CC BY
4Alexander Bannikov and Nikolay Petrov
Udmurt State University,
Faculty of Mathematics,
Universitetskay st. 1, Izhevsk, 426034, Russia E-mail: [email protected], [email protected]
Abstract. We consider two linear non-stationary problems of evasion of one evader from group of pursuers provided that players possess equal dynamic possibilities and the evader does not leave limits of some set. It is proved that if the number of pursuers is less than dimension of space the evader evades from a meeting on an interval [to, +ro).
Keywords: differential pursuit-evasion games.
1. Introduction
The important direction of the modern theory of differential games is linked with working out of solution methods of pursuit - evasion game problems with several players. Deriving both necessary and sufficient conditions of solvability of problems of evasion and pursuit under the initial data and game parameters thus is of interest. In the given research two non-stationary linear differential games with a simple matrix are considered.
2. The Non-stationary Problem with Simple Motion
In the space IRk(k > 2) a differential game of n +1 persons: n pursuers P1,...,Pn and a single evader E, is considered.
The motion law of each pursuer Pi has the form
±i = b(t)ui, ||wi|| < 1. (1)
The motion law of the evader has the form
y = b(t)v, ||v|| < 1. (2)
At t = to initial positions of pursuers x0,...,xn and initial position of evader y°, are set and x0 = y°, i = 1,...,n.
Here b : [to, to) ^ IR1 — measurable function.
It is supposed that evader E in the course of game does not leave convex set D (D c IRk) with a nonempty interior.
At b(t) = 1 and with the lack of phase restrictions the problem was considered
in Petrosyan L. A., 1966, Pshenichnii B. N., 1976, at b(t) = 1 with phase restrictions
the problem was considered in Ivanov R. P., 1978, Petrov N. N., 1984,
* This work was supported by the Presidium of the Russian Academy of Sciences under the program ”the Mathematical control theory”.
Kovshov A. M., 2009, Satimov N. Yu. and Kuchkarov A. Sh., 2001. The non-statio-
nary case under a condition n > k was considered in Bannikov A. S. and Petrov N. N., 2010.
Let a be some partition t0 < t1 < ■ ■ ■ < ts < ... of interval [t0, to), not having
finite condensation points.
Definition 1. The piecewise-program strategy V of the evader E which is defined on [0, to) that correspond to a partition a is the family of mappings {cl}^=0 which are putting in correspondence to magnitudes
(tl ,xi(tl), . . .,Xn (tl ),y(tl))
measurable function v = vl (t) which is defined for t G [tl, tl+1) and such that \\vi (t)|| < 1, y (t) G D, t G [tl ,tl+i).
Let’s designate the given game through r.
Definition 2. We say that in game r an evasion from a meeting occurs if a partition a of interval [to, +to) exists without having finite condensation points, strategy V of the evader E corresponding to partition a, such that for any trajectories x1(t),...,xn(t) of pursuers Pi,. ..,Pn takes place
xi(t) = y(t), t > to, i =1,...,n,
where y(t) is the trajectory of the evader E realized in the present state of affairs.
Theorem 1. If y0 G Int D, b is the function bounded on any compact set then the evasion from meeting is happening in the game r.
Proof. Because y0 G Int D, Dr (q) — full-sphere with the radius r with the center in the point q exists and y0 G Int Dr (q) C D.
Let e — distance from y0 to boundary Dr(q), Il = [t0 + l — 1,t0 + l], bl > 0 is such that \b(t)\ < bl for all t G Il,
t
rtj(r) = \t>T : J \b(s)\ds = j^-jY
T
Note that if t G Q(t) T,t G Il for some l, then
j + 1
Therefore
t
J |b(s)|ds < bl(t — t).
(3)
Let us define partition al of the segment for all segments Il and the number ml in the this way. Let us consider the segment I1. Let t01 = t0, j = 1, 2 ...,
T1 = J inf{t > Tj_ 1, t G (Tj_1)}, if Tj <t0 + 1 and Qj(t1_1) = %,
j [ t0 + 1, otherwise.
e
Then we assume that m1 = min{j : Tj = t0 + 1},a1 = {tq, ..., t^ i}.
