CC BY
4URAL MATHEMATICAL JOURNAL, Vol. 8, No. 1, 2022, pp. 76-89
DOI: 10.15826/umj.2022.1.008
MATRIX RESOLVING FUNCTIONS IN THE LINEAR GROUP PURSUIT PROBLEM WITH FRACTIONAL DERIVATIVES1
Alena I. Machtakova^, Nikolai N. Petrov^
Udmurt State University, 1 Universitetskaya Str., Izhevsk, 426034, Russian Federation
Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya Str., Ekaterinburg, 620108, Russian Federation
[email protected], [email protected]
Abstract: In finite-dimensional Euclidean space, we analyze the problem of pursuit of a single evader by a group of pursuers, which is described by a system of differential equations with Caputo fractional derivatives of order a. The goal of the group of pursuers is the capture of the evader by at least m different pursuers (the instants of capture may or may not coincide). As a mathematical basis, we use matrix resolving functions that are generalizations of scalar resolving functions. We obtain sufficient conditions for multiple capture of a single evader in the class of quasi-strategies. We give examples illustrating the results obtained.
Keywords: Differential game, Group pursuit, Pursuer, Evader, Fractional derivatives.
1. Introduction
The theory of two-player differential games, originally considered by Isaacs [20], has grown to be a profound and substantial theory that develops various approaches to the analysis of conflict situations [3, 14, 15, 19, 21, 22, 24, 36, 40]. The following methods for solving game problems were developed: the Isaacs method based on the analysis of some partial differential equation and its characteristics, the method of stable bridges, Krasovskii's rule of extremal aiming, Pontryagin's method based on alternating integration of convex sets, etc.
In [6, 7], Chikrii proposed a method of scalar resolving functions using Pontryagin's condition and, based on it, measurable choice theorems.
The method of scalar resolving functions was developed further to investigate linear and quasilinear group pursuit problems [2, 10, 18, 28-30, 38, 39]. In [8], Chikrii noted that scalar resolving functions attract the terminal set to the images of some multivalued maps. This attraction occurs in the conical hull of this set, which restricts the maneuverability of pursuers.
In [8, 11], for the analysis of two-player pursuit games, matrix resolving functions were proposed. In [26], matrix resolving functions were applied to studying the group pursuit problem described by a linear autonomous system of differential equations.
In the present paper, we consider matrix resolving functions in a linear group pursuit problem described by a system of differential equations with Caputo fractional derivatives. It should be noted that matrix resolving functions for solving group pursuit problems with fractional derivatives are used for the first time. Previously, scalar resolving functions were used in [23, 25, 27] devoted to this class of problems. We obtain sufficient conditions for multiple capture of a single evader.
1This work was supported by the Russian Science Foundation, project 21-71-10070.
The multiple capture of a single evader in the simple pursuit problem was considered in [4, 17]; [4] investigated it in a discrete setting. In [31, 32], the problem of multiple capture of a single evader was presented in the example of L.S. Pontryagin, and in [1, 33] it was considered in linear differential games.
2. Statement of the problem
Definition 1 [5]. Let f: [0, to) — Rk be an absolutely continuous function and a € (0,1). The Caputo derivative of order a of the function f is defined to be a function f of the form
= r(T^~a) SI (f=lT d*' Wkere ^ = LX
In the space Rk (k > 2), we consider a differential game G(n + 1) involving n + 1 players: n pursuers P1,... ,Pn and an evader E, which is described by a system of the form
D^zi = AiZi + Ui — v, Zi(0) = z0, Ui € Ui, v € V. (2.1)
Here i € I = {1,... ,n}, zi,ui,v € Rk, Ui and V are compact sets from Rk, a € (0,1), D(af is the Caputo derivative of order a of the function f, and A,i are constant square matrices of order k x k. Assume that z0 = 0 for all i € I. Define z° = {z0, i € I} to be the vector of initial positions.
Let v : [0, to) — V be a measurable function. Let us call the restriction of the function v to [0,t] the prehistory vt( ) of the function v at time t.
Definition 2. We will say that a quasi-strategy U of a pursuer P-i is given if a map U® is defined that associates a measurable function ui(t) with values in Ui to the initial positions z°, time t, and arbitrary prehistory of control vt( ) of the evader E.
