UDC 518.731.343.1

MATHEMATICS

Academician N. N. KRASOVSKII

ON A PURSUIT PROBLEM

In the present paper, which is related to the investigations ((^{1-14})), sufficient conditions are discussed under which the approximating extremal strategy (U_a^{(e)}) guarantees to the pursuer a convergence with the pursued at the time (\vartheta), or no later than the time (\vartheta).

Let the pursuing ((y[t])) and the pursued ((z[t])) motions be described by the equations

[
\dot y=f^{(1)}(t,y,u),\qquad \dot z=f^{(2)}(t,z,v),
\tag{1}
]

where (y) and (z) are (n)-dimensional phase vectors of the objects; (u) and (v) are (r)-dimensional control vectors constrained by the condition

[
u\in\mathcal U,\qquad v\in\mathcal V,
\tag{2}
]

the sets (\mathcal U) and (\mathcal V) being bounded and closed, and (f^{(i)}) continuous functions satisfying Lipschitz conditions with respect to (x). We shall call any absolutely continuous function (y[t]) or (z[t]) on the interval ([t_0,\vartheta]) satisfying the contingency ((^{15}))

[
\dot y[t]\in\mathcal F^{(1)}(t,y[t])\quad\text{or}\quad
\dot z[t]\in\mathcal F^{(2)}(t,z[t])
]

for almost all (t\in[t_0,\vartheta]) a motion (y[t]) or (z[t]). Here (\mathcal F^{(i)}(t,q)) denotes the convex hulls of the sets swept out by the vectors (f^{(i)}(t,q,w)), when (w) ranges over (\mathcal U) or (\mathcal V), respectively. Motions satisfying the initial conditions (y[t_]=y_) or (z[t_]=z_) will be denoted by the symbols (y[t,t_,y_]) or (z[t,t_,z_]). A meeting of the motions (y[t]) and (z[t]) is determined by the condition ((z[t]-y[t])\in\mathcal L), where (\mathcal L) is a given closed set. Introduce the (2n)-dimensional vector (x={y,z}). Denote by the letter (\mathcal M) the set in the space ({x}) specified by the condition ((z-y)\in\mathcal L). By the symbol (\rho(x,\mathcal M)) we denote the distance from (x) to (\mathcal M).

We shall say that, for a fixed initial position ({t_0,x_0}={t_0,y_0,z_0}), the approximating strategy ((^{14})) (U_a) guarantees convergence of the motions (y[t]) and (z[t]) at the time (\vartheta>t_0), if

[
\lim_{\delta\to 0}\sup\left[\sup_{z[t],\,y_\delta[t]}\rho\bigl(x[\vartheta],\mathcal M\bigr)\right]=0.
\tag{3}
]

Here (z[t]) is an arbitrary motion (z[t,t_0,z_0]); (y_\delta[t]) is the motion (y_\delta[t,t_0,y_0]) generated by the piecewise-constant control (u_\delta[t]) assigned by the strategy (U_a). If it is possible to construct a system of sets (\mathcal W(t)) ((t_0\le t\le\vartheta)), strongly (u)-stable ((^{14})) and satisfying the conditions (\mathcal W(\vartheta)=\mathcal M,\ x_0\in\mathcal W(t_0)), then the extremal ((^{14})) strategy (U_a^{(e)}) toward them, according to ((^{14})), will guarantee the convergence of (y[t,t_0,y_0]) and (z[t,t_0,z_0]) at the time (\vartheta). (In our case closed sets (\mathcal W(t)), by definition, are strongly (u)-stable if the following condition is fulfilled: whatever (t_\in[t_0,\vartheta)), (x_={y_,z_}\in\mathcal W(t_)), and (\delta\in(0,\vartheta-t_]) may be, for every motion (z(t,t_,z_)) one can choose a motion (y(t,t_,y_)) so that the inclusion (x(t_+\delta,t_,x_)\in\mathcal W(t_+\delta)) holds.)

Let (\mathcal W^(t,\vartheta)) ((t\le\vartheta)) be the set of all (x) for which the process (x(t)) absorbs the programmed set (\mathcal M) at the time (\vartheta) from the position ({t,x}) ((^{14})). As the sets (\mathcal W(t)) it is convenient to choose the sets (\mathcal W^(t,\vartheta)) provided they are strongly (u)-stable. We shall therefore indicate some sufficient conditions for strong (u)-stability of the sets (\mathcal W^*(t,\vartheta)).

