CC BY
10A. A. Azamov1 and B.T. Samatov2
1 Institute of Mathematics and Informational Technologies of Academy of Sciences of Uzbekistan E-mail: [email protected] 2 Namangan State University, Uzbekistan E-mail: [email protected]
Abstract. The notion of the strategy of parallel pursuit (briefly n-strategy) was introduced and used to solve the quality problem in ”the game with a survival zone” by L.A.Petrosyan. Further it was found other applications of n-strategy. In the present work n-strategy will be constructed in the cases when 1) a control function of Pursuer should be chosen from the space L2 and that for Evader should be chosen from L; 2) control functions both of players should be chosen from the space L2; 3) a control function of Pursuer should be chosen from intersection of spaces L2 and Lwhile that for Evader should belong to L.
Keywords: differential game, Pursuer, Evader, strategy, parallel pursuit, domain of attainability, survival zone AMS classification numbers: 91A23, 49N70
1. Formulation of the problem
Consider the differential game when Pursuer P and Evader E having radius vectors x and y correspondingly move in the space R". If their velocity vectors are u and v then the game will be described by the equations:
X = u, x(0) = xo, , >
y = v, y(0) = yo, ( )
where x, y,u,v € R", n > 1, xo and y0 are initial points of x and y correspondingly.
The family of all measurable functions u(-) : R+ ^ R" (controls of P) satisfying the following condition
o
is denoted by Up, Up C L2[0, to). The family of all measurable functions u(-) : R+ ^ R" satisfying the next condition
is denoted by Ua, Ua C Lœ[0, to). Further the family of all measurable functions satisfying both of conditions (2) and (3) will be denoted Up.
Analogously the family of all measurable functions v(-) : R+ ^ R" satisfying
TO
(2)
\u(t)\ < a almost everywhere (a.e.) t > 0,
(3)
TO
is denoted by Vp, Vp C L2[0, to) and satisfying the following condition
|v(t)| < a a. e. t > 0 (5)
is denoted by Vp, Vp C LTO[0, to). Besides the family of all measurable functions satisfying both of conditions (4) and (5) also will be denoted Vp.
In the theory of Differential Games an inequality of the forms (2) and (4) is usually called an integral constraint for control function (we will briefly say I-constraint). Analogously an inequality of the forms (3) and (5) is called a geometric constraint (briefly G-constraint).
The condition consisting of both inequalities (2)-(3) (or (4)-(5)) will be called a complex constraint (briefly C-constraint).
If U (correspondingly V) is one the introduced classes then there would be 9 possibilities of pairs (U,V) each of them generates the appropriate variant of simple motioned pursuit-evasion game. For brevity we will indicate this adding corresponding abbreviations to the word ”game” as a prefix. For example the pair (Ua, Vp) defines G-game, (Up, Vp) does IG-game and e. c.
We are going to study mainly the game with phase constraints for Evader being given by a subset L of R" which is called ”a Survival Zone” (for Evader naturally). (Notice that in the case L = 0 we have a simple pursuit-evasion game.)
Each pair (x0,u(-)) consisting of an initial position x0 and a control function u(-) € U (correspondingly (y0,v(-)), v(-) € V) generates the path by the formula
t t x(t) = x0 + | u(s)ds, (y(t) = y0 + /v(s)ds).
0 0
Further we will assume initial positions x0 and y0 are given such that x0 = y0 and y0 € L.
In the game ’with the Survival Zone L’ Pursuer P aims to catch Evader E, i.e. to realize the equality x(t) = y(t) for some t > 0, while E stays in the zone R" \ L. The aim of E is to reach the zone L before being caught by Pursuer or to keep the relation x(t) = y(t) for all t,t > 0. Notice that L doesn’t anyway restrict motion of P.
The Differential Game with ’a Survival Zone’ was suggested by R. Isaacs in (Isaacs, 1 965) for G-game. He solved it when L was a half-plane and formulated the Problem 5.9. 1 about the case when L is a disk. The game solved by L.A.Petrosyan for an arbitrary convex survival zone introducing the strategy of parallel pursuit (briefly n-strategy; the prefix n can be interpreted as meaning the word ’parallel’ and the initial letter of the family name of its author as well). About further studies, see (Azamov, 1 986) - (Azamov and Samatov, 2000), (Petrosyan, 1 977) - (Petrosyan and Dutkevich, 1969), (Satimov, 2003)- (Samatov, 2008).
