Abstract
Full Text
MATHEMATICS
L. A. PETROSYAN
ON A FAMILY OF DIFFERENTIAL GAMES OF SURVIVAL IN THE SPACE (R^n)
(Presented by Academician Yu. V. Linnik, 8 X 1964)
A family of antagonistic games is considered, each of which is a model of pursuit in some closed convex set (S \subset R^n). Two points, the pursuer (P) and the evader (E), possessing constant linear velocities (v) and (u) ((v>u)), move in the set (S), while having the possibility at every moment of time to change the direction of their motion. The evader (E) is considered caught as soon as the points (P) and (E) coincide. The aim of (E) is to reach the boundary of the set (S) before being caught by the pursuer. Before giving an exact definition of the game, let us define two sets of vector-functions.
Put (T=[0,\infty]). Let (\mathfrak C) be the set of vector-functions
[
\psi(x_1,\ldots,x_n,t)=[\psi_1(x_1,\ldots,x_n,t),\ldots,\psi_n(x_1,\ldots,x_n,t)],
]
defined on (R^n \times T) with values in (R^n); (\mathfrak P) is the set of vector-functions
[
\varphi[\psi(x_1,\ldots,x_n,t)]={\varphi_1[\psi(x_1,\ldots,x_n,t)],\ldots,\varphi_n[\psi(x_1,\ldots,x_n,t)]},
]
depending on (\psi(x,t)), with values in (R^n) and, for fixed (\psi), defined on (R^n \times T).
The following restrictions are imposed on the elements of the sets (\mathfrak C) and (\mathfrak P):
-
For every (\psi \in \mathfrak C) there exists a sequence (0=t_0<t_1<\cdots<t_m=\infty) such that the space (R^n \times T) can be divided into a finite number of regions (R_1,\ldots,R_{m-1}), each of which is representable in the form of a Cartesian product (R_k=R^n \times [t_k,t_{k+1})), (k=1,2,\ldots,m-1), and on each (R_k) ((k=1,2,\ldots,m-1)) the function (\psi \in \mathfrak C) and the corresponding (\varphi(\psi)\in\mathfrak P) are continuous together with their first partial derivatives.
-
For any (\varphi \in \mathfrak P) and (\psi \in \mathfrak C), the system of equations
[
\begin{aligned}
\dot{x}_i&=\varphi_i[\psi(x_1,\ldots,x_n,t)], \qquad &&i=1,2,\ldots,n;\
\dot{y}_j&=\psi_j(x_1,\ldots,x_n,t), \qquad &&j=1,2,\ldots,n,
\end{aligned}
\tag{1}
]
has a unique solution for any initial conditions (\xi,\eta \in S).
3.
[
\varphi_1^2+\cdots+\varphi_n^2=v^2;
]
[
\psi_1^2+\cdots+\psi_n^2=u^2,
]
where (v=\text{const}), (u=\text{const}), (v>u).
- Let ({x(t),y(t)}) be a solution of system (1) under the initial conditions (\xi,\eta\in S), and let
[
t_{S_E}=\inf{t:y(t)\in \overline S},
]
[
t_{S_P}=\inf{t:x(t)\in \overline S}.
]
Then for (t>t_{S_E}), (y(t)\in \overline S), and for (t>t_{S_P}), (x(t)\in \overline S).
For any (\xi,\eta\in S) we define a differential game of survival, which we agree to denote by (\Gamma(\xi,\eta)). The sets of vector-functions (\mathfrak C)
and (\mathfrak P), satisfying conditions 1–4, are the sets of pure strategies of players (P) and (E) in the game (\Gamma(\xi,\eta)). To each situation ((\varphi,\psi)), under initial conditions (\xi,\eta \in S), there correspond uniquely two trajectories ({x(t),y(t)}), which are the solution of system (1) and are called the trajectories of the pursuer (P) and the pursued (E).
Put
[
t_P=\min{t:x(t)=y(t)}.
