AN ANALOGUE OF THE π-STRATEGY IN PURSUIT AND EVASION DIffERENTIAL GAMES WITH MANY PURSUERS ON A SURFACE Текст научной статьи по специальности «Математика»

An Analogue of the ^-strategy in Pursuit and Evasion Differential Games with many Pursuers on a Surface

Atamurat Sh. Kuchkarov1,2 and Gafurjan I. Ibragimov2

1 Institute of Mathematics and Information Technologies,

Akademi of Sciences Republic of Uzbekistan,

100125, Durman yuli, 29, Tashkent, Uzbekistan E-mail: [email protected] 2 INSPEM and Dept. of Math., FS, UPM,

43400, Serdang, Selangor, Malaysia E-mail: [email protected]

Abstract In the present work we study the simple motion differential games of several objects on the surface with positive curvature. Using Jacobi field (Gromoll, Klingenberg, Meyer, 1968; Thorpe, 1979) it is constructed analogue of the strategy of parallel approach (Petrosjan, 1965), characterized with the property that straight lines through positions of the pursuer and the evader remains close to each other during the game. This strategy is used to solve a pursuit problem with many pursuers, in which the maximal speeds of all players are equal (similar pursuit game without phase constraints was studied by B.N.Pshenichniy, 1976). Moreover, it is proved that if the evader has advantage over pursuers in speed, then the evader can avoid contact with pursuers (in case players move on the plane the game was examined by F.L.Chernous’ko, 1976, and an evasion strategy in direction was constructed by him).

Keywords: differential games, pursuit, evasion, strategy, geodesic, Jacobi field, curvature.

1. Introduction

Phase constraints often arise in the real conflict control systems. The cases, when the object moves in a bounded set or on a surface in Rn (or along a curve), are examples of such constraints.

Different aspects of simple pursuit-evasion games under state constraints were studied in many works (see the reference). Particularly, in (Ivanov, 1980) a pursuit-evasion problem with many pursuers was solved on a compact set. In (Kuchkarov and Rikhsiev, 2001) a pursuit problem was solved when the evader moves on a strictly convex surface and the pursuer does along the space, with dynamical possibilities of the players being equal. In the paper of A.A.Azamov and A.Sh.Kuchkarov (2009), published in the second volume of the current series, a pursuit-evasion differential game and its different generalizations were considered when the evader moves along a given curve (generalization of Rado’s game ”Lion and Man”). In the paper (Ibragimov, 2002) a game problem was studied on a closed convex set with integral constrains on controls of players.

In simple pursuit differential games, that is games described by equations

x = u, y = v; x, y, u, v € Rn, |u| < p, |v| < a

strategy of parallel approach (called ^-strategy and characterized with the property that straight lines through positions of the pursuer and the evader are parallel),

which is one of the effective strategies, was applied by L.A.Petrosjan (1965, 1977) and others (see, for example, Pshenichnii, 1976; Azamov and Samatov, 2000) to solve different pursuit differential game problems.

If the pursuer is subject to state constraint, then ^-strategy, in general, may not be applicable. In the present paper we consider pursuit and evasion differential games when players move on a convex hypersurface.

When the evader doesn’t have an advantage in maximal speed, employing the Jacobi field we construct analogue of the ^-strategy. In particular, if the hypersurface is a subspace, then constructed strategy coincides with the ^-strategy. In realization of this strategy, geodesics passing through the positions of the pursuer and the evader change to a ”close” geodesic.

In the case where players have the same dynamical possibilities (analogue of Pshenichnii’s work, 1976) it is proved that in the differential game with many pursuers pursuit can be completed after several maneuvers when pursuers use this strategy.

In case when the evader has an advantage in maximal speed, an evasion strategy was constructed to avoid from many pursuers (analogue of Chernous’ko’s work, 1976).

2. Statement of the Problem

Let M be an n-dimensional surface (hypersurface) in Rn+1 (n > 2) from the class of smoothness C2; Mz be the tangent hyperplane to M at the point z € M; N be the field of unit normal vectors of M, i.e. vector N(z) is orthogonal to Mz at each point z € M.