Well known that control functions for P are not sufficient to solve a pursuit problem as they depend only on time-parameter t, t > 0 so the suitable types of controls should be strategies. There are different ways of defining such a notion. For us it is enough the following conception.
Definition 1. The map u : V ^ U is called the strategy for P if the following properties are held:
1° (admissibility). For every v(-) € V the inclusion u(-) = u[v(-)j € U is true.
2° (volterranianity). For every v1(-),v2(-) € V and t, t > 0, the equality v1 (s) = v2(s) a.e. on [0, t] implies u1(s) = u2(s) a.e. on [0, t] with uj(-) = u[v*(■)], i = 1, 2.
Definition 2. A strategy u(v) is called winning for P on the interval [0,T] in the simple game if for every v(-) € V there exists a moment t* € [0,T] that is to reach the equality x(t*) = y(t*).
Definition 3. A strategy u(v) for the player P is called winning on the interval [0, T] in the game with ”the survival zone” L if for every v(-) € V there exists some moment t* € [0,T] that
1) x(t*) = y(t*);
2) y(t) € L while t € [0,t*].
Definition 4. A control function v*(-) € V for the player E is called winning in the game with ”the survival zone” L if for every u(-) € U: 1) there exists some moment t,t> 0, that y(t) G L and x(t) ^ y(t) while t G [0, i]; or 2)x(t) ^ y(t) for all t > 0.
2. The nG-strategy
2.1. Definition of the nG-strategy
The main aim of the present paper is to construct analogies of n-strategy for IG-,I- and CG-games. In this section the definition and some properties of n-strategy settled by L.A.Petrosyan will be given for fullness of the exposition using analytical approach (see (Petrosyan, 1977)).
The G-game is a problem on simple pursuit with geometric constraints [ul< a and |v| < 3 for the control functions of the players P and E respectively .
Let z = x — y. In order to define the nG-strategy consider a state vector Z = (x, y) such that x = y and suppose E holds the constant control v,v € S p where Sp = {v : |v| < 3} is a ball. If P also applies the constant vector u, |u| = a, helping P to meet E at some moment T,T > 0, then Tu = Tv — z or
u = v — A£ (6)
where A = ^yT and £ = z/|z|. From (6) the following equation for A will be obtained: A2 — 2(£, v) A — a2 + v2 = 0, where < v, £ > is the scalar production of the vectors v and £ in R". It has the positive root Aa(z,v) = (£, v) + \JDq(z, v) if and only if
Dg(z, v) := (£,v)2 + a2 — v2 > 0. (7)
Substituting this value into (6) adduces to the formula
u = v — Ag(z, v)£ =: uG(z,v). (8)
Definition 5. The function uG(z,v) defined on the region (7) is called the strategy of parallel pursuit in the G-game or briefly the nG-strategy.
Let the initial state Z0 be fixed, z0 = x0 — y0 and E applies the arbitrary control v(-) € V p while P runs the nG-strategy. Then the dynamics of the vector z will be described by the Cauchy problem:
z = x — y= — Ag(z, v(t))£, z(0) = z0. (9)
Obviously, the conditions of the existence theorem of Caratheodory (Alekseev et al., 1979) for the problem (9) are valid therefore that has the unique solution. Denote it z(t,z0,v(-)) or simply z(t) and call a trajectory.
The following statement explains the term ’’parallel pursuit” for the nG-strategy (Petrosyan, 1965, 1977).
Theorem 1. For every z0, z0 = 0, and v(-) € Vp the formula z(t) = ^G(t)z0 is
t
true where AG(t) = 1 — tj-t f AG(zo, v(s))ds.
0
Corollary.
uG(z(t),v(t)) = ug(z0, v(t)), t > 0. (10)
Indeed, uG(z,v) is homogeneous on the variable z.
Henceforth ^G(-) will be called the approach function in the G-game while Ag(z, v) can be interpreted as the speed of approach.