]
In any situation the quantity (t_P) is uniquely defined and may be equal to some finite number or to infinity.
The payoff function is defined as follows:
[
K(\xi,\eta,\varphi,\psi)=
\begin{cases}
+1, & t_{S_E}\geq t_P,\
-1, & t_{S_E}<t_P,\
0, & t_{S_E}=\infty,\quad t_P=\infty.
\end{cases}
\tag{2}
]
The game (\Gamma(\xi,\eta)) was first considered by Isaacs ((^1)) under the name “the lifeline game.”
Definition. A strategy (\varphi^{\Pi}\in\mathfrak P) is called a (\Pi)-strategy if, in any situation ((\varphi^{\Pi},\psi)), the point (x(t)) lies on the straight line passing through (y(t)) and parallel to the straight line passing through (\xi) and (\eta), and (t_P<\infty).
Place the origin of coordinates at the point (\eta) and direct the coordinate unit vector (e_n) toward the point (\xi=(\xi_1,\ldots,\xi_n)). Let (\psi) be some strategy of (E). Then, from the definition of a (\Pi)-strategy, it follows that in the situation ((\varphi^{\Pi},\psi)) the trajectory (x(t)) is obtained as the solution of the system of equations
[
\dot x_i=\psi_i,
]
[
\dot x_n=-\left(v^2-\sum_{i=1}^{n-1}\psi_i^2\right)^{1/2},\qquad i=1,2,\ldots,n-1,
]
with initial conditions
[
x_i(0)=0,\quad x_n(0)=\xi_n,\quad i=1,2,\ldots,n-1.
]
Theorem 1. In the game (\Gamma(\xi,\eta)) there exists an equilibrium situation ((\varphi^0,\psi^0)) ((^2)), in which (\varphi^0) is a (\Pi)-strategy, while (\psi^0) is a solution of the functional equation
[
K(\xi,\eta,\varphi^{\Pi},\psi^0)=\min_{\mathfrak E} K(\xi,\eta,\varphi^{\Pi},\psi).
]
If additional restrictions (on angular accelerations) of the form
[
\left(\frac{d\varphi_1}{dt}\right)^2+\cdots+\left(\frac{d\varphi_n}{dt}\right)^2\leq w_P^2,
]
[
\left(\frac{d\psi_1}{dt}\right)^2+\cdots+\left(\frac{d\psi_n}{dt}\right)^2\leq w_E^2,
]
where (w_P) and (w_E) are certain fixed constants, are imposed on the classes of strategies of players (P) and (E), then Theorem 1 remains valid if (w_P\geq w_E).
Denote (z=(\xi,\eta)). Let (V(z)) be the value of the game (\Gamma(z)). At each point (z\in S\times S), (V(z)) may take one of two values, 0 or 1.
Definition. The set
[
\mathfrak B(S)={z:V(z)=1}
]
will be called the winning set of the pursuer (P).
Since the construction of the set (\mathfrak B(S)) completely describes the analysis of the family of games (\Gamma(z)) for (z\in S\times S), we shall undertake the study of this set. It turns out that the value of the games (\Gamma(z)) depends on two quantities: on the set (S) and on the ratio (\gamma=v/u). In accordance with this, below, instead of (\Gamma(z)), (V(z)), (\mathfrak B(S)), we shall write (\Gamma(S,\gamma,z)), (V(S,\gamma,z)), (\mathfrak B(S,\gamma)).
Lemma 1. Suppose the games (\Gamma(S_1,\gamma,z),\ldots,\Gamma(S_n,\gamma,z),\ldots) and the game
[
\Gamma\left(\bigcap_1^\infty S_n,\gamma,z\right).
]
Then
[
\mathfrak{B}\left(\bigcap_1^\infty S_n,\gamma\right)=\bigcap_1^\infty \mathfrak{B}(S_n,\gamma).
]
Lemma 2. The set (\mathfrak{B}(S,\gamma)) is closed.