Let us consider the segment I2. Let t2 = t0 + 1. For all j = 1, 2 ...,
Tj = inf{t > Tjj_1, t G Qj+mi (t]_1 )},
if
T.2 < to + 2 Qj+mi (t2_ 1) = 0
and Tj = t0 + 2, if appropriate conditions aren’t fulfilled.
Then we assume that
m2 = m1 +min{j : Tj = t0 + 2} a2 = {tq , . . .,Tl 2_mi }.
We assume that partition al-1 of the segment Il-1 and the number ml-1 are
defined. Let us consider the segment Il. Let t- = t0 + l — 1. For all j = 1, 2 ...,
Tj =inf{t > т■_l, t g Qj+mi_i (t2_1 )},
if
t1 <to + l and Qj+mi_i (t2_1) = 0
and Tj- = t0 + l, if appropriate conditions aren’t fulfilled.
Then we assume that
ml = ml_1+min{j:Tj=to+l}, al = {t-,...,Tml_mi-i}.
Note that in view of (3) the numbers ml exist for all l. As the partition a of the interval [to, to), we take such partition that contraction a on all segment Il matches with al. Let a = {t0 = t0 < t1 < ■ ■ ■ < Tr < ...}. We define strategy V of the evader in follows. v(t) = vj signb(t),t G [t.,Tj+1), where vj is defined from the following conditions
(vj ,xi(Tj) — y(Tj)) =0,i =1,...,n (vj ,y(Tj) — q) < ^ lvj1 = 1.
As n < k system has the solution.
Let us show that V is the evasion strategy. Let us consider the segment [t., Tj+1], Then from systems (1), (2) we have
t
'j) “L
y(t) = y('j)^y \b(s)\ds^vj,
T3
t
xi(t) = xi (Tj )^y b(s)ui(s)ds.
Tj
Therefore
|xi(t) — y(t)| = llxi('j) — y('j) — Mj(t)vj + Mj(t)ui(t)| >
> I- y(T3) - II - Mj(t) =
= ^IM^II2 -2Mj(t)(ai(Tj),vj) +M?(t) - Mj(t) =
= ^\\al{r0W + M]{t)-M0{t), (4)
t
where ai(tj) = xi(Tj) — y('j), Mj(t) = / \b(s)\ds,
Tj
J b(s)ui(s)ds, if Mj{t) + 0 Tj ,
0, if M2 (t) = 0
and ||Ui(t)|| < 1 for all t G [t.,Tj+1]. From (4) it follows that if xi(t.) = y(rj) then
xi(t) = y(t) for all t G [t., Tj+1], It means that if caption didn’t occur at the moment Tj then it wouldn’t on the segment [t. ,Tj+1 ].
As x0 = y° for all i then y(t) = xi(t) for all i,t > t0.
Let us prove now that y(t) G D for all t > t0. Let us consider the segment
[to,t]. Then
\W)-q\\ = \\y° + Ml(t)vl-q\\ =
= \J\\y° - q\\2 + 2(y° - q, v\)M\(t) + Mf(t) < yj(r -e)2
As M\(t) < | for all t G [ro, ti] then
\\y(t)-<l\\ <\j{r~e)2+ (|) <r“|-
Thus, y(t) G D for all t G [t0, t1 ].
Let us assume that inequality \\y(t) — </|| < r — -p-^ is proved for all t G
[Tj_1,Tj],j < l — 1. Let us prove that for all t G [t1_1,t1 ] inequality ||y(t) — q| <
r — jpj- holds true.
hit) ~ q\\ = \\y(ti-i) + Mi(t)vi - q\\ =
= \J\\y(ti-i) — q\\2 + 2(y(£i-i) — q,vi)Mi(t) + Mf(t) <
< ^J(r -y)2 + M2(t).
As Mi(t) < for all t G [t;_i,t;] then
ut)-4 < +
So we prove that V is the evasion strategy. □
3. Linear non-stationary problem of evasion in a cone
In the space IRfc(k > 2), a differential game of n +1 persons: n pursuers P1,...,Pn
and a single evader E, is considered.