Theorem 2. In the G-game nG-strategy is wining for P on the interval [0,Tg], where Tg = |z0|/(a — 3).
2.2. nG-strategy and Apollonian sphere
The first essential application of nG-strategy was brought out to the game with survival zone (Petrosyan, 1965, 1977) basing on an interesting link between nG— strategy and Apollonian sphere.
To expose this link let us consider the region
AG{x,y) = ^w\ \w - x\ > ^\w - y|j ;
what’s boundary is just Apollonian sphere. Bringing its equation to the form ^ — cG = Rg one easily can find the center cG (x, y) and the radius of Apollonian sphere (Azamov, 1986): CG(x,y) = (a2y — 32x)/(a2 — 32), R^(x,y) = aP^/^2 — 32^
It's easy to proof using (10) the following property
Theorem 3. AG(x(t),y(t)) = x(t) + ^G(t)[AG(x0,y0) — x0] for t € [0,t*], where t* = min{t : z(t) = 0}.
Theorem 4 (monotonicity of Apollonian Sphere (Petrosyan, 1965, 1977)).
If 0 < t1 < t2 then AG(x(t1),y(t1)) D Ag(x(t2),y(t2)).
Below in the section 6 the direct analytical proof of these assertions will be represented for more general situation.
2.3. G-game with ”a Survival Zone”
It has been noted above that Isaacs’ game with ”line of survival” was solved by L.A.Petrosyan using the method of approximating measurable controls by piecewise constant.
Theorem 5 (L.A.Petrosyan’s theorem). If a > 3 and L n AG(x0,y0) = 0
then the nG-strategy is winning in G-game with the Survival zone L.
P r o o f follows from Theorem 3.
Comment. The condition of Theorem 4 is necessary the player P to win. This is obvious for the condition a > 3. If Lf] AG(x0, y0) = 0, for example the intersection contains a point w, then E is able reaches w by the constant control v = 3(w — y0)/W — y01 and wins the game.
3. nJG-strategy
Now let us consider IG-game where controls of P should satisfy integral constraints while controls of E do geometric constraints. Our aim is to transfer constructions of L.A.Petrosyan to IG-game.
3.1. Definition of nIG-strategy
First of all we are to notice in IG-game as distinct from the G-game the current phase state Z should also include besides geometrical positions x, y (or reduced phase parameter z = x — y if this is enough) the current resource ~p of the player P defining by the formula
t
p(t) = p2 — j |u(s)|2ds.
0
Therefore in IG-game a state vector must be the triple ( = (x,y,p). Choosing the vector u and the number T from the conditions: Tu = Tv — z, T\u\2 =~p one gets the following quadratic equation for the variable A = yyT:
a2 — 2A(p/2+ (£,v)) + |v|' = 0 (11)
where p = ~p/\z\, £ = z/\z\.
Further in this section we will suppose p > 43. This implies the discriminant D(z,~p,v) := (p/2 + (£,v))2 — \v\2 is nonnegative and the equation (11) has the positive root AIG(z,p,v) = p/2+ (£,v) + \JD(z,p,v).
Though the obtained formula for AIG is some bulky it takes part in the following
Definition 6. The function ujG(z,~p,v) := v — AiG(z,~p,v)£ defined in the region p > 43 is called the nIG-strategy in the IG-game for the player P.
Let us suppose that a perty in IG-game starts from a initial state Zo = (x0, y0,p2) and P applies nIG-strategy while E may choose any control function v(-) € Vp. These data naturally generate the augmented trajectory x(t), y(t), ~p(t) describing the party in IG-game. Now put z(t) = x(t) —y(t). Then the path (z(t),~p(t)) will be found as the solution of Cauchy problem
j dz/dt = -XIG(z,p,v(t))z/\z\, z(0) = zo,
\ dp/dt = -XIG(z,p,v(t))p/\z\, p(0)=p2, ^ ^
which is valid while p > 43 and z = 0.
Theorem 6. If D(z(t),~p(t),v(t)) >0 a.e. on some interval [0,t*),t* > 0; then
a) p(t) = p0, where p0 = p2/|z0|;
b) uIG(z(t), p(t), v(t),) = uIG(z0, p2,v(t))-,
c) ^ig^o^^^u^))2 = p0Aig(z0, p2,v(t)).
