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UDC 517.977 Вестник СПбГУ. Прикладная математика. Информатика... 2024. Т. 20. Вып. 2
MSC 49N79, 49N70, 91A24
Differential game with a "life line" under the Gronwall constraint on controls
B. T. Samatov1, A. Kh. Akbarov2
1 Namangan State University, 316, Uychi ul., Namangan, 116019, Uzbekistan
2 Andijan State University, 129, Universitetskaya ul., Andijan, 170100, Uzbekistan
For citation: Samatov B. T., Akbarov A. Kh. Differential game with a "life line" under the Gronwall constraint on controls. Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 2024, vol. 20, iss. 2, pp. 265-280. https://doi.org/10.21638/spbu10.2024.211
We study the pursuit-evasion and "life line" differential games of one pursuer and one evader, whose controls are subjected to constraints given by Gronwall type inequalities. It is said that an evader has been captured by a pursuer if the state of the pursuer coincides with the state of the evader. One of the main aims of this work is to formulate optimal strategies of players and define guaranteed capture time. Here a strategy of parallel convergence (briefly, П-strategy) for the pursuer is suggested and proved that it is optimal for pursuit. To solve the "life line" problem we will investigate dynamics of the attainability domain of players by Petrosyan method, that is for the attainability domain, conditions of embedding in respect to time are given. This work grows and maintains the works of Isaacs, Petrosyan, Pshenichnyi, Azamov and other researchers.
Keywords: differential game, pursuer, evader, Gronwall constraint, strategy, parallel pursuit, attainability domain, "life line" game, the Apollonius sphere.
1. Introduction. In the theory of differential games, problems of pursuit-evasion occupy a special place due to a number of specific qualities. In the works [1, 2], this quality was clearly manifested in the construction of the fundamental theory of differential games and in a number of model problems. The book [3] contains specific game problems that were discussed in details and proposed for further study. One of these problems is the so-called "life line" problem that was initially formulated and studied for certain special cases in the book ([3], Problem 9.5.1). For the case when controls of both players are subject to geometric constraints, this game has been rather comprehensively studied by L. A. Petrosyan [4]. In the monograph [4], the notion of strategy of parallel pursuit (briefly, П-strategy) was introduced and used to solve the quality problem for the game with a "life line". This strategy proposed in a simple pursuit game with geometric constraints became the starting point for the development of the pursuit method in games with multiple pursuers (see e.g. [5-21]).
In the theory of differential games, control functions are mainly subjected to geometric, integral or mixed constraints (see [22-26]). However, differential type constraints on controls are also arisen in some applied problems such as ecological, technical problems [27, 28].
In the work [29], the concept of Gr-constraint on controls of players, which in a certain sense, generalizes geometric constraints, is introduced. The present work proposes Gronwall type constraints on controls of players for differential games of pursuit-evasion and for solution of the "life line" game by Petrosyan method. The main purpose of this work is to construct the П-strategy of pursuer, and to find the attainability domain of players, and also to give analytical solution of the "life line" problem in this case.
© St. Petersburg State University, 2024
2. Statement of problem. In the present paper, controls of the pursuer and evader are subjected to the following Gronwall constraints [29, 30]:
t
|u(t)| < p0 + pit + k J |u(s)|ds for almost every t > 0, (1)
0
t
|v(t)| < a0 + ait + kj |v(s)|ds for almost every t > 0, (2)
0
respectively, where p0, a0, pi, ai, k are given positive numbers.
Note that in (1) and (2) and in further constraints, as the norms of the control vectors u and v in the space R™, we will consider the usual Euclidean norm, i.e. |w| = + u\ + ... + «2, where wi,w2, ...un are the coordinates of the vector u in the space R™, and = v^i + + ... + here ...vn are the coordinates of the vector v in
the same space Rn .
Remark. For the cases pi = ai = 0, the pursuit-evasion and "life line" problems with Gronwall constraints on controls have been completely studied in the work [29]. Let motion equations of pursuer P and evader E be given by the followings:
x = u, x(0) = X0, (3)
y = v, y(0) = y0, (4)
correspondingly, where x, y, x0, y0, u,v G Rn, n > 2, x0 = y0.
Definition 1. A function u(-) = (ui(-), u2(-),..., u„(-j) is called an admissible pursuer control in game (3), (4) if it satisfies condition (1). Similarly, a function v(-) = (vi(-), v2(-),..., vn(-)) is called an admissible control of the evader in game (3), (4) if it satisfies condition (2).
The set of all admissible controls of the pursuer and the evader is denoted by the symbols UGr and Vcr, respectively. Then the pairs UGr and Vcr form the motion trajectories
c(t) = xo + j u(s)ds, y(t) = yo + j v(s)ds
of the pursuer and the evader, respectively.
Definition 2. A function u : R+ x Rn ^ Rn is called a strategy of the pursuer if u(t, v) is a Lebesgue measurable function with respect to t for each fixed v and is a Borel measurable function with respect to v for each fixed t.
