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SUMMARY:A.N. Rogalev (Institute of Computational Modeling SB RAS)
DTSTART:20210204T110000Z
DTEND:20210204T120000Z
DTSTAMP:20260423T024017Z
UID:mmandim/15
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/mmandim/15/"
 >Regularization of numerical estimation of the sets of solutions of ODEs  
 in stability problems on a finite time interval</a>\nby A.N. Rogalev (Inst
 itute of Computational Modeling SB RAS) as part of Mathematical models and
  integration methods\n\n\nAbstract\nThe sets of ODE solutions\, with initi
 al data belonging to the initial data regions\, have complex boundaries (b
 oundary surfaces in the dimension space). For the boundaries of the sets o
 f solutions (surfaces in the space of solutions)\, it is impossible to cho
 ose formulas of functions with the help of which it was possible to descri
 be the boundaries. As a result\, there are two possibilities — either to
  describe the values of the boundary surfaces in a set of discrete points 
 (on a grid)\, or to calculate their estimates of the maximum values in the
  directions of the coordinate axes\, or the maximum in any chosen directio
 n. The paper investigates and further uses the injectivity property of sol
 utions to ODEs. For linear systems of ODEs  the shift operator is linear a
 nd monomorphic (i.e.\, injective). These properties are also possessed by 
 the resolving operator\, which associates with the initial value the solut
 ion of the corresponding Cauchy problem (the entire solution\, not its val
 ue at a point) as an element of space.\n\nFor nonlinear ODE systems that h
 ave unique solutions in a certain region of initial data\, the boundaries 
 of the regions of initial data pass into the boundaries of the regions of 
 solutions at each specific moment in time. The class of such nonlinear ODE
  systems consists of systems whose solutions are uniformly bounded (Lagran
 ge stable). Preliminarily\, it is useful to construct a regularization of 
 estimates for the boundaries of the solution sets\, passing to the linear 
 approximation of the original system. Regularization is understood as find
 ing information about sets of exact solutions. This regularization establi
 shes  the values of compression / expansion in the given directions\, offs
 et along the time axis\, and rotation through some angle. Examples of stab
 ility studies on a finite time interval are given.\n
LOCATION:https://researchseminars.org/talk/mmandim/15/
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