The motion law of each pursuer Pi has the form
xi = a(t)xi + Ui, ||ui|| < 1. (5)
The motion law of the evader E has the form
y = a(t)y + v, ||v| < 1. (6)
i(t) =
At t = t0 we have the initial positions of the pursuers x1,...,x(n and the initial position of the evader y° and x0 = y°, i = 1,...,n.
Here a : [to, to) ^ IR1 is a measurable function.
It is supposed that the evader E in the course of the game does not leave convex cone
D = {y : y G IRk, (p., y) < 0,j = 1,...r},
where p1,. ..pr are the unit vectors IRk such that Int D = $.
The evader uses piecewise-program strategies.
At a(t) = a, a < 0 and lack of phase restrictions the problem was considered in Pshenichnii B. N. and Rappoport I. S., 1979, at a(t) = a, a < 0,n > k with phase restrictions the problem was considered in Petrov N. N., 1988; at a(t) = a, a <
0,n < k with phase restrictions the problem was considered in Petrov N. N., 1998, and at a(t) = a, a > 0,n < k with phase restrictions the problem was considered in Shuravina I. N., 2009.
Theorem 2. Let y° G Int D, a is a bounded function on any compact and n < k. Then the evasion from meeting occurs in the game r.
Proof. Let’s consider segment Ii,ti = t0 +1. In systems (5), (6) let’s make a change of variables
t t
f a(s)ds f a(s)ds
xi = etl wi, y = etl zl.
We will receive systems
t t _ f a(s)ds _ f a(s)ds
wi = e t‘ ui, Zl = e t‘ v. (7)
Let’s notice that xi(T) = y(T) at some i,T G Il if and only if wi(t) = z1(t). Besides y(t) G D if and only if zi(t) G D. Let further
t tl-1 _ f a(s)ds _ f a(s)ds
bi(t) = e tl ,Ki = e tl ,
Dr (q) - a full-sphere with radius r with the center in a point q y° C Dr (q) C D, £ - distance from y° to boundary Dr(q), q1 = K1q,r1 = K1r,£1 = K1£, qi = Kiqi_1, ri = Kiri_1, l > 2. Let’s notice that bi(t) > 0 for all t G Ii.
For the segment I1 on £1 and function b1 we will define number m1 and partition <ti under the scheme of the previous section. For the segment /2 on £2 = ^1+2 an<^ function b2 we will define number m2 and partition a2 and so on. For segment Ii on £i = ^iei~j^2 and function bi we will define rrii and partition <j;. As a partition a of interval [t0, to) we take such partition, which contraction on any segment Ii coincides with ai.
Let Tj, Tj+1 G ai. We set strategy V of evader E on [t. , Tj+1 ] supposing v(t) = vj, where vj is defined from following system
(vj,zZ (t2 ) — w\(Tj ))=0, i ^...^ (vj ,qi — z l('j)) > 0, hj || =1.
As n < k then vj is always exists. Let’s show that strategy V guarantees evasion from meeting.
1. Let’s show that zi(t) = wi(t) for all i,t G Ii. Let tj,Tj+1 G ai. From sys-
tems (7) we have
zi(t) = zi(Tj) + J bi(s)ds • vj, Tj
t
'ji(t) = wi(Tj) + J bi(s)ui(s)ds.
Therefore for all t G [t. , Tj+1)
IIz\t) - wi{t)II > IIz\r0) + Mj(t)vj - w'(r,-)|| - Mj(t) = ^(«'(^))2 + 2Mj{t){vj, aiirj)) + - Mj(t)
= ^(«'(^))2 - (Mj(t))2 - Mi{t) > 0 if a\(Tj) ^ 0.
Here
t
Mj(t) = J bl(s)ds, ai(Tj) = zi('j) — wi(t2).
Tj
Therefore if capture didn’t occur at t. then it wouldn’t occur on segment [t., Tj+1]. As z0 — w0 =0 for all i then we prove that zl(t) = wi (t) for all i,l,t G Il.