Definition 3. It is said that a strategy u = u(t, v) guarantees capture at time moment T(u) if at some time t* G [0, T(u)] an equality x(t*) = y(t*) is satisfied for any control v(-) G VGr of the evader, here x(t) and y(t) are the solutions of the initial value problem
x = u(t,v(t)), x(0) = x0, y = v(t), y(0) = y0,
where t ^ 0.
Definition 4. A function v : R+ ^ Rn is called a strategy of evader if v(t) is a Lebesgue measurable function with respect to t.
Definition 5. We say that a strategy v(t) is called winning for evader in the Gr-game of evasion on [0, if for any control of pursuer u(t) £ UGr the condition x(t) = y(t) holds for all t ^ 0, here x(t) and y(t) are the solutions of the initial value problems
x = u(t), x(0) = xo,
/ = v(t), y(0) = yo-
We use the following statement.
t
Lemma (see [31]). If \u(t)\ < a + /(/3 + 7|w(s)|)eis, then |w(i)| < ¿(e^ - l) + ae~<t,
o 7
where w(t), t ^ 0, is a measurable function, and a, ft are given non-negative numbers and Y is a given positive number.
By this Lemma, if u(-) £ UGr and «(•) £ VGr, then
|u(t)| < p(t), t > 0, (5)
|v(t)| < V(t), t > 0, (6)
where
^) = ^(efci-l)+Poefci, ^(0)=po, (7)
m = ^(ekt~ l)+*oekt, V(0) = o-o. (8)
It can be easily checked that the converse is not true, that is, the inequalities (5) and (6) do not imply (1) and (2). To define the notions of optimal strategies of players and optimal pursuit time, we consider two games.
The goal of the pursuer P is to capture evader E, i.e. achievement of the equality x(t) = y(t) (Pursuit problem) and the evader E strives to avoid an encounter (Evasion problem), i.e., to achieve the inequality x(t) = y(t) for all t > 0, and in the opposite case, to postpone the instant of encounter as long as possible.
This paper is devoted to solving the following problems for Gronwall type constraints on controls.
Problem 1. Solve Pursuit problem in the game (3), (4) with the Gronwall type constraints (1) and (2) (briefly, Gr-game of Pursuit).
Problem 2. Solve Evasion problem in the game (3), (4) with the Gronwall type constraints (1) and (2) (briefly, Gr-game of Evasion). Problem 3. Solve the differential game of "life line".
3. A solution of the Pursuit problem. In this section, we construct the optimal strategy for pursuer and give a solve of the Pursuit problem.
To construct a strategy for pursuer, first we assume that pursuer knows t, v(t) at the current time t.
Definition 6. If S0 ^ 0, ¿i ^ 0, then the function
UGr(t, v) = v - r(t,v)£o (9)
is called a IIGr-strategy of the pursuer in the Gr-game of pursuit, where r(t,v) = (v,£o) + V(v^o)2 + V2(t) - \v\2, £o = z0/\z0\, 60 = po ~ (To, ¿1 = Pi ~ o~i, z0 = x0 - yo, (v,£o) is the scalar product of the vectors v and £0 in the Euclidean space Rn. Note that
|UGr(t,v)| = ^(t). (10)
Indeed, if we square equalities (9) on both sides, then we get
|uGr (t, v)|2 = |v|2 + r(t, v)(r(t, v) - 2(v, Co ».
From here and from the form of the scalar function r(t,v), it is easy to calculate equality (10). Let us now check the admissibility of strategy (9) for every admissible function v(t) G Rn, t > 0. From inequality (1) and from equalities (7) and (10) we have
t t Po + Pit + kj |uGr(v(s),s)|ds = po + pit + kj <f(s)ds =
0 t
= P0+Plt + kJ \^(eks - 1) + p0efcs] ds = <p(t) = luGrMt),t)\,
0
which proves the admissibility of strategy (9).
Proposition 1. If ¿0 ^ 0, ¿i ^ 0, then the function r(t,v) is continuous and nonnegative for all (t, v) e [0, to) x Rn.
Proposition 2. For every z0, z0 = 0, and v(-) e VGr, there exists a scalar function R(t, v(-)) such that z(t) = z0R(t, v(-)), where z(t) = x(t) — y(t).
Proposition 3. Let $(t) = A{ 1 - ekt) + Bt + 1, where A = B = If
S0 ^ 0, > 0 or S0 > 0, ^ 0 is valid, then the function $(t) is monotone decreasing in t,t ^ 0, and there exists unique positive root of the equation
$(t)=0 (11)
with respect to t. Here we call a guaranteed capture time the positive root of equation (11) and denote it by TGr.
We prove the statements.
Theorem 1. If 50 > 0, > 0 or ¿0 > 0, > 0 is valid, then the nGr-strategy guarantees capture in the Gr-game of pursuit on the time interval [0,TGr].