Let (x_={y_,z_}\in \mathfrak W^(t_,\vartheta)). Choose (t^>t_) ((t^<\vartheta)) and define (\mathcal Y(x_)) as the totality of all points (y(t^,t_,y_)) which are obtained by ranging over all possible motions (y(t,t_,y_)). The set (\mathcal Y(x_)) is bounded and closed. Fix some motion (z^0(t,t_,z_)) ((t_\le t\le t^)). Denote (\hat z^0=z^0(t^,t_,z_)). Let the point (x^={y^,\hat z^0}\notin \mathfrak W^(t^,\vartheta)). Then one can single out a certain set (\mathcal Z_{[t_,\vartheta]}(x^)) of motions (z(t,t_,z_)) which for (t_\le t\le t^) coincide with the motion (z^0(t,t_,z_)) and at the same time have the property that for each of them it is impossible to choose a motion (y(t,t^,y^)) effecting an encounter ((z(\vartheta)-y(\vartheta))\in\mathcal L). Further, for each (z(t,t_,z_)\in\mathcal Z_{[t_,\vartheta]}(x^)) one can choose some set (\mathcal Y_{[t_,\vartheta]}(z(t))) of motions (y(t,t_,y_)) effecting an encounter ((z(\vartheta)-y(\vartheta))\in\mathcal L). The totality of all motions (y(t,t_,y_)\in\mathcal Y_{[t_,\vartheta]}(z(t))), corresponding to all (z(t)\in\mathcal Z_{[t_,\vartheta]}(x^)), determines a certain set of points (y=y(t^,t_,y_)). We denote this set by the symbol (\mathcal Y(x^)). Obviously, (\mathcal Y(x^)\subset \mathcal Y(x_)).

Lemma 1. If, for all sufficiently small (\delta=t^-t_>0), for all possible points (x_) the sets (\mathcal Y(x_)) are convex, and if all sets (\mathcal Y(x^)) can be chosen closed, convex, and upper semicontinuous with respect to inclusion under variation of (x^), then the sets (\mathfrak W^(t,\vartheta)) are strongly (u)-stable.*

Suppose, to the contrary, that the lemma is false. Then for some point (x_={y_,z_}\in\mathfrak W(t_,\vartheta)) one can fix a motion (z^0(t,t_,z_)) such that all points (x^={y,z^0(t^,t_,z_)}) with (y\in\mathcal Y(x_)) will lie outside (\mathfrak W^(t^,\vartheta)). But then, in accordance with the preceding, one can construct a mapping of the points (y^\in\mathcal Y(x_)) to the sets (\mathcal Y(x^)\subset\mathcal Y(x_)). According to theorem ((^{16})), in this case there is found a point (y^0) satisfying the condition (y^0\in\mathcal Y(x^0)), where (x^0={y^0,z^0}). But by the construction of the sets (\mathcal Y(x^)) this is impossible, since it follows that at (t=t^) there passes through the point (x^0) a motion (x(t,t_,x_*)) which both arrives at (\mathcal M) at the instant (\vartheta), and at the same time in no way can arrive at (\mathcal M) at this instant. The contradiction proves the lemma.

The choice of the sets (\mathcal Y(x^)) which appear in the lemma is stipulated with a certain degree of arbitrariness. Making Lemma 1 more concrete, one can choose as the set (\mathcal Z_{[t_,\vartheta]}(x^)) the totality of all motions (z(t,t^,z^0)) solving the problem for the program maximin
[
\max_{z(t,t^,z^0)}\min_{y(t,t_,y^)}\rho(x(\vartheta),\mathcal M),
]
and as (\mathcal Y_{[t_,\vartheta]}(z(t))) choose all motions (y(t,t_,y_)) effecting an encounter ((z(\vartheta)-y(\vartheta))\in\mathcal L). Then the sets (\mathcal Y(x^)) will necessarily be closed and upper semicontinuous with respect to inclusion (in (x^)). In this case the conditions of Lemma 1 reduce only to the requirement of convexity of (\mathcal Y(x_)) and (\mathcal Y(x^)). Finally, in the case where system (1) is linear and the sets (\mathcal U,\mathcal V), and (\mathcal L) are convex, the conditions of Lemma 1 are replaced by a single condition of convexity of the set ({v(t)}) of all those controls (v(t)) ((t^\le t\le\vartheta)) which solve the problem for the program maximin
[
\max_v\min_u\rho(x(\vartheta,t^,x^),\mathcal M).
]
This condition is obviously fulfilled if the condition of uniqueness of the vector (l^0), indicated in papers ((^{13,14})), is fulfilled. Thus, the conditions of Lemma 1 in this particular case reduce to the already known conditions of strong (u)-stability of the sets (\mathfrak W^(t,\vartheta)).

We shall say that, for a fixed initial position ({t_0,x_0}={t_0,y_0,z_0}), an approximating strategy (U_a) guarantees the approach of the motions (y[t]) and (z[t]) by the instant (\vartheta>t_0), if
[
\lim_{\delta\to0}\sup\left[
\sup_{z[t],\,y_\delta[t]\, t_0\le t\le\vartheta}
\left(\inf \rho(x[t],\mathcal M)\right)\right]=0.
\tag{4}
]