P r o o f is immediately followed from the equation (12).
This statement allows to redefine nIG-strategy in more simply way putting
uig(v) = v — Aig(zo, p2, v)£o.
3.2. IG-game without phase constraints
Let an initial position and control v(-) € Vp of E be given and P run nIG-strategy. Then one has the generated trajectory z (t) and can define the approach function in IG-game by the following way
t
AIG(t) = l--—f \IG(z0, p2,v(s))ds.
|zo| J
0
Denote TIG = \z0\/( f - ¡3 + \Jx - poP).
The following statements expresses an analogy of the properties of nG-strategy for IG-game.
Theorem 7. If 43 < po then for every v(-) € Vp the following properties are true
a) z[t) = AIG(t)z0; p(t) = AIG(t)p2;
b) there is t* € [0, Tig] such that AIG(t*) = 0 and AIG(t) > 0 while t < t*.
Theorem 8. If 43 < po then in the simple motion IG-game the niG-strategy is winning on [0,Tig].
P r o o f of these theorems in more general situation will be given in the section 5.
3.3. Pascal’s snail and Cartesian’s oval
As in G-game in order to solve IG-game with a survival zone following L.A.Petrosyan we will try to find the set of all points where P is able to reach earlier than E supposing they start their motions from some current positions x, y respectively.
Let E moves holding a constant vector v and P aims to catch it by a constant control u as well spending over his resource ~p. Let w be a point where P meets E. The set of all such points will be described by the following relations: | w —
x\ = T\u\, \w — y\ = T\v\, T\u\2 = ~p. Excluding T and |w| one gets the surface |w||w — x\2 = ~p\w — y| that bounders the searching region
AiG(x, y,~p) = {w : \w-x\2 > {~p/¡i)\w-y\)
For n = 2 the boundary of the region AIG will be Cartesian oval (if 43 < p) or the inner loop of Pascal's snail (if 43 = p). It is interesting to remember here that such curves appeared in G-game with a survival zone considered under terminal condition of /-capture (Petrosyan and Dutkevich, 1969).
3.4. IG-game with ”a Survival Zone”
Now let P hold the nIG-strategy while E can apply any control v(-) € Vp. Naturally an initial position is being considered.
Theorem 9. The following properties take a place:
A. AIO(x(t), y(t), p(t)) = x(t) + AIG(t)[AIG(x0, yo, p2) - x0\.
B. If 0 < ti < t2 then AIG(x(ti),y(ti),~p(ti)) D AIG{x{t2),y{t2),p{t2)).
The statement will be also proven below in the section 5. It implies
Theorem 10 (Petrosyanian type theorem). If 43 < p0 and
Lf] Aig(xo, yo, p2) = 0 then niG-strategy is winning in IG-game with ”the Survival Zone” L (for E).
Comment. The condition of the last theorem is necessary, too, because one can easily notes that if a) 43 < p0 and L p| AIG(x0,y0,p2) = 0 or b) 43 > p0 then the Player E would win the game holding simply some constant control v* , |v*| = 3.
4. ni-strategy
4.1. Definition of ni-strategy
Now we consider I-game where admissible controls of both players should satisfy integral constraints as u(-) € Up, v(-) € Va (see (Ushakov, 1972)). Since considerations principally repeat those provided for G- and IG-games here they will expose somewhat shorter. In I-game a state vector should be a quadruple ( = (x,y,p,a) including current values of the rest of resources of the players P and E respectively, i.e.
t t p(t) = p2 — J \u(s)\2ds, a(t) = a2 — J |-y(s)|2<is.
00
Arguments, like above, lead to the following system to build the n-strategy for the /-Game: Tu = Tv — z ~p—T[u\2 = a — T\v |2, T> 0 oiu = v — A£, |m|2 —|w|2 = A(5 where A = \z\/T, S=(p — a)/\z\. Hence
A = S + 2{£,v). (13)
Obviously, the formula (13) has sense when S + 2(£, v) > 0. If S + 2(£, v) < 0 we put A = 0. Unifying two cases gives the next formula
Ai(z, S, v) := max{0, S + 2(£, v)}.