Recall that the geodesic connecting the points x and y is an absolutely continuous curve y : [0; t] ^ M, t > 0, x = y(0), y = y(t) that has the shortest length among the curves on M connecting x and y. If there is a geodesic Y : [0; t] ^ M, t > 0, connecting the points x and y, then its length is called distance between points x and y denoted d(x,y). In the sequel, we assume that all differentiable curves y : [a, 3] ^ M have unit velocities Iy'(t)| = 1, t € (a, (3), i.e. all differentiable curves are considered to be given by natural parametrization.

The following assumption will be needed throughout the paper: for any point z € M and vector w € Mz, |w| = 1 there exists a geodesic y : [0, ro) ^ M such that y(0) = z, Y(0) = w.

Motions of the group of pursuers P = {P1, P2, ..., Pm} and the evader E are described by the equations

Pi : xi = ui, xj(0) = xi0; E : y = v, y(0) = y0, y0 = xo, (1)

in M, where xi, y € M, ui € MXi, v € My; ui, v are control parameters.

Definition 1. The Borel measurable function u(t) = (u1(t), u2(t),..., um(t)), |ui(t)| < pi, t > 0, (respectively v(t), |v(t)| < a, t > 0) is called a control of the group of pursuers P (the evader E) if for the solution x1 (t), ..., xm(t) ( respectively y( t)) of the initial value problem

xi = ui(t), xi(0) = xio, i =1, 2, ..., m, (y = v(t), y(0) = yo)

the inclusions

ui(t) € MXi(t) (v(t) € My(t))

(2)

hold almost everywhere on t, t > 0, where pi > 0, a > 0.

The set of all controls of the group of pursuers (of the evader respectively) denoted by Up (Ve).

Definition 2. A function U = (U1, U2, ...,Um), where

Ui(x1, x2,...,xm, y, v) ^ Bpi(xi), (x1, x2,..., xm, y, v) € Mm+1 x Ba(y),

is called a strategy of the group of pursuers P if

1) for an arbitrary control v(-) € Ve of the evader the initial value problem

{y = v(t), y(0) = Уo,

(3)

xH = Ui(x1, x2, ...,xm, y, v(t)), xi(0) = xio, i = 1, 2,...,m, has a unique absolutely continuous solution (x(t),y(t)) = (x1(-), x2(-),..., xm(),

y(0);

2) the function u(t) = X(t),t > 0, belongs to Up.

The function xi (•) is called a trajectory of the pursuer Pi generated by the triple: strategy U, the initial position (x1o, x2o,..., xmo, yo), and the control v(-). Here Ba(x) = {z € Mx : lz — xl< a} is a ball on the tangent space Mx.

Definition 3. Let there exists a number T and a strategy U of the group of pursuers P such that for any control v(-) of the evader E the trajectories x1 (•), x2( ) ..., xm( ), y( ) generated by the strategy U, the initial position {x1o, x2o,..., xmo, yo}, and the control v(-) satisfy the equality xi(t) = y(t) at some i € {1, 2,,..., m} and t € [0, T]. Then we say that pursuit can be completed for the time T in the game (1)-(3).

Definition 4. A mapping V : Mm+1 ^ Rn is called a strategy of the evader E if there is a partition A : 0 = to < t1 < ... < such that

1) tn ^ro as n ^ro;

2) for any control u(-) = (u1(^), u2(-),..., um( )) € Up of the group P the initial value problem

{y V (t, x1 (ti ), x2 (ti ), ..., xm (ti ), y(ti )), ti < t < ti+1, y(0) 0,

(4)

xj = uj(t), xj(0) = xjo, j = 1, 2,...,m,

has a unique absolutely continuous solution (x(-),y(-));

3) the function v(t) = y(t), t > 0, belongs to the set Ve.

Definition 5. If the exists a strategy V of the evader E and a partition A such that for any control u( • ) of the group P the solution of the system (4) satisfies the inequalities xi(t) = y(t), t > 0, i = 1, 2,..., m, then we say that evasion is possible in the game (1)-(4).

3. Evasion from many pursuers

In this section we consider two evasion differential games. In the first one maximal speeds of the pursuers don’t exceed that of the evader and the number of pursuers less than or equal to the dimension of the surface. In the second one the evader has advantage over the pursuers in speed, and the number of the pursuers is arbitrary.