2. Let’s introduce that for all natural l, t G Il following inequalities hold
Iz\t) - qi\\ < n - —t G [Tj,Tj+1], {rhTj+1 G ai). (8)
Let’s consider the segment I1 = [t0,t1] and partition a1 of this segment.
|| z1 (to) — q1H = HK1y0 — Kq = K1(r — £) = r1 — £1.
Let t G [tq,T11]. Then
\\z\t) = ll^o) + M^{t)v\ - gi|| =
\J\\zl(to) ~ <7i||2 + 2M01(t)(z1(t0) - qi,v{) + (Mi(t))2 <
£\ 2 E\
<\j(n~ei)2 + ^J
because Mg(t) < y for all t G [tq , r/] owing to a choice r/. The further proof of an inequality (8) for I1 is similar to the proof of a corresponding inequality from the previous section.
Let us assume that equality (8) is proved for all Il,l < s. Let us prove an inequality for Is+1. Owing to the supposition inequalities holds true
s £s
z>>(ts) — qs|| < rs —
s
About Some Non-Stationary Problems of Group Pursuit with the Simple Matrix 53 Then
llz s+1(ts) — qs+1 y = ||Ks+1(y(ts) — qs)| =
£s
= Ks+l\\zS (ts) — (/s|| < Ks + l(rs-------—) = rs+i — £s + l.
ms + 2
Therefore the proof of the inequality (8) for Is+1 is similar to the proof (8) for I1. Therefore we have
zl(t) C Dn (qi) C D
for all t G Il and for l. Therefore we have y(t) G D for all t > t0. Thus specified strategy V is the evasion strategy. The theorem is proved. □
Theorem 3. Let y0 G Int D,
0 G {z1,...,z‘n,P1,...,Pr}.
Then evasion from meeting occurs in the game r.
The proof of this theorem is similar to the proof of the stationary case Petrov N. N., 1988.
4. Evasion problem from group inertial pursuits
In the space IRk(k > 2) a differential game of n +1 persons: n pursuers P1,...,Pn and a single evader E, is considered.
The motion law of each pursuer Pi has the form
xi(t) = a(t)ui(t), ui G U.
The motion law of the evader E is
y(t) = a(t)v(t), v G U,
and at t = t0 initial positions of pursuers xi(t0) = x0, xi(to) = x0 and initial position of evader
y(to) = y°, y(to)= y0, x0 = y0, i = 1,...,n,
are set.
Here x^ y, ui, v G IRk, U C IRk — convex compact, 0 G Int U; a(t)— bounded measurable function integrated on any compact subset of axis t, a(t) = 0 almost everywhere on interval [t0, +to). Measurable functions ui(t), v(t) accept values at t > t0 from set U.
Let’s designate the given game through r = r(n,z(to)), where
z(t) = (z1(t),z1(t),...,zn(t),zn(t)), zi(t) = xi (t) — y(t), i = 1,...,n.
At a(t) = 1 resolvability sufficient conditions of the local evasion problem have been got by Prokopovich P. V., Chikrii A. A., 1989.
Sufficient conditions of resolvability of the local evasion problem for a stationary control example of Pontryagin have been got by Chikrii A. A.,Prokopovich, P. V., 1994.
In this section, using ideas of Chikrii and Prokopovich, the conditions of resolvability of the local evasion problem have been got in a non-stationary case.
Definition 3. Positional counter-strategy V of evader E is the measurable mapping
[to, +to) x IR2nk x Un ^ U.
Then at set controls ui(t) of pursuers Pi, i = 1,...,n strategy V defines control v(t) = V(t, z(t), u1(t),..., un(t)), which will measurable function.
Definition 4. We say that in differential game r from initial position z(t0) local
evasion problem is solvable, if for all measurable functions
ui(t), t > to, ui G U, i = 1,...,n,
strategy V of evader E exists such that zi(t) =0 for all t > t0, i = 1,...,n.
We consider that controls of pursuers are formed on the basis information about position z(t) of differential game.
Theorem 4. If the condition 0 G co{z°, ..., zn} is satisfied then in game r local evasion problem from initial position z(t°) is solvable.