Proof. Let v(-) e VGr be an arbitrary control of the evader, and let the pursuer use the nGr-strategy. Using the equations (3) and (4) we get the initial value problem
z = uGr(t, v(t)) — v(t) = —r(t, v(t))&, z(0) = z0.
From this, we can see that
z(t)= R(t,v(-))z0, (12)
where
t
1
r( s,v(s) )ds.
R(t,v(-)) = 1 - -i- /r(s,v(s))ds.
Ы J
0
Now we will study the behavior of the function R(t, v(-)) of t. Using the definition of the function r(t, v), we have
t
R(t,v(')) < 1 " щ/W(v(sUo)2+^(s) ~ Ж12 "\(v(s),^)\]ds.
Since the function f(t,q) = \J+ <f>2(t) — \v(t)\2 — q, q G R, is monotone decreasing with respect to ? for every t > 0. Therefore, by the inequality |(v(t),£o}| ^ |v(t)| < ^(t), and also from (7) and (8) we have
t
1 Г 1
R(t, v(-)) <1-R</ " 1>{*)]ds = 1 " Щ
о
f + T)«"-1'-*'
ф(*)
or
R(t,v()) < $(t). (13)
According to Proposition 3, there is some time TGr such that $(TGr) = 0. Consequently, from (13) there exists time t*, 0 < t* < TGr, that R(t*,v()) = 0, and hence z(t*) =0 by (12).
Next, we will prove the admissibility of the strategy (9) for all t, t > 0. It is easy to check that the equality is valid
y>(t) = ky(t) + pi.
Integrate both sides of this equality
t
y(t) = po + Pit + k J ^(s)ds.
o
Take into account of (10)
t
|uGr(t,v(t)| = po + pit + kj |uGr(s,v(s))|ds.
0
This finishes the proof of Theorem 1. □
Theorem 2. If conditions of the Theorem 1 hold, then for any control of the pursuer the strategy of the evader v(t) = —^(t)£o, t ^ 0, guarantees to keep the inequality x(t) = y(t) on the time interval [0,TGr). P r o o f. Let 0 < t < TGr. Then
t t (x(i) - y(t), Co) = |yo - xo| - J(v(s),£o)ds + J(u(s),^o)ds >
о 0
t t > |yo - xo| + J ^(s)ds - J ^(s)ds > 0.
Hence, x(t) = y(t), 0 < t < TGr. This completes the proof. □
Theorems 1 and 2 allow us to conclude that TGr is the optimal pursuit time, the nGr-strategy is an optimal strategy for pursuer and v(t) = —^(t)£0 is an optimal strategy for the evader E.
4. A solution of the Evasion problem. In the present section, the Evasion problem is considered as a control problem from the point of view of the evader E. To solve this problem we suggest a strategy for the evader E and give a definition of solution of evasion.
Definition 7. We call a strategy of the evader the following control function:
VGr(t) = -V>(i)£o, t > 0, (14)
in the Gr-game of evasion.
We prove the following statement.
Theorem 3. If ¿0 ^ 0, ¿1 ^ 0, then the strategy (14) is winning for the evader in the Gr-game of evasion.
Proof. Let ¿0 < 0, ¿1 < 0, and «(•) € UGr. Suppose that the evader implements strategy (14) for all t > 0. Obviously, vGr(t) € VGr. Then for any u(t) we obtain
|z(t)| >
t t t t zq — J vGr(s)ds — J |u(s)|ds = |zo| + j ^(s)ds — J |u(s)|ds.
Using the inequality |u(t)| < y(t) obtained
|z(t)| > *(t),
where tf(t) = |zo| + ((¿1 + k^o)/k2)(1 - efci) + (¿i/k)t. If ¿0 < 0, ¿1 < 0, and k > 0, then
dt k V k
for all t > 0.
This implies that the function ^(t) is monotone increasing on [0, to). Hence it follows that ^(t) > |z0| > 0. This completes the proof of Theorem 3. □
5. The differential game with "life line". Here we are going to study mainly the game with phase constraints for the evader being given by a subset M of Rn which is called the "life line" (for the evader naturally). (Notice that in the case M = 0 we have a simple game.)
In the the Differential Game with "life line" the pursuer P aims to catch the evader E, i.e. to realize the equality x(t) = y(t) for some t > 0, while the evader E stays in the zone Rn \ M. The aim of the evader E is to reach the zone M before being caught by the pursuer P or to keep the relation x(t) = y(t) for all t, t > 0. Notice that M doesn't restrict motion of the pursuer P. Further we will assume initial positions x0 and y0 are given such that x0 = y0 and y0 € M.
Definition 8. A strategy uGr(t, v) of the pursuer P is called winning on the time interval [0,TGr] in the game of "life line" if for every v(-) € VGr there exists some moment t* € [0,TGr] that x(t*) = y(t*) and y(t) € M while t € [0,t*j.