Theorem 1. Let pi < a = 1, i =1, 2, ..., m, and m < n. Then evasion is possible in the game (1)-(4).

Proof. The ball B(p,r) is called convex if for any x, y € B(p,r) there is a unique geodesic y : [0, a] ^ M with the ends x, y and the length d(x, y) that completely belongs to B(p, r). The ball B(p, ro) is called strongly convex if all balls B(p, r), 0 < r < ro, are convex.

It is known [Gromoll, Klingenberg, Meyer, 1968] that for any r > 0 there exists a number ro = ro(B(p, r)) > 0 such that all balls B (q, ro), q € B(p, r) are strongly convex. Let ro = B(yo, 1). Set k = 1 + [1/ro] and consider the partition of the segment [0, 1] by

to = 0, ti+1 = ti + 1/(2k), i = 1, 2,..., 2k — 1.

We proceed by induction. We show that if xi(t) = y(t),i = 1,2,...,m on [0, ti], l € {0,1,..., 2k — 1}, then there exists a control v = v(t), |v(t)| <1, ti < t < ti+1, such that pursuit will not be completed on [tl,tl+1]. Let

J(l) = {i < m : d(xi(ti),y(ti)) < 1/k}.

It follows immediately from the definition that all balls B(p, 1/(2k)), p € B(yo, ro), are convex and, in addition, for each i € J(l) there exists a unique geodesic Yi : [0,a.j] ^ M with the ends Yi(0) = y(ti) and Yi(ai) = xi(ti) (note that y(t) € B(yo, 1) for all t € [0,1]). As dim M = n, n > m, then there exists a vector ui, in the tangent hyperplane to M at y(ti) such that (ui, Afi(0))) < 0, |^| = 1, i € J(l), where {•, •) is the inner product Rn+1.

We construct a geodesic yo : [ti, ti+1] ^ M that satisfy the conditions Yo(ti) = y(ti), Yo(ti) = ui. It is clear that [Gromoll, Klingenberg, Meyer, 1968] such a geodesic exists and is unique. We define the control v(t) of the evader on the interval [ti, ti+1) by v(t) = Yo(t).

Then

t t

d (xi (ti), xi(t)) = J lxi(r) l dr = J |ui(r)| dr < pi(t — ti), i = 1, 2,...,m,

ti ti

t

d(y(ti), y(t)) = J |^(r)| dT = t — ti, ti < t < ti+1. ti

Using triangle inequality gives

d(xi(t),y(t)) > d(xi(ti),y(ti)) — d(xi(ti),xi(t)) — d(y(ti),y(t))

> 1/k — Pi(t — ti) — (t — ti) > 0

for all i e J(l) t e [ti, ti+l] (recall ti+l — ti = 1 /(2k)).

Now we assume that i e J(l). Then by the definition of the set J(l) and convexity of the ball B(y(ti+i), 1/(2k)) we have

{7*(t) : ti <t < ti+i} n B(y(ti+i), 1/(2k)) = 9,

d(y(ti+i),Xi(ti)) d(Yo(tk+i), Yi(tk+i)) > 1/(2k).

Therefore from the triangle inequality for all t e [ti, ti+l] we obtain

d(xi(t), y(t)) > d(y(ti+i), Xi(t)) — d(y(ti+i), y(t))

> d(y(ti+i,Xi(ti)) — d(xi(ti+i),Xi(t)) — d(y(ti+i)),y(t)) (6)

> 1/(2k) — (t — ti)pi — (ti+i — t)/Pi > 0.

Combining (5) with (6) yields xi(t) = y(t) for all t e [ti ,ti+l] and i e {1, 2,...,m}. Hence, by induction we obtain that for all t e [0,1] and i e {1, 2,..., m} relations Xi(t) = y(t) hold.

Repeated application of this argument enables us to conclude that none of the points Xi(t) coincides with y(t) on the intervals [1, 2], [2, 3],.... The proof of the theorem is complete.

Theorem 2. Let pi < a, i = 1, 2, ...,m. Then evasion is possible in the game (1)

-(4).

