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\section*{Problem Statement\footnote{ Arkadii A. Chikrii, ``Soft Landing of Moving
Objects'', Cybernetics Institute, Ukraine, Kyiv-1998}}
\bigskip
Let a motion of the pursuer evolve in three-dimensional Euclidean space
$R^{3}$ and its dynamics be subject to the equation
\begin{equation}
\label{eq1}
\ddot {x} + \alpha \dot {x} = \rho u,\;{\left\| {u} \right\|} \le 1
\end{equation}
\noindent
where $x = (x_{1} ,x_{2} ,x_{3} )$ are geometric coordinates of the object.
Here $x_{1} $, $x_{2} $ denote coordinates in the horizontal plane and
$x_{3} $ a height. Vectors $\dot {x} = {\textstyle{{dx} \over {dt}}}$ and
$\ddot {x} = {\textstyle{{d^{2}x} \over {dt^{2}}}}$ are velocity and
acceleration, respectively; $\alpha $ - friction coefficient; $\rho > 0$ -
resource coefficient; $u$ - control, which is chosen in a unit ball centered
at the origin of $R^{3}$; ${\left\| {x} \right\|} = \sqrt {\left( {x,x}
\right)} $, where by $\left( { \cdot , \cdot} \right)$ is denoted a scalar
product of vectors.
It is assumed that control $u(t)$, $t \ge 0$, is Lebesgue measurable
function of time. For simplicity'sake and in view of possible practical
applications, it may be assumed that function $u(t)$ is piecewise-continuous
or even piecewise-constant.
The evader moves in the horizontal plane and his motion is described by the
equation
\begin{equation}
\label{eq2}
\ddot {y} + \beta \dot {y} = \sigma \vartheta ,\;{\left\| {\upsilon}
\right\|} \le 1
\end{equation}
\noindent
where $y = (y_{1} ,y_{2} )$ are coordinates of the object. Vectors $\dot
{y}$ and $\ddot {y}$ denote velocity and acceleration of the evader at point
$y$, $\beta > 0$ - coefficient of friction, $\sigma > 0$ - coefficient of
resources, and $\vartheta $ - control of the evader, taking its values in a
flat circle centered at the origin. In the sequel we shall sometimes write
$\tilde {y} = (y_{1} ,y_{2} ,0)$ or even $\tilde {y} = (y,0)$ in order to
treat $y$ as vector in $R^{3}$.
The game (\ref{eq1}), (\ref{eq2}) will be analyzed from the pursuer's point of view. His
goal is to achieve ``soft meeting'' with the evader at a finite instant of
time:
\begin{equation}
\label{eq3}
{\left\| {x - \tilde {y}} \right\|} \le \varepsilon _{1} ,\;{\left\| {\dot
{x} - \dot {\tilde {y}}} \right\|} \le \varepsilon _{2}
\end{equation}
\noindent
where $\varepsilon _{1} $, $\varepsilon _{2} $ are positive numbers,
specifying the required proximity of the players. The hyperplane $\{y_{3} =
0\}$ stands for state constraint of the pursuer. The pursuer is allowed to
move in this hyperplane, not intersecting it.
Without loss of generality, the initial state of the pursuer is assumed to
lie in the upper halfspace, that is $x_{3}^{0} = x_{3} (0) > 0$.
To simplify the treatment, we set $\varepsilon _{1} = \varepsilon _{2} = 0$,
that is we shall study the precise ``soft landing''. Note, that it is easy
to pass from this problem to the problem (\ref{eq1})-(\ref{eq3}) and the solution of problem
(\ref{eq1})-(\ref{eq3}) immediately follows from the solution of problem on precise ``soft
landing''.
For the sake of convenience, let us reduce the second order system (\ref{eq1}), (\ref{eq2})
to a system of first order but yet of larger dimension with the help of
introduction of new variables
\[
z_{1} = x,\;z_{2} = \dot {x},\;z_{3} = \tilde {y},\;z_{4} = \dot {\tilde
{y}}
\]
Differentiating the above equalities in time and taking into account the
equations (\ref{eq1}), (\ref{eq2}) we obtain an equivalent system
\begin{equation}
\label{eq4}
{\begin{array}{l}
\dot {z}_{1} = z_{2} \\
\dot {z}_{2} = - \alpha z_{2} + \rho u \\
\dot {z}_{3} = z_{4} \\
\dot {z}_{4} = - \beta z_{4} + \sigma \vartheta \\
\end{array}}
\end{equation}
In the strict sense (\ref{eq4}) is a system of 12$^{th}$ order, but, in fact, only
of 10$^{th}$, since although vectors $z_{1} $, $z_{2} $, $z_{3} $, $z_{4} $
are three-dimensional, $z_{3} $ and $z_{4} $ have zero third components.
Thus, the pursuer strives to bring a trajectory of system (\ref{eq4}) to a linear
subspace
\begin{equation}
\label{eq5}
M_{0} :\;z_{1} = z_{3} ,\;z_{2} = z_{4}
\end{equation}
\noindent
or to a certain its neighbourhood for any admissible counteraction of the
evader. In order to formulate the problem (\ref{eq4}), (\ref{eq5}) in a more general form
and to develop a general approach for solution of the linear game problem we
shall present the motion of a conflict-controlled process in the form
\begin{equation}
\label{eq6}
\dot {z} = Az + u - \vartheta ,\,u \in U,\,\vartheta \in V,\,z \in R^{n}
\end{equation}
\noindent
where $A$ is a square matrix of order $n$, $U,\,V$ are nonempty compacts,
and the terminal set is a cylindrical set
\begin{equation}
\label{eq7}
M^{\ast} = M_{0} + M
\end{equation}
Here $M_{0} $ is a linear subspace of $R^{n}$ and $M$ is a compact from the
orthogonal complement $L$ to $M_{0} $ in $R^{n}$. By $\pi $ is usually
denoted the operator of orthogonal projection from $R^{n}$ to $L$.
One can easily see that the problem of ``soft landing'', formulated in the
form (\ref{eq1})-(\ref{eq3}) or (\ref{eq4}), (\ref{eq5}), is a specific case of the differential game (\ref{eq6}),
(\ref{eq7}). Pontryagin's condition for the problem (\ref{eq6}), (\ref{eq7}) means the nonempty of
set-valued mapping
\begin{equation}
\label{eq8}
W\left( {t} \right) = \pi e^{tA}U{\mathop { -} \limits^{\ast} } \pi e^{tA}V
\ne \emptyset \;\forall t \ge 0
\end{equation}
The availability of information on a current state of the game to the
pursuer will be specified separately for each particular method, presented
in the paper. Denote states of the players (\ref{eq1}), (\ref{eq2}) by
\[
\bar {x} = (x,\dot {x}) = (z_{1} ,z_{2} ),
\quad
\bar {y} = (\tilde {y},\dot {\tilde {y}}) = (z_{3} ,z_{4} )
\]
\end{document}