Definition 7. The function uI(z,S,v) := v — AI(z,S,v)£ is called nI-strategy in the I-game.
Let the initial state Z = (x0,y0,p2,a2) be given and z0 = x0 — y0. Suppose P applies nI-strategy while E may choose any control function v(-) € Va. These data naturally generate the augmented trajectory x(t),y(t),~p(t),a(t) describing the party in I-game.
Now put z(t) = x(t) — y(t) and S(t) = Q(t)/\z(t)\, where 0(t) = ~p(t) — a(t), 8(0) = 0o = p2 — a2. Notice ¿o = ¿(0) = 6q/\zq\. Then the path (z(t),0(t)) will be defined as the solution of Cauchy problem
{dz/dt = —Xi(z, 5, v(t))z/\z\, z(0) = zo, dd/dt = —Xi(z,5,v(t))Q/\z\, 0(0) = Q0.
The problem (14) has the unique solution while z = 0.
Theorem 11. If 5(t) > 0 a.e. on some interval [0,t*),t* > 0, then
a) 5(t) = 5o;
b) ui(z(t), 5(t), v(t)) = ui(zo, 5o, v(t));
c) \ui(zo,5o,v(t))\2 = \v(t)\2 + Xi(zo,5o,v(t))5o.
Proof is easily follows from the equation (14)
Thanks to the property b) the function
ui (v) = ui (zo, 5o, v) = v - Xi (zo, 5o, v)£o can be used as fff-strategy instead of uI(z, 5, v).
4.2. I-game
Here the following function
t
1
Ar(t) = 1 - J A/(z0, S0, v(s))ds
(14)
0
services as the measure of approach of P to E.
Theorem 12. z(t) = Aj(t)z0, 6(t) = Aj(t)60.
Theorem 13. If p > a then nj-strategy is winning in the interval [0,Tj] where Tj = \zo\2/(p - a)2.
Proof. Applying the inequality of Cauchy to the scalar production in (13) gives the estimation Aj(t) < 1 — max{0, Sot — 2a^/t]/\zo\. That’s why there exists t*,
(^) <t* <T/ such yl/(t*) = 0.
Further we are to verify admissibility of the corresponding realization of nj-strategy when E runs his admissible control v(-) G Va. Thus
if it it
J \uj(v(s))\2ds = J |v(s)|2ds + So j Xj(v(s))ds < a2 + ¿o\zo\ = p2,
where t* = min{t : Aj(t) = 0} Q.E.D.
4.3. I-game with a "Survival Zone”
Let the current state ( = (x,y,p,a), x ^ y, be considered and E be supposed moving by a constant velocity v. Consider the situation when P tries to catch E also by a constant control u unless the players spend their resource fully. If w is the point where P meets E and T is the time of occurrence of such event then \w — x\=T\u\, \w — y\ = T\v\, T\u\2 = p, T\v\2 = a.
In the case ~p > a the exclusion of T, |w| and |-y| from the relations just written gives the equation of the Apollonian sphere \w — cj\ = Rj where cj = y — <r£q/5q and RT = yj~p a/So-
Denote
Ai (Z)
{w : |w — C[ | < i?/} if p > a, {w : (2w — x — y, z) < 0} if ~p = a.
And now consider the party in the I-game when P holds nI-strategy while E applies some his admissible control v(-) G Vg accounting some initial position is fixed.
Theorem 14. Ai(Z(t\)) D Ai(Z(t2)) for 0 < t\ < t2.
The Proof is elementary and differs in the cases p > a and p = a but requires long calculations that’s why will be omitted here.
Theorem 15 (Petrosyanian type theorem). If p > a and Ln Ai(Zo) = 0 then the ni-strategy is winning for P in the I-Game with ”the Survival Zone L.”
Comment. If either a) p > a and L n Ai (Zo) = 0 or b) p < a then there exists a control function v* (t) G Va being considered as a particular sort of strategies of E is winning for E in the I-Game with ”the Survival Zone L”.
5. ^cG-strategy
Further it will be considered the game (1) under the following conditions
u(-) G UP, v(-) G Vg (15)
called complex-geometric constraints. As we have come to an agreement the game (1), (15) will be called CG-game and studied from the view of the theory of L.S. Petrosyan.