Definition 9. A control function v*(-) € VGr of the evader E is called winning in the game of "life line" if for every u(-) £ UGr : there exists some moment t,t> 0, such that y(t) G M and x(t) ^ y(t) while t G [0, t), or x(t) ^ y(t) for all t > 0.
5.1. Dynamics of the attainability domain. Suppose that ¿0 > 0, ¿1 > 0. In consequence, a set of capture points may consist of some finite set. We will construct the attainability domain under these conditions.
Assume that the evader E chooses any control function v(-) € VGr and the pursuer P applies the strategy (9). Define for each control the following trajectories of the evader
E and pursuer P:
t t y(t) = yo + J v(s)ds, x(t) = xo + J UGr(s,v(s))ds
0 0
on interval t G [0, t] respectively, where t is a pursuit time. Now we generate the sets
Bp(t) = Bp(x(t),y(t)) = {p : |p - x(t)| > h(t)|p - y(t)|} , (15)
Bp(0) = Bp(xo,yo) = {p : |p - xo| > h(0)|p - yo|} (16)
for the pair (x(t),y(t)), where
[ ) m - 1) + *0e** ' ^ a0 '
Here it is obvious that the relation y(t) G BP(t) holds for each t G [0,t].
Proposition 4. If ¿o > 0, ¿i ^ 0, then for the scalar function h(t) the relation h(t) > 1 holds on the time interval [0, t] .
Theorem 4. The set (15) is equivalent to
Bp(t) = x(t) + R(t, v(-))[a(t, zo)S + c(t, zo)] (17)
for all t G [0, t], here S is the unit ball whose center is on zero point in Rn and
h(t)\z0\ h2(t)z0 a(t, z0) = TTu\-7' c(Mo) = --
h2(t) - 1' v ' u/ h2(t) - 1" Proof. From (15) we get
Bp (t)= x(t) + Bp (t), (18)
here Bp(t) = {p : |p| > h(t)|p + z(t)|}. Now we will present that B*P(t) is a ball. For this purpose, square both sides of the inequality
|p| > h(t)|p + z(t)|,
and after that simplify the last result, i.e.
|p|2 > h2(t) (|p|2 + 2(p,z(t)> + |z(t)|2)
or
(h2(t) - 1)|p|2 + 2h2(t)(p,z(t)> + h2(t)|z(t)|2 < 0. (19)
Divide both sides of (19) by the expression h2(t) - 1
2 2 h2(t){p,z(t)) fe2(t)|z(t)|2 H + h2(t)-l + h*(t)~ 1 ^ ( )
Add the expression h2\t)-i J both sides of (20), and write it down as
W2 + 2/ hHt)z(t)\ | fh?(t)z(t)\2 ^fh?(t)z(t)\2 h?(t)\z(t)\*
h2(t) - 1/ Vh2(t) - iy ^ \h2(t) - 1/ h2(t) - 1"
After some simplification we can generate the result
/>2(i)z(i)
p +
mw)\ ^ h?(t) - г
h2(t) -1 Hence we have set
Bp(t) = {P : |p - c(t, z(t))| < a(t, z(t))} = c(t, z(t)) + a(t, z(t))S,
where B*p(t) is the ball whose center is on the point c(t,z(t)) = — and whose
radius equals a(t, z(t)) = ■ Then from formula (12) we obtain
. . h2(t)z(t) . .. h2(t)zo
h2(t) - 1 V ' V "h2(t) - 1'
h2(t) -1 v ' v "h2(t) -1' In consequence, the (18) can be written in the form
Bp (t) = x(t)+ R(i,v(^))
/i(t)|z0|_s,_ /i2(t)z0
h2(t) - 1 h2(t) - 1
or in the form (17) which finishes the proof. □
Now we are going to show monotony of the set BP (t).
Theorem 5 (Petrosyan type theorem [10]). Let: a) p0 > <r0, pi > a1, and
b) p1a0 ^ po^i. Then the set BP (t) is monotone in relation to the inclusion while t € [0, т], i.e. Bp(ti) D Bp(t2) for 0 < ti < t2.
Proof. First, by (17) we determine the derivative of the support function (see [32]) F(Bp(t), p) of the set Bp(t) for any p € Rn and |p| = 1, that is,
jtF{BP{t),p) = jtF{x{t) + R{t,v{-))[a{t,z0)S + c{t,z0)],p) =
= Jt [(x(t), p) + R{t, v(-))[a(t, z0)F(S, p) + (c(t, z0), p)]] =
= (X(t),p) + R(t, v(-))[a(t, zq) + (c(t,zo),p)] + + R(t,v(^))[a(t,zo) + (C(t, zq), p)] = Фl(t,p) +Ф2(t,p),
where
Фl(t, p) = (X(t), p) + R(t, v(-))[a(t, zq) + (c(t, zq), p)], Ф2(t, p) = R(t,v(0)[a(t, zq) + (C(t,zo),p)]. Now we prove that the inequality
^-F(BP(t), p) = $i(t, p) + Ф2(*, p) < 0 dt
is true on t € (0, т].