Proof. Let 0 G co{z°,... ,zn}. Based on the convex sets separation theorem unit
vector p and number £ > 0 exist such that
max (z°,p) < — 2£. (9)
i=1,...,n
Let us designate
n(t) = . I^n ||zi(t)||, (10)
i=1,...,n
S = min{l,£, y/r](to)}. (11)
1. Let max(z°,p) < 0. Let us suppose
1,...,n
v«>=v,(o={vr^ > 0, (i2)
where vp,v_p G U are vectors such that (vp,p) = C(U,p), (v_p, —p) = C(U, —p). Then for all i = 1,...,n we have
(zi (t),p) =
t
= (z°,p) + (t — t°)(z°,p) + J(t — s)a(s)(ui(s) — vp(t),p) ds < -2£(t —1°) < 0
to
at t > t°. The local evasion problem from initial position z(t°) is solvable.
2. Let us assume that (z°,p) > 0 and (z°,p) < 0 for all i G {2,... ,n}. Let us
describe evasion maneuver which guarantees the solvability of evasion problem for the such initial position z(to). Let K = min{l, y^}, t\ = KJi = K^- < r/(to). Let’s suppose
v(t) = vp(t), t G [t°, +TO)\[t1,t1 + J1),
where t1 is either first moment, when for the first time equalities are fulfilled llzi(ti)ll = $1 and (zi(t1),p) > 0, or +to, and if t1 < +to, then on interval [ti,ti + r\), we choose control v(t) in special way.
This way chosen control of evader E the pursuer Pi, i G {2,...,n} does not influence a game course. Really, at any controls ui(t), i = l,...,n, we have
t
(Zi(t),p) = (Zi(to),p) + J a(s)((ui(s),p) - (v(s),p)) <
to
<-2$ + 2c\U\r1 < -S, t > t0. (13)
Therefore
t
(zi(t),p) = (zi(to),p) + J(zi(s),p) ds < (zi(to),p) < 0, t > to, i = 1. (14)
to
So we have at t > t0 || zi (t) | =0, i = 1.
Let us notice that at t = t1 |z1(t1)| = S1, therefore at any controls u1(t) and v(t) on segment [t1,t1 + t1]
tl+Tl
(z1(t1 + T1),p) = (z1(t1),p)+ J (z 1(s),p) ds < $1 - $T1 =0. (15)
tl
Therefore, at t = t1 + t1 position z (t1 + T1) of differential game corresponds to previous case 1:
(zi(t1 + T1),p) < -S,
(zi(t1 + T1),p) < 0, i = 1,...,n.
Thus,if on any controls u1(s) we can specify control v(s), s G [t1,t1 + t1 ), such that ||z1 (t)| = 0 at t G [t1,t1 + T1], then the solvability of local evasion problem for initial position z(to) in the case 2 will be proved.
Let us assume that
(z1(t1),z1(t1)) = -lz1(t1 )||||z1(t1)||. (16)
Vectors z1(t1) and z1(t1) are linearly dependent, therefore unit vector ^ exists such that
(^,z1(t1)) = (^, z1 (t1)) = 0. (17)
Let e1 G (0,T]_) is some number such that at arbitrary control u1(s), v(s), s G [t1,t1 + £]_], inequality (z1(s),p) > 0 is fulfilled. On the segment [t1,t1 + e1] we choose control v(s) so that
(*,v(s))= ( C(U’"»’ <*u1(s)) < (18)
( " \ -C(U, -v), (V,u1(s)) > 0. ' '
Let us show that the number y1 G (0,e]_) exits such that
(z1(t1 + Y1),z1(t1 + Y1)) = -^z1(t1 + Y1)||z1(t1 + Y1)|. (19)
At t > t1 let us consider the functions
h(t) = (V, z1(t)) = (t - s)a(s)(V, u1 (s) - v(s)) ds,
tl
t
(20)
f2(t) = (V,z1(t)) = J a(s)(V,u1(s) - v(s)) ds. tl
The functions f1(t), f2(t), t1 < t < t1 + e1, satisfy to the set of equations
f1(t) = f2(t), (21)
f 2(t) = a(t)(V,u1(t) - v(t)),
and f 1 (t 1) = f2(t2) = 0.