Let p = (a, ¡3, p) G P, where
P = {p G R+ : f (3, p) < a, p > 43, a > 0,p > 0},
p p2 p2 f(3,n) = 2 - VT “M/3, >1=]z~oP Z° = X° ~ y° ^ °'
Then for all v, |w| < /3, the following functions are defined correctly
Ag(v) =< Zo, v > +v/< Zo, v >2 +a2 - \v\2, Zo = jf^y,
Xig(v) = % + < Zo, v> +•'/(2+ < v >)2 - \v\2,
Xcg(v) = min {XG(v),XIG(v)} , (see sections 2.1 and 3.1).
Definition 8. The function
uco(v) = v — \co(v)£o (16)
is called the strategy of parallel pursuit in CG-game (briefly, nco-strategy) .
The following theorem reinforces theorems 2 and 8.
Theorem 16. If p € P then the ncG-strategy is winning for P on [0,Tcg] where Too = max {Tg, Tig}.
Proof. An initial position takes part in the condition p € P. Suppose that the player P runs nCG-strategy while E may apply ani his admissible control v(-) € Vp. They generate appropriate paths P : x(t) and E : y(t). Putting z(t) = x(t) — y(t) the definition of the nCG-strategy (see (16)) implies
z(t) = Aco(t)zo, (17)
t
where AGG(t) = 1 — 177/ AGG(-y(s))cis. The formula (17) explains the term ”parallel
I z° I 0
pursuit” for the nCG-strategy. One can write the inequality
t
AGG(t) < 1 - 7—r min ACG(v(s))ds N OeVßJ
lCG s , _ , ,
v(
on the right side of that it is sitting the problem of minimizing called an elementary control problem (Alekseev et al., 1979, p.360). Solving it, one gets the following estimation
Aco{t) < 1 - —r min Acg(v).
\zo\ I v |<¡3
Because of the condition p € P it is obvious
min A cg(v) = min \a- ß, 7; ~ ß + \ ~.------------vß \ ■
\v\<ß I 2 V 4 I
Henceforth Acg(t) < 1-t/TcG. As AcG(t) is continuous in [0,TGG] and ACG(0) = 1 there exists t* € [0,TCG] such that AcG(t*) = 0 or x(t*) = y(t*). Let t* be the
minimal value of such t*. Notice u = 0 for t > t*.
Now we are to prove admissibility of ffCG-strategy on [0, t*]. For that notice
, ( N |2 i \v - AG(v)io ? if Ag(v) < XIG(v), ( )
|UCG(v)l n \v - A/g(v)£o\2 if Aig(v) < Xg(v), (18)
die to the definition of ffCG-strategy. Revealing squares in (18) and due to the
definitions of Ag(v) and Aig(v) one can rewrite (18) in the next form
|U (v) |2 = \ a2 if AG(v) < AIG(v), (19)
|Ucg(v)\ \ i^Aig(v) if A/g(v) <Ag(v). (19)
But here the inequality Aig(v) < Ag(v) implies the estimation ¡iAig(v) < a2. Thus,
\ucg(v)\2 < a2, i.e. ucg(v(-)) € Ua.
We should also show that ucg(v()) G Up on [0,t*]. For that let us divide the interval [0,t*] into two parts by following way
E< = {s : Ag(v(s)) < Aig(v(s))},E> = {s : Ag(v(s)) > Aig(v(s))}.
It is clear that this sets are measurable. We have
t*
J |ucg(v(s))|2^s = J a2ds + ^ J AiG(v(s))ds.
0 E< E>
Taking into account a? < ^Ag(v(s)) for s G E< and p G P we get
t* t*
J |ucg(v(s))|2ds < n J min{AG(v(s)),AiG(v(s))}ds =
00
= p2 (1 - AcG(t*)) = p2
Q.E.D.
6. CG-game
We continue to suppose that the conditions of Theorem 1 take place and consider the situation when the player E moves from a position y with a constant velocity v, |v| < ¡3 while P holding nCG-strategy from a position x having some current quality of his resource ~p. Let w be a point where P would meet E. The set AcG(x,y,~p) of all such points can be described by the following manner
a) if XG(v) < XIG(v) then ACG(x,y,p) = AG(x,y);
b) if XIG(v) < XG(v) then ACG(x, y,~p) = Aig(x, y, p).