To do this, we first show that $1(t,p) < 0. Square the inequality |v(t)| < ^(t) and multiply both sides of the result by the expression
\v(t)\2h2(t) < v2(t)
h2(t) - 1 ^ h2(t) - 1'
Make some simplification
(21)
According to definition of the function r(t, v(t)) (see Definition 6) we can express the equality ^2(t) - |v(t)|2 = r(t, v(t))[r(t, v(t)) - 2(v(t),£o>], and make some substitutions in (21)
Mt)\2 < ^0\[r(t,v(t)) -2{v(tlto)]
or
I , o U .uiM <- l!Mil) (00\
K*)l +MtMt))^^pr < T^FT- (22)
Add the expression to both sides of (22), and rewrite it again
r2(t,v(t)) r2(t,v(tj)
^ (h2(t) - l)2 + h2(t) - r 1 J
The fact that the left-hand side of (23) consists of quadratic standard form of the sum of two vectors, and as a consequence of simplifying the right-hand side of (23) we obtain
уУг> + Ш7\-
h2(t) - 1
h(t)r(t,v(t)) (0As
^ h2(t) — 1 • [ '
It is obvious that for any vector p G Rn, |p| = 1 the inequality
. . r(t, v(t)) \ is valid.
According to this and from (24), we have
m , r(t,v(t))r
=» (»(<). p) ->•(<.»(»)) - (£o,(>) < -^Щ^ГГ =>■
^ (x(t),p) + R(t,v(-))[a(t,Z0) + (c(t,Z0),p)j < 0. (25)
Formula (25) means that (t,p) < 0.
Now we are going to present that the inequality $2(t, p) < 0 holds. Because of R(t, v(-)) > 0 on t € (0, t], it is enough to prove that
d(t,zo) + (c(t,zo)^) = J (26)
First, compute the first derivative on the right-hand side of (26)
h2(t) V_ -2h(t)h'(t)
h2(t) — 1) (h2(t) — 1)2'
On the other hand, using (7), (8) and based on the condition b) of Theorem 5 we have the following:
dh(t) _ ¥<(t)m - <p{tW{t) _ (Pla0 - po<Ti)efct , .
dt ~ ~ V>2W '
Consequently, this inequality (^jJqjf^J ^ 0 is satisfied in the interval (0, г].
Now multiply both sides of the inequality (£o,p) < 1 (for any p € Д", |p| = 1) by the expression — (у^тщг- j
Add i j to both sides of (28), and obtain
Compute the right-hand side of (29) and in relation to (27) we obtain this result
( Kt) У / fe2(t) y= h'(t) \h?(t)-lj \h?(t)-lj (h(t) +1)2 ^ '
From this, we generate the following relation for the left-hand side of (29):
Multiply both sides of (30) by |z0| and we get (26). Hence it follows that Ф2(^p) < 0. This completes the proof. □
5.2. A solution of the game with "life line". It has been noted above that Isaacs' game with "life line" was comprehensively solved by L. A. Petrosyan using the method of approximating measurable controls by piecewise constant.
Theorem 6. If Theorem 5 is valid and Mp| BP(0) = 0, then the nGr-strategy (9) is winning in the game (1)-(4) with "life line".
Proof. The proof directly follows from Theorem 5. □
Now we define a set in the form
Be(t) = {P : |P - xo| > x(t)|P - yo|}
for all t G (0, TGr], where
t
J <^>(s)ds
x(t) =
о_ = (Pi +kp0)(ekt - 1) -kPlt
« ~ (<7i + ko~o)(ekt - 1 )-ka1t
J ^(s)ds
0
and TGr is the first positive root of the equation (11). It's obvious that lim^o x(i) =
Theorem 7. Let the conditions po > ao, p1 > a1 and p1ao ^ po^1. Then x(t) is increasing on interval t G (0,TGr] and the set BE (t) is decreasing with respect to t G (0, TGr], i.e., an inclusion BE(t1) D BE(t2) holds for any t1? t2 G (0, TGr] and 0 < t1 ^ t2.