If it is possible to construct a system of sets (\mathfrak W(t)) ((t_0\le t\le\vartheta)) satisfying the conditions (\mathfrak W(\vartheta)=\mathcal M), (\mathcal M\subset\mathfrak W(t)), (x_0\in\mathfrak W(t_0)) and (u)-stable ((^{14})), then, according to ((^{14})), the extremal strategy (U_\varepsilon^{(e)}) for them will guarantee the approach of (y[t,t_0,y_0]) and (z[t,t_0,z_0]) by the instant (\vartheta). Let (\mathfrak W^0(t,\vartheta)) ((t\le\vartheta)) be the set of all (x={y,z}) for which the process (x(t)) falls-

gives the program set (\mathcal M) at the moment (\vartheta) from the position ({t,x}) ((^{14})). As the sets (\mathcal W(t)) it is convenient to choose the sets (\mathcal W^0(t,\vartheta)) ((t_0\le t\le \vartheta)), under the condition of their (u)-stability. We indicate some sufficient conditions for the (u)-stability of the sets (\mathcal W^0(t,\vartheta)). For this purpose one should repeat the construction of the sets (\mathcal U(x_)) and (\mathcal U(x^)) described above, but here with the sole difference that the sets (\mathcal W^) are replaced by (\mathcal W^0), and the meeting condition ((z(\vartheta)-y(\vartheta))\in\mathcal L) is replaced by the condition (\min_t\rho(x(t),\mathcal M)=0) (for (t^\le t\le\vartheta) in the first case and for (t_*\le t\le\vartheta) in the second). Analogously to Lemma 1, the following assertion is now proved.

Lemma 2. If, for all sufficiently small (\delta=t^-t_>0), for all possible points (x_) the sets (\mathcal U(x_)) are convex, and if all the sets (\mathcal U(x^)) can be chosen closed, convex, and upper semicontinuous with respect to inclusion (in (x^)), then the sets (\mathcal W^0(t,\vartheta)) are (u)-stable.

Making Lemma 2 more concrete, one may take as the set (\mathcal Z_{[t_,\vartheta]}(x^)) the collection of all motions (z(t,t^,z^0)) that solve the problem of the program maximum
[
\max_{z(t,t^,z^0)}\min_{y(t,t^,y^)}\min_t\rho(x(t),\mathcal M)
]
((t^\le t\le \vartheta)), and as (\mathcal U_{[t_,\vartheta]}(z(t))) take all motions (y(t,t_,y_)) that realize a meeting ((z(t)-y(t))\in\mathcal L) for (t_\le t\le\vartheta). Then the conditions of Lemma 2 reduce only to the requirement of convexity of (\mathcal U(x_)) and (\mathcal U(x^*)).

A consequence of the results from paper ((^{14})) and Lemmas 1 and 2 is the following assertion:

Theorem. Suppose that, for all sufficiently small (\delta=t^-t_>0), for all possible points (x_) the sets (\mathcal U(x_)) are convex. If, moreover, for the sets (\mathcal U(x^)) constructed on the basis of (\mathcal W^(t,\vartheta)) or (\mathcal W^0(t,\vartheta)) ((t_0\le t\le\vartheta)) can be chosen closed, convex, and upper semicontinuous with respect to inclusion (in (x^)), then, under the condition (x_0\in\mathcal W^(t_0,\vartheta)) or (x_0\in\mathcal W^0(t_0,\vartheta)), the strategy (U_a^{(\Theta)}), extremal to the sets (\mathcal W^(t,\vartheta)) or (\mathcal W^0(t,\vartheta)), guarantees the convergence of the motions (y[t]) and (z[t]) at the moment (\vartheta) or by the moment (\vartheta), respectively.*

Sverdlovsk Branch
of the V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
3 XII 1969

CITED LITERATURE

(^{1}) W. H. Fleming, Contrib. to the Theory of Games, 3, Princeton, 1957, p. 407.
(^{2}) D. L. Kelendzheridze, Avtomatika i telemekh., 23, No. 8, 1008 (1962).
(^{3}) L. S. Pontryagin, PMM, 26, No. 5, 960 (1962).
(^{4}) C. R. Wazewski, Adv. in Game Theory, Ann. Math. Studies. 113 (1964).
(^{5}) R. Isaacs, Differential Games, Moscow, 1967.
(^{6}) E. F. Mishchenko, L. S. Pontryagin, DAN, 174, No. 1, 27 (1967).
(^{7}) L. S. Pontryagin, DAN, 175, No. 4, 764 (1967).
(^{8}) B. N. Pshenichnyi, Avtomatika i telemekh., No. 1, 65 (1968).
(^{9}) B. N. Pshenichnyi, DAN, 184, No. 2, 285 (1969).
(^{10}) M. S. Nikol’skii, Vestn. Mosc. Univ., ser. matem. i mekh., No. 3, 65 (1969).
(^{11}) P. Varaiya, SIAM J. Control, 5, No. 4, 153 (1967).
(^{12}) N. N. Petrov, DAN, 190, No. 6 (1970).
(^{13}) N. N. Krasovskii, A. I. Subbotin, PMM, 33, No. 4, 698 (1969).
(^{14}) N. N. Krasovskii, A. I. Subbotin, DAN, 190, No. 3 (1970).
(^{15}) A. F. Filippov, Vestn. Mosc. Univ., ser. matem. i mekh., No. 2, 25 (1959).
(^{16}) S. Kakutani, Duke Math. J., 8, 457 (1941).