in relation to the state-of-the-art:

Geometric Control Theory started in late 70s’ with the use of Lie bracket (R. Hermann, P. Brunovsky, H. Hermes, A. Krener, C. Lobry, H. Sussmann). Transitivity and accessibility criteria were found, the local conrollability problem was understood, existence and uniqueness of realizations criteria were given, functional expansions were introduced and used.

The most important for applications class of linear systems was geometrically characterised (Jakubczyk and Respondek 1980) (feedback linearizable systems). A challenging open problem is to characterize flat systems. In a different setting the problem was already stated by E. Cartan and D. Hilbert. New developments in understanding the geometry of non-integrable (non-holonomic) distributions, especially identification of the role of abnormal curves, give hopes that such criteria will be found. Collaboration of mathematicians from Poland and France would be highly desirable in this context.