UDC 517.92

MATHEMATICS

Academician L. S. PONTRYAGIN

A LINEAR DIFFERENTIAL GAME OF EVASION

Here a strengthening of the results of paper (1) is given.

A linear differential game is considered

[
\dot z = Cz - u + v + a .
\tag{1}
]

Here (z) is the phase vector of the game, belonging to a given vector Euclidean space (R) of finite dimension; (C) is a given linear mapping of the space (R) into itself; (a) is a given constant vector from (R); (u) is the control of pursuit; (v) is the control of evasion; (u) and (v) are vectors from (R), but they are not arbitrary, and satisfy the conditions: (u \in P), (v \in Q), where (P) and (Q) are given compact convex subsets of the space (R). The game is considered finished when (z) reaches a given vector subspace (M) of the space (R).

The aim of the game is to prevent its termination; for this purpose, at each instant of time (t) we choose the value (v(t)) of the control (v), using the functions (z(s)) and (u(s)), known to us on the interval (0 \leq s \leq t). Such are the rules of the game.

Let (L) denote the orthogonal complement in (R) to (M); let (\nu) denote the dimension of (L), and let (\pi) denote the operation of orthogonal projection from (R) onto (L). Let (A) and (B) be two subsets of the space (L). We shall write (A \overset{*}{\subset} B) if there exists a vector (x \in L) such that (x + A \subset B).

Evasion theorem. If (\nu \geq 2) and there exists a real number (\mu > 1) such that the relations

[
\dim \pi e^{\tau C} Q = \nu; \qquad
\mu \pi e^{\tau C} P \overset{*}{\subset} \pi e^{\tau C} Q
\tag{2}
]

hold for all sufficiently small real positive values of the parameter (\tau), then, acting according to the rules of the game, we can prevent its termination throughout the whole time (0 \leq t < \infty), if, of course, the initial state (z_0) does not belong to (M). Moreover, we can conduct the game in such a way that for the distance of the point (z(t)) to (M) the estimate (3) holds.

To write the estimate, denote by (\xi) the distance of the point (z) to (M), and by (\eta) its distance to (L). Then the estimate holds

[
\xi(t) > \frac{c \xi_0^{k}}{[1+\eta(t)]^{m}},
\tag{3}
]

provided only that (\xi_0 \leq \varepsilon). Here (c,\varepsilon) are positive constants, and (k) and (m) are natural numbers, depending only on the game, but not on its course.

Let us give a more detailed description of the process of evasion. A parallel translation of either of the sets (P) and (Q) in the space (R) can be compensated by a change of the vector (a); using this, we can arrange that the sets (P) and (Q) belong respectively to vector subspaces (U) and (V) of the space (R), with (\dim P = \dim U), (\dim Q = \dim V); in addition, for simplicity, let us suppose that (\dim V = \nu).

Further, we may assume that, instead of (2), the ordinary inclusion holds

[
\mu \pi e^{\tau C} P \subset \pi e^{\tau C} Q .
\tag{4}
]

Define the linear mappings (f_\tau) and (g_\tau), respectively, of the spaces (U) and (V) by the formulas

[
f_\tau=\pi e^{\tau C};\qquad g_\tau=\pi e^{\tau C}.
\tag{5}
]

It turns out that the mapping

[
h_\tau=g_\tau^{-1}f_\tau
\tag{6}
]

is an analytic function of the parameter (\tau) for all small values of (\tau), although this is not true for the mapping (g_\tau^{-1}), so that a linear mapping (h_0) of the space (U) into the space (V) is defined. It is also possible to achieve, by a parallel shift of the set (Q), that there exists a sufficiently small positive number (\delta) such that if a vector (w\in V) satisfies the inequality (|w|\leq\delta), then the vector (v), defined by (7), belongs to (Q):

[
v=h_0(u)+w,\qquad \text{where } u\in P.
\tag{7}
]

To each point (z_0\in R), for which (\xi_0\leq 1), there is assigned a time interval of length

