Positional Control in a Pursuit Game with Two Weak Pursuers and One Evader

A model game with two pursuers and one evader is studied. Three inertial objects move in a straight line. The dynamics of the pursuers $P_1$, $P_2$ and the evader $E$ are the following:
\begin{equation} \label{patsko1} \begin{array}{lclcl} \ddot{z}_{P_1} = a_{P_1}, & \qquad & \ddot{z}_{P_2} = a_{P_2}, & \qquad & \ddot{z}_E = a_E, \\[1ex] \dot{a}_{P_1} = (u_1 - a_{P_1})/l_{P_1}, & & \dot{a}_{P_2} = (u_2 - a_{P_2})/l_{P_2}, & & \dot{a}_E = (v - a_E)/l_E,\\[1ex] |u_1| \leqslant \mu_1, & & |u_2| \leqslant \mu_2, & & |v| \leqslant \nu, \\[1ex] a_{P_1}(t_0) = 0, & & a_{P_2}(t_0) = 0, & & a_E(t_0) = 0. \end{array} \end{equation}
Here, $z_{P_1}$, $z_{P_2}$, $z_E$ are the geometric coordinates of the objects; $a_{P_1}$, $a_{P_2}$, $a_E$ are their accelerations generated by the controls $u_1$, $u_2$, $v$. The time constants $l_{P_1}$, $l_{P_2}$, $l_E$ define how fast the controls affect the system. It is assumed that $\mu_i < \nu$, $\mu_i/l_{P_i} < \nu/l_E$, $i = 1$, $2$. Such a set of parameters corresponds to the case of weak pursuers.
Fix a time instant $T$. At this instant, the deviations of the evader from the first and second pursuers are measured: $d_{P_1, E}(T) = z_{E}(T) - z_{P_1}(T)$, $d_{P_2, E}(T) = z_{E}(T) - z_{P_2}(T)$.
Suppose that the pursuers act together. Join them into one player $P$ which is called the first player. It has a vector control $u = (u_1, u_2)$. The evader is regarded as the second player. The payoff function is defined as follows:
\begin{equation} \label{patsko4} \varphi = \min\Bigl\{\bigl|d_{P_1,E}(T)\bigr|, \ \bigl|d_{P_2,E}(T)\bigr|\Bigr\}. \end{equation}

At any instant $t$ both players know exactly all phase coordinates $z_{P_1}$, $\dot{z}_{P_1}$, $a_{P_1}$, $z_{P_2}$, $\dot{z}_{P_2}$, $a_{P_2}$, $z_{E}$, $\dot{z}_{E}$, $a_{E}$. The first player chooses its control to minimize the payoff; the second player maximizes it.
The talk gives results of numerical construction of the level sets of the value function (the maximal stable bridges) for the considered game in the three-dimensional space $t$, $x_1$, $x_2$. Here, $x_1$ ($x_2$) is the deviation of the evader from the first (second) player forecasted from the current time instant to the termination instant $T$ under zero players' controls.
The main part of the talk deals with construction and justification of the first player control in game by means of switching lines that depend on the time. The method provides a result for the first player, which is close to thte optimal one and stable with respect to small errors of numerical constructions and inaccuracies of measurements of the current phase state.