We denote A*CG(x,y,~p) the union of sets AG(x,y) and AjG(x,y,~p).
Now consider the party when P applies nCG-strategy (16)and some control
v(-) G V/3, on the interval [0,t*] is chosen by E till the time-moment t* = min{t :
Aca{t) = 0}
Let x(t), y(t) are the current positions of the players and ~p(t) is the current resource of the pursuer P defining as
t
P(t) = P2 ~ J \uca(v(s))\2ds, p(0)=p2, t> 0. (20)
0
We are going to study the dynamics of the attainable domains
AcG(t) = x(t) + AcG(t)(AcG(0) - xo), ACG(t) = Ag(t) U AiG(t),
where
Aa(t) = AG{x{t), y(t)), AIO(t) = AIO(x(t), y(t), p(t)),
Ag(0) = Ag(xo, yo), Aig(0) = Aig(xo, yo, p2),
Acg(0) = Ag(0) U Aig(0), Acg(0) = ACg(0).
Theorem 17. A*CG(t) C ACG(t) while t G [0,t*].
Proof. Obviously
ACG(t) — x(t) = (AG(t) — x(t)) U (AIG(t) — x(t)).
Using formulas (17), (20) and the inequality ~p(t) > p2Aca(t) we have Ag(t) — x(t) = {w : |w| > (a/3)lw + AcG(t)zol} = AcG(t)(AG(0) — xo), AIO(t) - x(t) = {w : \w\2 > (p(t)//3)\w + ACa{t)z0\} C C {w : Wl2 > (p2AcG(t)/3)lw + AcG(t)zo\} = AcgW(Aig(0) — xo), those imply the desired result. Q.E.D.
Theorem 18. The set-valued function coAcG(t) is monotonically nondecreasing on the interval [0,t*] in the sense of order by inclusion. (coA denotes the convex hull of the set A.)
Proof. Using the condition |v| < 3 and relations (18)-(19) we obtain lv — Ag(v)^oI> alvl/3 if Ag(v) < Aig(v)
and
\v - A/g(«)£o| > \JpXio{v)\v\/¡3 if XG(v) > XIG(v).
In its turn this assertions are equivalent to the following ones
Izolv + AG(v)yo G Ag(v)Ag(0) if Ag(v) < Aig(v)
and
Izolv + AiG(v)yo G Aig(v)Aig(0) if Ag(v) > Aig(v).
Hence |zo|v+ACG(v)yo G Acg(v)Acg(0). Now using the support function C(A,p) = sup < w,p > (Blagodatskiy, 1979) it easy to see
wEA
< Izolv,p > —Acg(v)C(Acg(0) — yo,p) < 0 for all p, lpl = 1. Hence
< v - Xcg(v)£o,P > -TJ,-rAGG(i;)C'(AGG(0) - x0,p) = C{ACa{t),P) < 0 lzol dt
Q.E.D.
Theorem 19 follows the main result of the paper concerning to CG-game with a Survival Zone, being denoted L as before.
Theorem 19 (Petrosyanian type theorem). If p G P and coAcg (0) H L = 0
then a strategy Ucg(v) for the player P is winning on the interval [0,Tcg] in the game with ”the survival zone” L.
Comment. Here the property of necessity of the condition 'p G P and coACG(0) H L = 0’ is true as well. Moreover as in the case p G P and ACG (0) H L = 0 and so in the case p G E = R+ \ P the player E is able to solve the evasion problem using some constant control v*, |v* | = 3.
The conclusion. It has been considered the pursuit-evasion game with different constraints for both Pursuer and Evader described by the simplest differential equation (1) only. The circumstance allowed us to construct winning strategies explicitly and to solve nontrivial problem of pursuit when a Survival Zone for Evader is present. Even for the simplest dynamics (1) there are problems staying open. For example we don’t know how to solve pursuit problem for GC-,CI-,IC-,GI- and C-games with a Survival Zone.
Finally the authors thank prof. L.A.Petrosyan and prof N.A.Zenkevich for attention and support.
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