Proof. We introduce an orthogonal cartesian coordinate system, z(1), z(2),..., z(n+l) at the point O = y0 of Rn+1, where axis Oz(n+l), coincides with normal vector to M at O. In this coordinate system hypersurface M can be given by the equation

z(n+l) = f (z), z = (z(l), z(2), ..., z(n)).

at a neighborhood \z — y0\ < r, r > 0, of the point y0. Fix the numbers e > 0 and r0 e (0, r) so that

\df(z)/dz(i')\ <e< mini 2/(1 + n), min (1 — pi)! (7)

I l<i<m I

for all z, \z — y (0) \ < r0 and i e {1, 2,...,n}. As the functions df (z)/dz(i) are continuous and df(0)/dz(i) = 0,i =1, 2, ...,n, then such numbers e and r0 exist.

Let n be orthogonal projection operator from Rn+1 onto the hyperplane z(n+1) = 0. Then for any control v(-) e VE and corresponding solution y(t), t > 0, of the equation y = v(t) with initial state y0 we have

\ny(t)\ = VKt)|2 - \d,f(ny(t))/d^

\v(t)\2 —{^ E (df(ny(t))/dz(i))(dy(i) (t)/dt)^j > (8)

> J |w(t)|2 - e2 \dy(i')(t)/dt\SJ > ^^(t)!2 - e2n\v{t)\2 > (1 - e)|w(t)|.

Now taking \v(t)\ = 1 yields \ny(t)\ > 1 — e. Therefore \nXi(t)\ < \aci(t)\ < pi < 1 — e (see, (7)).

Let

y = ny, v = y, Xi = nx'i, ui = ~x, 0 < i < m.

Then

y = ny, v = V = ny = nv; Xi = nxi, ■ui = Xi = nXi = nui, 0 < i < m. (9)

Now we consider an evasion game of the point V (the evader) from the points X1,X2, ...,Xm (the pursuers) in Rn. We take v and ul, u2, ..., um as control parameters, which satisfy inequalities (see, (8) - (9)): \v\> 1 — e > pi >\ui\, 1 < i < m. If y0 = xi0, then, clearly, v(0) = Xi(0) for all i, 1 < i < m.

At this time, if the evader V uses the evasion method in direction (Chernous’ko, 1976), then evasion is possible from the points X1, X2,..., Xm on the interval [0, r/2] (to this end, it is sufficient to apply ” the evasion maneuver along narrow corridor” in the direction (1,0,..., 0) on that interval. After that the evader V applies the mentioned maneuver to avoid of points X1,X2, ...,Xm in direction ( —1,0,..., 0) on the interval [r/2, 3r/2].

Repeated application of this maneuver on intervals [3r/2, 5r/2], [5r/2, 7r/2], [7r/2, 9r/2] and so forth ensures the evader V to avoid of points X1,X2, ...,Xm. Of course, during the motion of the point y the point y remains in the circle \y—y0\ < r0 on [0, to), i.e. (7) and (8) holds for all t > 0. The relations y(t) = Xi(t), t > 0, i = 1, 2,...,m, imply that y(t) = xi (t) for all t > 0 and i = 1, 2,...,m. Proof of the theorem is complete.

Note that Xi and ui (respectively V and V) are defined uniquely by xi and ui (y and v)

4. Pursuit differential game

4.1. Strategies of Pursuers

In this subsection, we assume that the Riemann curvature with respect to any two dimensional plane W,W C Mx (curvature in two dimensional direction) at any point x e M is non negative and bounded by some number 1/R, R > 0 [Gromoll, Klingenberg, Meyer, 1968].

Lemma 1. Let y : [0,a] ^ M be an absolutely continuous curve on M; f : [0,a] x [0, /3] ^ M be a smooth mapping that satisfy conditions

1) f(t, 0) = y(t);

2) for any fixed t the curve f (t,r), 0 < r < 3, is a geodesic;

3) the vector field fr(t,r)\r=o is parallel along the curve y(t), 0 < t < a (fr =

df/dr and {■, ■)).

Then

{frt,ft)\r=0 = 0, 0 < t < a. (10)

Recall that such a mapping f : [0, a] x [0,3] ^ M is called the Jacobi rectangle.

Proof. Since the vector field fr(t, r)\r=o is parallel along the curve y(t), 0 < t < a, the vector field d(fr(t, r)\r=0)/dt is orthogonal to M along y(t), 0 < t < a(Thorpe, 1979). It is clear that the vector ft(t,r) is tangent to M. Hence, (10) holds.