Let (z^0=(\xi_1^0,\xi_2^0,\eta_1^0,\eta_2^0)\in S\times S); then by the section of the set (\mathfrak{B}(S,\gamma)) at the point (\xi^0=(\xi_1^0,\xi_2^0)) we shall mean the set
[
\mathfrak{B}_{\xi^0}(S,\gamma)={z:z=(\xi_1^0,\xi_2^0,\eta_1,\eta_2)\in\mathfrak{B}(S,\gamma)},
]
and by the section of (\mathfrak{B}(S,\gamma)) at the point (\eta^0=(\eta_1^0,\eta_2^0))—the set (\mathfrak{B}{\eta^0}(S,\gamma)={z:z=(\xi_1,\xi_2,\eta_1^0,\eta_2^0)\in\mathfrak{B}(S,\gamma)}). We shall find the form of the sets (\mathfrak{B}\xi(S,\gamma)) and (\mathfrak{B}_\eta(S,\gamma)) in the case when (S) is a closed half-plane.
(\mathfrak{B}_\xi(S)) is given by the inequality in polar coordinates with origin at the point (\xi)
[
\frac{ruv}{v^2-u^2}+\frac{rv^2}{v^2-u^2}\cos\varphi-b\geq 0,
\tag{3}
]
where (b) is the distance from the point (\xi) to the boundary.
The set (\mathfrak{B}_\eta(S,\gamma)) is given by the inequality with origin at the point (\eta), in polar coordinates,
[
\frac{ruv}{v^2-u^2}+\frac{ru^2}{v^2-u^2}\cos\varphi-c\leq 0,
\tag{4}
]
where (c) is the distance from the point (\eta) to the boundary of the set (S).
Let (S) be an arbitrary convex closed set; then it is representable as a countable intersection of half-planes (S_k), (S=\bigcap_1^\infty S_k).
Hence, by the lemma, (\mathfrak{B}(S,\gamma)=\bigcap_1^\infty \mathfrak{B}(S_k,\gamma)); passing to the section at the point (\xi), we obtain (\mathfrak{B}\xi(S,\gamma)=\bigcap_1^\infty \mathfrak{B}\xi(S_k,\gamma)), and passing to the section at the point (\eta), we obtain
[
\mathfrak{B}\eta(S,\gamma)=\bigcap_1^\infty \mathfrak{B}\eta(S_k,\gamma).
]
Thus, the sets (\mathfrak{B}\xi(S,\gamma)) and (\mathfrak{B}\eta(S,\gamma)) can be approximated by sets of the form (\bigcap_1^N \mathfrak{B}\xi(S_k,\gamma)) and (\bigcap_1^N \mathfrak{B}\eta(S_k,\gamma)). However, these sets are sections of the winning set for an (N)-vertex polygon, which are easily obtained by combining inequalities (3) and (4) for its sides and applying Lemma 1.
The following theorem describes the change of (\mathfrak{B}(S,\gamma)) with the change of (\gamma=v/u) for fixed (S).
Theorem 2. 1) Let (\gamma_k>\gamma,\ k=1,2,\ldots,\ \lim_{k\to\infty}\gamma_k=\gamma); then
[
\mathfrak{B}(S,\gamma)=\bigcap_1^\infty \mathfrak{B}(S,\gamma_k).
]
2) Let (\gamma_k\leq\gamma,\ k=1,2,\ldots,\ \lim_{k\to\infty}\gamma_k=\gamma); then
[
\mathfrak{B}(S,\gamma)=\bigcup_1^\infty \mathfrak{B}(S,\gamma_k).
]
Leningrad State University
named after A. A. Zhdanov
Received
5 X 1964
REFERENCES CITED
- R. Isaaks, Differential Games of Kind, Recent Advances in Game Theory, Princeton, 1961, p. 185–194.
- R. D. Luce, H. Raiffa, Games and Decisions, IL, 1961.