Proof. First of all, we will prove that x(t) is increasing, i.e. x'(t) ^ 0 under the conditions of Theorem 7. For this purpose, calculate the derivative of x(t)
dX(t) = fc2(p0ai -pi*o){ekt - 1 - kekH) dt [(cri + ka0)(ekt - 1) - ka\t]2 '
Now, analyze a sign of expression l(t) = ekt - 1 - kektt for every t G (0, TGr], i.e.: a) limt^0/(i) = 0; b) ^ = -k2ektt < 0. So, l(t) < 0 on interval t G (0,TGr],
Therefore, x (t) ^ 0 is true if p1ao > po^1. Since x(t) is increasing on t G (0,TGr], we can write a relation x(t) > 1 on that interval. From (15), (16), we have
BE(t) = so - £[t] ,z0 + S. (31)
x2 (t) - 1 x2(t) - 1
Then we determine the character of the derivative of support function F(BE(t),p), when |p| = 1 and t G (0, TGr]:
= - + (^t)'n -
(x2(t) - 1)2Л^и"~' (x2(t) - 1)2' = (2x(i)(eo,M)-x2w-i)- x'(t)N
(x2(t) - 1)2 = -leox(i) - Ml (x2(t)_1)2 <
□
Theorem 8. Let ¿o > 0, ¿1 > 0, p^o > po^1 and M[)Be(TGr) = 0. Then for evader E there exists some control which is winning in the game (1)-(4) with "life line".
Proof. Let p € Mp| BE(TGr) and the evader E implements the control v*(t) = ^(t)v, «*(•) € VGr, where v = (p - y0)/|p - y0|. Since |v*(t)| = ^(t) for all t > 0, then substituting this into inequality (2) we get formula
t t a"0 + 0"it + k j |v*(s)|ds = ^0 + ^lt + k j ^(s)ds =
= ctq + ait + k
(efcs - 1) + a0eks] ds = ф(Ь) = K(t)|,
which proves the admissibility of the control v*(t) for all t ^ 0. Then time of the achievement of the point p is n for evader and we have
- -J |v*(s)|ds = J ^(s)ds = |p — yo|,
(32)
and from of Theorems 1 and 7, it follows, that n ^ TGr. We suppose that for the pursuer exists a certain control function «*(•) € UGr that x(t) = y(i) holds and i < 77. If z(t) = x(t) - y(t) and z(0) = z0, then from Z(t) = u*(t) - v*(t) we have
z(t) = z0 + J (u* (s) — v*(s))ds = 0
From this, it follows that
г г г
zq — j v*(s)ds |u*(s)|ds ^(s)ds
^ |zq|2 — 2 J ^(s)ds(zo ,v) + ^ J ^(s)dsj < ^ J ^(s)dsj ^
- 2 -
^ ^J ^(s)dsj (x2(t) — 1) + 2 J ^(s)ds(zo,v) — |zo|2 > 0 ^ -
=> J '<P(s)ds > № := _ xW{z0,Vy + |z0|2(x2(t) - 1) - (zo,^)].
(33)
If x(t) is increasing for t € (0, TGr], then it is easy to check that f (t) is decreasing function on (0, TGr]. Consequently from 77 < 77, it follows that f (n) ^ f (i).
Sincep € BE(TGr) and n ^ TGr, then from Theorem 7 we havep € BE(TGr) C BE(n). Hence we obtain
|p - X0| > x(n)|p - y01 ^ ^ |z0 - (p - y0)|2 > X2(n)|p - y0|2 ^
2
^ |zo|2 - 2{z0,p - yo) + |p - yo|2 > x2(n)|P - yo|2 ^ ^ 0 > (x2(n) - 1)|P - yo|2 + 2|p - yo|(zo, v) - |zo|2 ^ ^ |p - yo| < f (j) < f (t)-
t n
Then from the last inequality and from (32), (33), we have f ^(s)ds > f ^(s)ds or
oo
t > n, though this contradicts our supposition. □
Theorem 9. Let ¿o ^ 0, ¿1 ^ 0. Then for the evader E there exists some control which is winning in the game (1)-(4) with "life line".
Proof. Let the evader use the control (14), and let the pursuer choose an arbitrary control w(-) € Ugi- Then, similar to the proof of Theorem 3, under the conditions of the current theorem we again derive |z(t)| ^ ^(t) ^ |zo| > 0 for any t £ [0, from which
we infer z(t) = 0, i.e. x(t) = y(t) (see Proposition 2). Therefore, by virtue of Definition 9 in the game (1)-(4) with a "life line" the evader E is also considered to be winning. The proof is complete. □
6. Example. Assume that the game (1)-(4) is described as (see Figures 1 and 2)
u, xo
t
(0,0), \u(t)\ ^2 + 2y/2t +J \u(s)\ds, t > 0,
(34)
y = v, yo = (0, - 1), |v(t)| ^ /3 + 21 + / |v(s)|ds, t > 0.
J o
У
( м
4 \ \ 74 \
/ 2 \ \ \
■ 0 Р , \
-16 -14 -12 -10 -8/' -6 -4 -2 0 2 4 6 8\ 10 12 14 16 18' 20 22
1 -2 Е \
BP{ 0) -4 1
-6 1 /
-12
(35)
Figure 1. The representation which the ncr-strategy is winning in the game (34), (35) with "life line"
Then based on Theorem 1, we can obtain Tqr = 0.37. In accordance with Theorem 4, we generate the set Bp(0) = {p : \p — c\ < o, c = (0, —4), a = 4 + 2a/3}- A set of points p = (p^p2) on a boundary of BP(0) consists of the circle
дВР(0) = {(рър2) : pj + (P2 + 4)2 = 28 + lGv^}.