[
\theta=\theta_0/(1+\eta_0),
\tag{8}
]

where (\theta_0>0) is a constant depending on the game. In addition, to the same point (z_0) there is assigned a value of the vector (w=w(z_0)), satisfying the condition (|w(z_0)|\leq\delta), such that the control (v(t)), defined from the control (u(t)) on the interval (0\leq t\leq\theta) by formula (7), i.e., by the relation

[
v(t)=h_0(u(t))+w(z_0),
\tag{9}
]

together with the control (u(t)) gives a motion (z(t)), (0\leq t\leq\theta), (z(0)=z_0), for which the inequalities

[
\xi(\theta)>\varepsilon/[1+\eta(\theta)]^k,\qquad \varepsilon\leq 1;
\tag{10}
]

[
\xi(t)>c'\xi_0^k/[1+\eta(t)]^{2k-1},
\tag{11}
]

are satisfied, where (\varepsilon) and (c') are positive constants depending on the game.

Inequality (10) provides a basis for considering, in the space (R), the hypersurface (S) defined by the equation

[
\xi=\varepsilon/[1+\eta]^k.
\tag{12}
]

The hypersurface (S) divides the space (R) into two regions: the inner region (S_-), containing (M), and the outer region (S_+).

If during some part of the game the point (z(t)) is outside the surface (S), then we do not concern ourselves with the choice of the control (v), and only at the moment of time (t_0), when the point (z(t_0)\in S), do we switch on, for the time (\theta) (see (8)), the special evasion control (v(t)) specified by formula (9), taking (z_0=z(t_0)). At the end of this time interval the point is again outside the surface (S) (see (10)), and the process repeats again. On the interval of time (t_0\leq t\leq t_0+\theta) we have the inequality

[
\xi(t)>c\varepsilon^k/[1+\eta(t)]^{k^2+2k-1},
\tag{13}
]

which is easily derived from inequality (11). In inequality (13) there is a constant (c), depending on the game, with (0<c<c'). If at the very beginning of the game the point (z_0=z(0)) lies inside or on the surface (S), then the evasion control (9) is switched on immediately on the time interval (0\leq t\leq\theta) (see (8)), and then on this interval inequality (11) holds, while at its end the point (z(\theta)) is already outside the surface (S) (see (10)).

Thus, throughout the whole game, with the possible exception of the first time interval (0\leq t\leq\theta), the point (z(t)) either lies outside the surface (S), or satisfies inequality (13). On the first time interval it may satisfy inequality (11). Coarsening the inequalities-

(11) and (13), as well as the condition that the point (z(t)) remains outside the surface (S), we obtain estimate (3).

To illustrate the result, let us consider a pursuit process in a Euclidean vector space (E) of dimension (\nu \geqslant 2), in which there is a pursuing point (x) and an evading point (y). The motions of these points are given by the equations

[
\ddot{x}+\alpha \dot{x}=u,\qquad |u|\leqslant \rho;\qquad
\ddot{y}+\beta \dot{y}=v,\qquad |v|\leqslant \sigma,
\tag{14}
]

where (\alpha,\rho,\beta,\sigma) are positive numbers, and (u,v\in E) are control vectors; the pursuit process ends when (x=y). Elementary computations show that if (\sigma>\rho), then the conditions of the evasion theorem are satisfied, and the point (y) can at all times move away from the point (x). In the case where the opposite inequality (\rho>\sigma) holds, it follows from the results of note (2) that in the space of initial states ((x_0,\dot{x}_0,y_0,\dot{y}_0)) there is an open set (\Omega) such that, if the initial state ((x_0,\dot{x}_0,y_0,\dot{y}_0)\in\Omega), then pursuit always terminates.

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR
Moscow

Received
18 XII 1969

CITED LITERATURE

(^1) L. S. Pontryagin, E. F. Mishchenko, DAN, 189, No. 4 (1969).
(^2) L. S. Pontryagin, DAN, 174, No. 6 (1967).