As 0 G Int U then C(U, q) > 0 for all q = 0. Therefore \a(t)(V, u1(t) - v(t))\ > 0 almost everywhere on [t1,t1 + e1] and f2(t) ^ 0 on any segment [a, 3] C [t1 ,t\^ + e1], t1 < a < 3. Therefore the set G = {t G (t1,t1 + £1)\f2(t) = 0}is the nonempty open set. Therefore G = [J(aj, 3j), when {(aj, 3j)} is no more than countable system of
not intersected intervals.
Let (aj0, 3j0) is the some interval of this system. Then we have
f2 (ajo ) = f2 (3jo ) = 0, f2(t) = 0, t G (ajo , 3jo ).
If f1(ajo) = 0 then the relation (15) is fulfilled at t1 + y1 = ajo. If f1(ajo) = 0 then
f1(t) = f2(t) = 0, t G (ajo ,3jo ).
Therefore f1(t)f2(t) > 0 on (ajo ,3jo). So at t1 + y1 = t, when t is some arbi-
trarily chosen number from an interval (ajo ,3jo), the relation (15) takes place.
Thus, control v(s) at s G [t1,t1 + e1] is defined according to a rule (14). Then we have
||z1(t)|| =0, t G [t1,t1 + £1],
and at some moment t = t1 + y1 the relation (15) will be fulfilled.
If
(z1(t1),z1(t1) = -||zl(tl)||||Zl(tl)||, (22)
then we suppose y1 = 0. Let’s notice that (z1(t1 + Y1),p) > 0 and (z1(t1 + Y1),p) < -S, therefore from the inequality (15) linear independence of the vectors z1(t1 + 71) and z1(t1 + 71) follows. Further for all s G [t1 + Y1,t1 + T1) we will suppose v(s) = u1(s). Then we have
z1(t) = z1(t1 + Y1) + (t - t1 - Y1)z 1(t1 + Y1) (23)
at t1 + y1 < t < t1 + T1, therefore |z1(t)| = 0 at t G [t1 + Y1,t1 + t1 ].
Thus on any measurable function u1(s) G U it is possible to construct such
measurable function
v(s) = V(s,z(s),u1(s),...,un(s)), v(s) G U, s G [t1,t + T1),
that || z1 (t) | = 0 at t G [t1,t1 + t1 ]. The solvability of local evasion problem is proved in the case 2.
3. Let (z0,p) > 0 for all i G I' = {1,...,s}, s < n, and (z0,p) < 0 for i G {1,...,n}\I'. We define such numbers Tj, Sj, j = 1,...,Nl, N1 < s, that
Tj > Tj+l, $j > $j+1, j 1,..., N1 1,
and if at t' > t0 for some i G {1,...,s} equality !zi(t')|| = Sj hold and (zi(t'),p) >
then (zi(t' + Tj),p) < 0 for any controls ui(s), v(s) defined on [t', t' + Tj].
Time moment ti > t0 when equality n(t) = Si the first time is carried out and number I G {1, ...,s}, exists such that Hze(ti)|| = Si, (zi(ti),p) > 0, we name the i-th meeting moment. Without reducing a generality, we suppose that at t = ti we have |zi(ti)| = Si and (zi(ti),p) > 0. it means that pursuers are numbered in that order in what their meeting with the evader happens.
Let us suppose
v(t) = vp(t), t G [to, +^)\T, T = y [ti,ti + Ti), (24)
i=1,...,Ni
At t = t0 we construct sequences {T'l1}i'=1, {S\}°=1 in follows:
i S i i S2
r\ = K^ Sl = Srl = K¥TI. (25)
Numbers t1, i = 1,..., N-\_, will be defined so that t1 < t1 , i = 1,..., N1, therefore
SS £ Ti < Ci = K ^ — 2|cT|c'
i=1,...,N1 1 1
Then for any control ui(s), i = 1,...,n, defined on [t0,t] and v(s) defined on [t0 ,t] n T we have inequalities
(zi(t),p) = (z0,p) + J a(s)(ui(s) - v(s),p) ds+
[to,t]nT
+ j a(s)(ui(s) - vp(t),p) ds <—2S + 2\U\c^(T) <
[to,t]\T
< —2S + 2\U\c£l < —S, t G [to, +^), i = 1,...,n. (26)
t + T
Therefore (zi(t' + t),p) = (zi(t'),p) + / (zi(s),p) ds < (zi(t'),p), t' > to, t > 0.
t!