The following lemma is the result of the Rauh’s comparison theorem (Gromoll, Klingenberg, Meyer, 1968).

Lemma 2. Let X be the Jacobi field along the geodesic 7 : [0, to) ^ M. If {X'(0),X(0)) =0 and X(0) = 0, then 0 < \X(r)\ < \X(0)\ for all r e (0,nR/2).

Lemma 3. Let d(x,y) < nR/2, 7 : [0,ro] ^ M be a geodesic such that 7(0) = y and j(d(x, y)) = x, and X be a Jacobi field along 7 such that X(0) = v, \v\ < a < p, {X'(0),X(0)) = 0. Then there exists a number a(x,y,v) > 0 such that

Uo(x, y, v) = X(d(x,y)) — a(x,y,v)Y(d(x,y)), \Uo(x,y,v)\x = p. (11)

Proof. We obtain from Lemma 2 that \X(r)\ < \X(0)\ = \v\ < a < p if 0 < r < nR/2. Therefore there exists a number a(x,y,v) that satisfies the relations (11).

Remark 1. In case M is Euclidean space the strategy U0(x,y,v) coincides with the strategy of parallel approach (Petrosjan, 1965). If M is n-dimensional sphere, then it can be shown that the function U0(x,y,v) is defined for all possible triples

(X, y, v).

Theorem 3. Let (x0,y0) e M2, d(x0,y0) < nR/2; v(t),t > 0, be a control of the evader E. Then

1) there exists a solution (x(t),y(t)) of the initial value problem

'V = v(t), y(0) = y0, X = U0(x,y, v(t)), x(0) = X0, (12)

on the maximal time interval [0,a) where x(t) = y(t);

2) if x(t) = y(t), 0 < t < a and <p : [0, a] x [0, nR/2] ^ M is the Jacobi rectangle such that y(t) = (p(t, 0), and for each fixed t e [0, a] the curve f(t, t) is geodesic on M through x(t), then the vector field Vt = <pr(t,r)\r=0 is parallel along the curve y(t) , 0 < t < a and

J-(d(x(t), y{t))) = ~{V\ v(t)} - \/p2 - \v(t)\2 + KV^vit)}]2. (13)

Proof. 1). Let v(t),t > 0, be an arbitrary evader’s control, y(t), t > 0 be the solution of the initial value problem y = v(t), y(0) = y0, f (0,r) = ^(0,r) and f : [0, to) x [0, nR/2] ^ M be the Jacobi rectangle that satisfy hypothesis of Lemma 1.

Recall that for each fixed t the vector field ft(t,r) is the Jacobi field [Gromoll, Klingenberg, Meyer, 1968] along the curve f (t,r), 0 < r < nR/2 that satisfy the condition ft(t, 0) = v(t), and the vector field fr(t,r) is tangent one. Therefore Lemmas 1 and 2 imply that

0 < \ft(t,r)\ < \ft(t, 0)\ = \v(t)\ < a < p

if 0 < r < nR/2. Hence, the solution r(t) of the initial value problem

r = ft, fr) + \J{(ft, fr)f ~ 4 ((/t)2 - /92)j , r( 0) = d (x0, yo),

exists and doesn’t increase on the time interval [0, 9), where r(t) = 0. Set z(t) = f (t,r(t)), z(0) = X0, t e [0,9).

Then \z(t) \ = p and

z(t) = ft (t, r(t)) + fr (t, r(t)) r(t), t e [0, 9).

Then it is not difficult to verify (see, Lemma 3) that

ft (t, r(t)) + fr (t, r(t)) r(t) = U0 (z(t), y(t), v(t)) and the function z(-) satisfies the equation

z(t) = U0 (z(t), y(t), v(t)), z(0) = X0.

Therefore by uniqueness of the solution we have 9 = a,

x(t) = z(t), d (z(t),y(t)) = r(t), y(t,r) = f (t,r), (14)

and the field r(t, r)\r=0 is parallel along the curve y(t), t e [0, a].

2). Let Xt(r) be a Jacobi field orthogonal to the curve p(t,r) along p(t,r) for every fixed t e [0,a] and

Xt(0) = v(t) — (7t(0), v(t)) Yt(0), Yt(r) = fr(t, r).

