Second International Conference on Game Theory and Management

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The first part of presentation is devoted to constructing the maximal stable
bridges in the three-dimensional (by the target set) linear differential game.
In the second part, the procedure elaborated is applied to building an adaptive
control in problems where there is no *a priori* geometric constraint for the second
player's control.

Maximal stable bridges construction

In the original game, the phase vector can have any dimension *P*, *Q* onto the controls of the first and second
players are segments.

For constructing the maximal stable bridge in the approximating game, we divide
the time axis into a grid {*t _{i}*} with the step Δ to the left from the termination
instant

The construction of the time sections is implemented in a backward procedure:
in the first step, on the basis of the target convex polyhedron, a convex polyhedron
*W*_{1} is built; in the second step, on the basis of
the polyhedron *W*_{1},
a convex polyhedron *W*_{2} is built, and so on.

The construction of each next section *W*_{i+1} of the
bridge includes operations with convex polyhedrons, namely, constructing
the algebraic sum (Minkowski sum) *F*_{i} of the polyhedron
*W*_{i} and the segment
*P _{i}*

For a continuous, positive homogeneous, and piecewise linear function
*γ* : *l* → *γ*(*l*)*S*, we introduce a grid (graph)
*G*(*γ*),*γ*.
At each node of *G*(*γ*),*γ* is written.

When constructing the support function
*ρ* (·, *W*_{i+1})*W*_{i+1}*ρ*(·, *W _{i}*)

The grid
*G*(*ρ*(·, *F _{i}*))
is the result of overlapping the grids

We recompute values of the support function
*ρ*(·, *F _{i}*)

To pass to the support function of the polyhedron *W*_{i+1},
it is necessary to
exclude the nodes of the graph
*G*(*ρ*(·, *F _{i}*)),

To implement the convexification, a special iterative procedure was elaborated.
Collect "suspicious" links into a list.
Initially, we put there that links of the grid
*G*(*ρ*(·, *F _{i}*)),

To elaborate the algorithm for constructing the maximal stable bridge, we used ideas from the cited papers.

The scheme of the algorithm for constructing the stable bridge is presented. It is based on the stepwise procedure for building the time sections. The iterative procedure of convexification is the most expensive in computations of each next section.

In the case *k* = 0,*k* = 1,*u*, the second player governs
the control *v*.

In a two-dimensional game with fixed termination instant, the terminal
payoff function *γ* is taken as
*γ*(*x*) = max{|*x _{1}*|, |

To construct the epigraph of the value function, consider a game of the third
order with the target set *M* as the cut-off
(at a level c^{*}) of the epigraph of the
payoff function for the two-dimensional game.
Then, a *t*-section^{*})
of the epigraph of the
value function for the two-dimensional game at the same instant *t*.

Here, the target set of the three-dimensional game is presented: side view and bottom view.

Here, the *t*-section*c*-axis.

Adaptive control

We consider a problem of constructing an adaptive control in linear
systems with unknown level of the dynamic disturbance.
The aim of useful control is to guide
*n* selected components of the phase vector to a convex bounded
target set at the fixed terminal instant *θ*.
The useful control obeys a geometric
constraint.
The dynamic disturbance (the second player's control) is assumed to
be bounded, but the level of its constraint is unknown *a priori*.

Here, the requirements to the adaptive control are presented.

An ordered family of the stable bridges is considered. Each bridge is defined by a triple: the constraint for the first player's control, the constraint for the second player's control, and the target set.

The main bridge *W*_{main} corresponds to the triple *P*,
*Q*_{max},*M*.
Here, the sets *P* and *M* are given by the problem formulation,
and the set *Q*_{max} is chosen by us and can be treated
as a constraint for the disturbance, which
is assumed to be "reasonable"
in the problem under consideration.
We assume that each of the sets *P*,
*Q*_{max},*M*
contains the origin of its space.

For the case *n* = 2, the algorithms for constructing the adaptive control were
worked out earlier. The picture of
the embedded system of bridges is
presented just for such a case.

In this work, we consider the case of a three-dimensional target set.
Therefore, the *t*-sections

Now, let us present the simulation results for some dynamic system (we do not write it). To construct the adaptive control, we use the principle of extremal aiming well-known in the theory of differential games. The adaptive control is used for the first player; a random disturbance plays the role of the second player's control.

The following sequence of slides shows the behavior of the phase state of the
system with respect to the *t*-sections

In the initial interval of time, the phase point is out of the main bridge
(it is the red colored point).
Therefore, for constructing the adaptive
control, *t*-sections*t*-sections*t*-section*t*-section*t*-section*k* < 1.

At the left low corner of each slide, the corresponding instant is presented. The neighbor slides with the same instant differ from each other by the point of view. In passage from one instant to the next, the point of view does not change.

A.G.Ivanov, A.V.Ushakov

Institute of Mathematics and Mechanics

Ekaterinburg, Russia

iagsoft@imm.uran.ru