And it means that approaching with the pursuer Pi, i G{1,..., n}\I' will not occur. Without limiting a generality, we consider further that N1 = n, i.e. approaching occurs with each pursuer.
Note that, if at the time instant t = t' for some i G {1,...,n} we have the relations
Uzi(t')U = Sl, (zi(t'),p) > 0,
then for any controls ui(s), v(s) defined on segment [t',t' + t}] we have
t + T?
(zi(t' + Tl),p) = (Zi(-t'),p) + (Zi(s),p) ds<5\ - St[ =0. (27)
Let t\ = t}, 6i = 6}. Then 11 > to. Let us define the evasion maneuver recurrently. Let us have at the time instant t = ti we have the relations ||zi(ti)y = 6i, (zi(ti),p) > 0, with defined number Ti and monotonically decreasing sequences of positive numbers [t({6f}'f=i.
Let us suppose (zi(ti), Zi(ti)) = —||zi(ti)||||Zi(ti)||. The number ei G (0,Ti) exists so that at arbitrary uf(s), I = 1,...,n, v(s), s G [ti,ti + ei], we have inequalities
min ||zf(tj + t)|| > $l+ 1, (zi(s),p) > 0. (28)
T e[0,Ei]
The vectors zi(ti), Zi(ti) are linear dependent, therefore there is an unit vector ■0i that
(0i, zi (ti)) (0i, Zi(ti )) 0.
We choose control v(s) on segment [ti,ti + £i] so that
(A, v(s)) = ( C<UM’ 5 0' (29)
( ( )) \ -C(U, -0i), (0i,Ui(s)) > 0. ( )
Then we have Yi G (0,£i) so that
(zi(ti + Yi), Zi (ti + Yi)) = -Hzi(ti + Yi)HHZi(ti + Yi)|.
From (28) and inequality (Zi(ti + Yi),p) < —6 we have linear independence of vectors Zi (ti +Yi) and Zi(ti +Yi). If (zi(ti), Zi(ti)) = —||zi(ti)||||Zi(ti)|| then we assume that Yi =0.
According to the reasonings in the case 2, the evader control v(s) on interval [ti + Yi,,ti + U) it is necessary to consider ui(s) as equal.However if i < n then on [ti + Yi,ti + Ti,) approaching with pursuers Pi, i + 1,...,n, might occur. Therefore
v(s) = Ui(s), s G [ti + Yi,ti + Ti)\ y [tj,tj + Tj),
j=i+1
if i < n and
v(s) = Un(s), [tn + Yn,tn + Tn).
Let i < n and tf G [ti + Yi,ti + Ti), I = i + 1,...,n. The evader will approach with pursuers Pi+1 ,...,Pn so close and bypass them for such a short period of time, that on the trajectory zi(t) on segment [ti + Yi, ti + Tj,] at any controls ui(s), s G [ti + Yi, ti + Ti] next relations holds true:
(zi(ti + t),Zi(ti + t)) = —||zi(ti + t)||||Zi(ti + t)||, t G [Yi, Ti], (30)
min ||Zi(t)| > a.i, (31)
te[ti+ji,ti+Ti]
ai > 6i+i. (32)
From (32) we have, that approach of evader with each pursuer can occur no more than once.
Let Hi(ti+Yi +t), t G IR1, be the line which is passing through points zi(ti +Yi) + TZi(ti + Yi) and Zi(ti + Yi). On the basis of linear independence of vectors zi(ti + Yi) and Z-i (ti + Y-i) we have at any t vectors zi (ti + Yi) + tZi (ti + Yi-) and Zj, (ti + Yi-) linearly are independent. Therefore at any t we have
f (t)= mm J^l > °. (33)
xeHi(ti+Yi+T)
(f (t ) =
II ((1 - r)||6||2 - (a, b)) (a + (r - 1)6) + IIa + I ||a + 6||2
a = Zi (ti + Yi), b = Zi(ti + Yi))
What is more the function f (t) is continuous.Let us define the number at the time instant t = ti + Yi,
Pi = min f (t ). (34)
t E[0,Ti-Yi]
If v(s) = ui(s) at s G [ti + Yi, ti + Ti] then the corresponding trajectory is
zit = zi (ti + Yi) + (t — ti — Yi)zi(ti + Yi)
and
Zi(t) = Zi(ti + Yi), t G [ti + Yi,ti + Ti].