To prove (13) we decompose the vector ft(t, 0) = v(t) in orthogonal vectors

Xt(0) and Yt(0) :

v(t) = Xt (0)+ Ai(t) (f)' (0).

Then

U0 (x(t),y(t),v(t)) = Xt (d (x(t),y(t)))+ A2 (t) (y*)'(d (x(t), y(t))),

where ____________________________

A2{t) = -\j p2 - \X* (d(x(t),y(t)))\2.

It can beeasily checked that

r(t) = A2 (t) — Ai(t). (15)

By (11) and (12) we obtain

Ai(t) = {Yt(0),v(t)) = {¥r(t,r)\r=0 , v(t)) = {Vt,v(t)) ,

A2(t) = -\Jp2 - |X‘(0)|2 = -v/p2-\v(t)\2 + \l(t).

Combining the last formulas with (14) and (15) gives (13). The proof of the theorem is complete.

In its turn equality (13) implies

d(x(t), y{t))) <0- p.

Integrating the last inequality yields

d (x(t), y(t)) < d (X0, V0) + (a — p) t.

Hence, the following corollary is true

Corollary 1. If m = 1, (x0,V0) e M2 and d(x0 ,V0) < nR/2, then pursuit can be completed in the game (1)-(4) with the help of the strategy U0(x,y,v) for the time d(x0,V0)/(pi — a).

4.2. Pursuit differential game with many pursuers

If at least for one i, pi > a, then, clearly, pursuit can be completed from any initial positions. Therefore, a pursuit game is intensional if maximal speeds of all players are equal. That is why we consider the case pi = a =1 and the pursuit game (Pshenichnii, 1976)

Theorem 4. Let geodesics Yi : [0, ro] ^ M satisfy conditions 7i(0) = yo, Yi(0) = ei, yo € int conv{ei, e2, ..., em}, and xio = Yi(ri), 0 < ri < Rn/2, i = 1, 2, ..., m, be initial positions. Then pursuit can be completed in the game (1)-(4) with the help of the strategies Ui = Uo(xi, y, v), i =1, 2,..., m, for the time

m /

T = d(xio, yo)/ 2 min max (v, ei)

* ^ \ \v\ = 11<i<m

i=1

where v € My0.

Note that the condition yo € int conv{ei, e2,..., em} implies that m > n and

min max (v,ei) > 0, v € My0.

\v\ = 1 1<i<m

Proof. Let v(-) be an arbitrary control and y( ) be corresponding trajectory of the Evader E; Fy(t)(e) be a parallel shift of the vector e € My0 to the point y(t) along the curve y(t), t > 0. Let the pursuers use the strategies Ui = Uo(xi, y, v),

i = 1, 2,,..., m. Then according to (13) we obtain

/ m \ m j----------------------------------t:

ft £ d(xt(t),y(t)) = - £ V1 - \v(t)|2 + \(Fy{t)(et),v(t))\2-

\i=1 J i=1

m m

(Fy(t) (ei),v(t)) <-Y, (K Fy{t](ei),v(t)) I + (Fy(t) (ei),v(t))) , i=1 i=1

where v(t) = v(t)/|v(t)|, if v(t) = 0. In case v(t) = 0, v(t) is any unit vector tangent to M at y(t). As a parallel shift along a curve doesn’t change the inner products, then

m \

J2d(xi(t),y(t))) < - maxmS(Fy(t)(ei),v(t)) + KFy(t)(ei),v(t))D i=1 j -l-m

<—2 min max (v,ei) (16)

\ v \ = 1 1<i<m

whenever d(xi(t),y(t)) > 0 for all i, where v is a unit vector tangent to M at yo. Now we integrate (16) to obtain

mm

V'd (xi(t),y(t)) < V'd (xio, yo) — 2t min max (v,ei) .

\ v \ = 1 1<i<m

i=1 i=1

Hence, by the time T an equality xi(t) = y(t) holds at least for one i. Proof of the theorem is complete.

Remark 2. If the initial positions of the players don’t satisfy hypothesis of Theorem 4, then the pursuers using another strategies can take the necessary position and then apply the strategy Uo.

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