Вестник СПбГУ. Прикладная математика. Информатика... 2024. Т. 20. Вып. 2
t
У
3
2 М
0 Р \ X
-8 -7 -6 -5 -4 -3 -2 -1 . 1 2 3 4 5 \ 6 7 8 9 10 11 12 13 14
-1 Е
-2
-3
WGr)
-4
-5
-6
-7
Figure 2. The representation which the evader wins in the game (34), (35) with "life line"
Using (31) and Theorem 7, the set Be(TGr) = {p : |p - c| < a, c = (0, -3.323), a = 2.779} is constructed. A set of points p = (pi,p2) on a boundary of BE(T0r) consists of the following circle:
BBe(Tor) = {(pi,p2) : Pi + (p2 + 3.323)2 = (2.779)2}.
We wish to thank professors A. A. Azamov and G. I. Ibragimov for discussing this paper and for providing some useful comments.
References
1. Pontryagin L. S. Izbrannye trudy [Selected works]. Moscow, MAKS Press, 2004, 551 p. (In Russian)
2. Krasovskii N. N., Subbotin A. I. Game-theoretical control problems. New York, Springer, 2011, 517 p.
3. Isaacs R. Differential games. New York, John Wiley and Sons, 1965, 385 p.
4. Petrosyan L. A. Differential games of pursuit. Singapore, World Scientific Publ., Series on optimization, 1993, 326 p.
5. Pshenichnyi B. N. Simple pursuit by several objects. Cybernetics and System Analysis, 1976, vol. 12, no. 5, pp. 484-485.
6. Pshenichnyi B. N., Chikrii A. A., Rappoport J. S. Effektivnyj metod resheniya differencial'nyh igr so mnogimi presledovatelyami [An effective method for solving differential games with many pursuers]. Dokl. Akad. Nauk SSSR, 1981, vol. 256, no. 3, pp. 530-535. (In Russian)
7. Azamov A. O zadache kachestva dlya igr prostogo presledovaniya s ogranicheniem [On the quality problem for simple pursuit games with constraint]. Serdica Bulgariacae Math., 1986, vol. 12, no. 1, pp. 3843. (In Russian)
8. Azamov A. A., Samatov B. T. The n-strategy: Analogies and applications. The Fourth International Conference Game Theory and Management. St. Petersburg, 2010, vol. 4, pp. 33-47.
9. Blagodatskikh A. I., Petrov N. N. Konfliktnoe vzaimodejstvie grupp upravlyaemyh ob'ektov [Conflict interaction of groups of controlled objects]. Izhevsk, Udmurt State University Press, 2009, 266 p. (In Russian)
10. Chikrii A. A. Conflict-controlled processes. Dordrecht, Kluwer, 1997, 403 p.
11. Grigorenko N. L. Matematicheskie metody upravleniya neskol'kimi dinamicheskimi protsessami [Mathematical methods of control for several dynamic processes]. Moscow, Moscow University Press, 1990, 198 p. (In Russian)
12. Petrosyan L. A. Ob odnom semejstve differencial'nyh igr na vyzhivanie v prostranstve Rn [About some of the family differential games at a survival in the space Rn]. Papers of Academy sciences of USSR, 1965, vol. 161, no. 1, pp. 52-54. (In Russian)
13. Petrosyan L. A. Igry presledovaniya s "liniej zhizni" [Pursuit games with "a survival zone"]. Vestnik of Leningrad State University, 1967, vol. 3, no. 13, pp. 76-85. (In Russian)
14. Petrosyan L. A., Dutkevich V. G. Igry s "liniej zhizni", Sluchaj l-zahvata [Games with "a survival zone", occation l-catch]. Vestnik of Leningrad State University, 1969, vol. 3, no. 13, pp. 31—38. (In Russian)
15. Petrosyan L. A., Rikhsiev B. B. Presledovanie na ploskosti [Pursuit on the plane]. Moscow, Nauka Publ., 1991, 96 p. (in Russian)
16. Petrosyan L. A., Mazalov V. V. Game theory and applications. II. New York, Nova Sci. Publ., 1996, 219 p.
17. Samatov B. T. On a Pursuit-Evasion problem under a linear change of the pursuer resource. Siberian Advances in Mathematics, 2013, vol. 23, no. 10, pp. 294—302.
18. Samatov B. T. The pursuit-evasion problem under integral-geometric constraints on pursuer controls. Automation and Remote Control, 2013, vol. 74, no. 7, pp. 1072—1081.
19. Samatov B. T. Problems of group pursuit with integral constraints on controls of the players. I. Cybernetics and Systems Analysis, 2013, vol. 49, no. 5, pp. 756—767.
