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This talk was presented at the conference
**"Viscosity Solutions and Applications" &
"Analysis and Control of Deterministic and Stochastic
Evolution Equation", Bressanone-Brixen, Italy,
July 3-7, 2000.**
The paper is devoted to the analytical and numerical
study of a time-optimal game problem in the plane. This problem
is a game extension of the model brachistochrone problem.

Let

When considering the brachistochrone problem in the book "Differential games", Isaacs introduced a disturbance influencing the dynamics of the system. The disturbance can be interpreted as a second player whose objective is to increase the time of attaining the terminal set. Isaacs considered the first quadrant as the state space. The terminal set was the positive semiaxis

Lidov pointed out to an error in the Isaacs' solution. Later on, Chigir has improved the solution of Isaacs. The horizontal line

Let the vectograms of the players are of the following form: a circle of radius

Using the Isaacs' method, we obtain three families of characteristics that are constructed taking into account the boundary condition on the terminal set. The characteristics of the first, second, and third family emanate from the vertical part of the boundary, from the right upper vertex of

The singular dispersal line

It was the description of the solution structure for

For this slide,

If

If

If

If

These are results of our study based on the construction of the
singular line. The solution is not trivial. We have a code for
the computation of the singular line. The full exact proof for the
described solution structure is not completed yet. Nevertheless,
we are sure that our results are correct because they are in
agreement with the independent computation of level sets of the
value function. This independent computation utilizes an
algorithm based on the backward propagation of fronts.

Here, level sets of the value function are presented for

Here, a fragment of the previous picture is shown. More fronts for the same time interval are presented.

On this slide, a fragment of the collection of level sets of the value function is shown for

If

This picture is done for

If

The computation is done for

L.V. Kamneva, V.S. Patsko

Institute of Mathematics & Mechanics Ural Branch of RAS

S.Kovalevskaya str.16

620219 Ekaterinburg, Russia

e-mail:

kamneva@imm.uran.ru

patsko@imm.uran.ru

V.L.Turova

Center of Advanced European Studies and Research

Friedensplatz 16

53111 Bonn, Germany

e-mail:

turova@caesar.de