It is clear that at any t G [ti + Yi,ti + Ti]
M(t)H>l3i, Hz°(t)H>l3i. (35)
Let us assume that on segment [ti + Yi,ti + Yi) we have the final system of
intervals [tr,tr + tr), r = 1,...,l, I > 1, such that
M U [tr,tr + Tr)) <£i+i, (36)
r=1,...,l
6+i = min {\jrl + . (37)
Let us show if
v(s) = ui(s), s G [ti + Yi, ti + Ti)\ U [tr ,tr + Tr),
r
= 1,
control v(s) is defined on the set |J [tr,tr + tr) and control ui(s) is defined on
r=1,...,f
segment [ti + Yi,,ti + Ti] are chosen arbitrarily, so corresponding trajectory zl(t), t G [ti + Yi,ti + Ti], such that
\\4(t)-zum<^ (38)
ll^)-^°WII<|, (39)
2
Let t =1. It is clear that
I|z1(t) - z?(t)|| < \U|c((t 1 )2 + 2t 1Ti) < \U\c(g+1 + 2^i+1 n).
Analogically, for t G IN, t G [ti + Yi,ti + Ti] we have
(40)
owing to relations (36), (37).
Thus the inequality (38) is proved. The inequality (39) implies from the definition of the number &+1.
Let us show for any t G [Yi,Ti]
such that zi(ti + tq) = -qZi(ti + tq). Owing to inequalities (38), (39) we have
z° (ti + tq) = zi(ti + tq) + x, Z° (ti + tq) = Z-(ti + tq) + y,
S — {a — («1, a.2) G IR \a.1 + «2 — 1}.
According to (34),
min I«1(zf(ti + tq) + x) + a2(Zei(ti + tq) + y)|| > Pi.
On the other side for af = y^-,
IIai(zei(ti + tq) + x) + a*^(Zei(ti + tq) + y)|| =
(4(U + t), zi(ti + t)) = -IIzi(ti + t)||zi(ti + t)||.
(41)
Let us assume the opposite, there are
TQ G [Yi, Ti], q G IR1, q > 0,
(42)
(43)
i.e. vectors z°(ti + tq), Z°(ti + tq) can be shown as
where x, y G &-S, S C IR" full-sphere with the radius 1 and with center in the origin. Let
Therefore the inequality (41) holds true for any t G [7^, t].
Let us define sequence {Tj+1}(f=i+1 at the instant moment t = ti + Yi on 6+1 in follows:
6 T i+1
Ti+1 — min-fri+1 Ti+k — i+1 h > 9
Ti+1 — nimiTi > 2 i+1 ~ 2fc_1’ -
It is clear that
n-i
Y.T+k <i+i. k = 1
We construct new sequence on the basis of the previous sequence
r s;i+k\to 0i+k = XTi+k
{0i+1 }k=1, 0i+1 = 0Ti+1 •
Let (5j+i = Tj_|_i = t*^1. Then we have (5j+i < r*^1 < In respect
that (42) we have inequalities (31), (32).
Thus we choose the control v(s) which is defined at s G [ti,ti + 7^ from rela-
tions (24) and
v(s) = Ui(s), s G [ti + Yi,ti + Ti)^ y [tp,tp + Tp),
p=i+1,...,n
if i < n,
v(s) = Un(s), s G [tn + Yn,tn + Tn).
Then we have ||^i(t)y = 0 at t G [ti,ti + t\] and (zi(t),p) < 0 at t > ti + t\, i = 1,...,n.
Note that the approaching of evader with pursuers can occur not later than
max (z0,p)
i=1,...,n
0 ^----------x-----•
0
The theorem is proved. □
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