20. Samatov B. T. Problems of group pursuit with integral constraints on controls of the players. II. Cybernetics and Systems Analysis, 2013, vol. 49, no. 6, pp. 907—921.
21. Samatov B. T. The П-strategy in a differential game with linear control constraints. Journal of Applied Mathematics and Mechanics, 2014, vol. 78, no. 3, pp. 258—263.
22. Dar'in A. N., Kurzhanskii A. B. Control under indeterminacy and double constraints. Differential Equations, 2003, vol. 39, no. 11, pp. 1554-1567.
23. Kornev D. V., Lukoyanov N. Yu. On a minimax control problem for a positional functional under geometric and integral constraints on control actions. Proceedings of the Steklov Institute of Mathematics, 2016, vol. 293, pp. 85-100.
24. Satimov N. Yu. Metody resheniya zadachi presledovaniya v teorii differencial'nyh igr [Methods of solving of pursuit problem in differential games]. Tashkent, National University of Uzbekistan Press, 2003, 240 p. (In Russian)
25. Ibragimov G. I. A game of optimal pursuit of one object by several. Journal of Applied Mathematics and Mechanics, 1998, vol. 62, no. 2, pp. 187-192.
26. Ibragimov G. I. Optimal pursuit with countably many pursuers and one evader. Differential Equations, 2005, vol. 41, no. 5, pp. 627-635.
27. Aubin J. P., Cellina A. Differential inclusions. Set-valued maps and viability theory. Berlin, Heidelberg, New York, Tokyo, Springer-Verlag, 1984, XIII, 342 p.
28. Pang J. S., Stewart D. E. Differential variational inequalities. Mathematical Programming. Series A, 2008, vol. 113, no. 2, pp. 345-424.
29. Samatov B. T., Ibragimov G. I., Hodjibayeva I. V. Pursuit-Evasion differential games with the Gronwall type constraints on controls. Ural Mathematical Journal, 2020, vol. 6, no. 2, pp. 95-107.
30. Samatov B. T., Akbarov A. Kh., Zhuraev B. I. Pursuit-Evasion differential games with Gr-con-straints on controls. Izvestiya Instituta matematiki i informatiki Udmurtskogo gosudarstvennogo universi-teta [Processing of Institute for Mathematics and Informatics of Udmurt State University], 2022, vol. 59, pp. 67-84.
31. Gronwall T. H. Note on the derivatives with respect to a parameter of the solutions of a system of differential equations. Annals of Mathematics Second Series, 1919, vol. 20, no. 4, pp. 292-296.
32. Blagodatskikh V. I. Vvedenie v optimal'noe upravlenie [Introduction to optimal control]. Moscow, Vysshaya shkola Publ., 2001, 239 p. (In Russian)
Received: January 7, 2024. Accepted: March 12, 2024.
A u t h o r s' i n f o r m a t i o n:
Bahrom T. Samatov — Dr. Sci. in Physics and Mathematics, Professor; [email protected] Adakhambek Kh. Akbarov — PhD in Physics and Mathematics; [email protected]
Дифференциальная игра с «линией жизни» при ограничениях Гронуолла на управления
Б. T. Саматов1, A. X. Акбаров2
1 Наманганский государственный университет, Узбекистан, 116019, Наманган, ул. Уйчи, 316
2 Андижанский государственный университет, Узбекистан, 170100, Андижан, Университетская ул., 129
Для цитирования: Samatov B. T., Akbarov A. Kh. Differential game with a "life line" under the Gronwall constraint on controls // Вестник Санкт-Петербургского университета. Прикладная математика. Информатика. Процессы управления. 2024. Т. 20. Вып. 2. С. 265-280. https://doi.org/10.21638/spbu10.2024.211
Изучается дифференциальная игра с «линией жизни» для одного преследователя и одного убегающего при управлениях удовлетворяющих неравенств типа Грануолла. Убегающий считается пойманным со стороны преследователя, если состояние убегающего совпадает с состоянием преследователя. Одна из основных целей настоящей работы — построение оптимальных стратегий для игроков и определение оптимального времени поимки. Для преследователя предлагается стратегия параллельного сближения (короче, П-стратегия) и доказывается ее оптимальность. Для решения задачи с «линией жизни» исследуется динамика области достижимости игроков методом Петросяна, т. е. найдены условия монотонности по включению относительно времени этой области достижимости. Отметим, что работа продолжает исследования Айзекса, Петросяна, Пшеничного, Азамова и др.
Ключевые слова: дифференциальная игра, преследование, убегание, ограничение Гро-нуолла, стратегия, параллельное преследование, область достижимости, игра с «линией жизни», сфера Аполлония.
Контактная информация:
Саматов Бахром Таджиахматович — д-р физ.-мат. наук, проф.; [email protected] Акбаров Адахамбек Хасанбаевич — канд. физ.-мат. наук; [email protected]