Definition 3. An m-fold capture (a capture for m = 1) occurs in the game G(n + 1) if there exist a time T > 0 and quasi-strategies U\,...,Un of pursuers P\,... ,Pn such that, for any measurable function v(-), v(t) € V, t € [0,T], there exist times r\,...,rm € [0,T] and pairwise different indices h,...,im € I such that zil (ri) = 0 for all l = 1,... ,m.
The aim of this paper is to obtain conditions for the solvability of the pursuit problem.
Assumption 1. For all i € I, it is true that 0 € P| (Ui — v).
vev
In what follows, we assume that Assumption 1 holds. We introduce the following notation:
^ B i
which is a generalized Mittag-Leffler function [16], where B is a square matrix of order k x k, p > 0, and p € R1; A= {(t,r) : t > 0, 0 < t < t}, J = {1,...,k},
gi(t,t) = (t-t)a~1E1(Al(t-t)a,a), r^t, g(t,t) = 0,
a
h{t) = EL{MaA)z°l, Wi(t,t,v)=gi(t,t)(Ui-v),
a
Wi(t,r)=p| Wi(t,T,v), i € I, 0 < t < t,
vev
where (t, t) € A and v € V.
Consider an arbitrary diagonal square matrix Li of order k x k of the form
Aii 0 . .0
Li = 0 Ai2 . .0 = diag(Aii,Ai2 ,...,Aik )
U 0. . Aifc/
We identify the matrix Li with the vector (Aii,..., Aik), understand the inequality Li > 0 coordi-natewise, and introduce the multivalued maps
Mi(t,T,v) = {Li : Li > 0, -Lifi(t) € Wi(t,T,v)}, (t,T) € A, v € V.
By Assumption 1, for all i € I, v € V, and t,T such that 0 < t < t, the sets Wi(i,T, v) are not empty and 0 € Mi(t, t, v). By the properties of the parameters of the conflict-controlled process (2.1), the maps Mi(t, t, v) are measurable in t [12]. Then the maps Wi(t, t) are measurable in t [12].
Define the scalar functions
A°(t,T,v)= sup min Ay(t, t, v), (t,T) € A, v € V. (2.2)
Li&Mi(t,r,v) jeJ
Assumption 2. For all (t, t) € A and v € V, the supremum in (2.2) is attained.
Assuming that the supremum in (2.2) is attained, we define the sets
M*(t, t, v) = {Li(t,T,v) € Mi(t, t, v) : A°(t,T,v) = min Ay(t,T,v)}.
j
It follows from [12] that, under the above assumptions, Mi(t, t, v) and M*(t,T, v) are measurable in (t, v) and closed-valued for any t > 0. By the measurable choice theorem [35, Theorem 20.6], for each i € I in M*(t, t, v), there exists at least one selector measurable in (t, v) for any t > 0. We
fix these selectors L*(t, t, v) and define A*(t,T, v) = min A*j(t,T, v). Next, define
j
Q(m) = {(ii,...,im) : ii,___,im € I and are pairwise different},
¿(t,T) = inf max min A*(t,T, v). vev Aen(m) ieA '
3. Sufficient conditions for capture
Lemma 1. Suppose that Assumptions 1 and 2 hold and
t
j 0
lim / ¿(t, s)ds = 0
Then there exists a time T > 0 such that, for every measurable function v(-), v(t) € V, t € [0, T], there is a set A € Q(m) such that the following inequalities hold for all l € A, j € J:
I A*,-(T,s,v(s))ds > 1. Jo
Proof. Let v(-) be an arbitrary measurable function, v: [0, to) — V. Then the inequalities
K3(t,s,v(s)) > K(t,s,v(s)) hold for all t > 0, s € [0, t], l € I, and j € J. Therefore, the inequalities
J Af(t,s,v(s))ds >J Af (t, s, v(s))ds 0 0
hold for all t > 0, l € I, and j € J. In addition
ft
AeQ(m) leA Jo AeQ(m) Jo
Since, for any nonnegative numbers üa(A € Q(m)), one has
1
max a a > tt- a a, where C,
AeQ(m) ~ C™ A ^ , v ' n Aeü(m)
m n
n!
(n — m)! m!
it follows from (3.2) that
t t
max min / AUt, s,v(s))ds > —— / > min XUt, s, v(s))ds > Aefi(m) leA J n - C? ~
0 0 Aen(rn)
t t
^ —— / max min Xf(t, s, v(s))ds > —— / §(t,s)ds. - C™ J Aen(m) /eA - c™ J
0
Since
t
J 5(t, s)ds = +to,
there exists T > 0 such that
T
m Cn
J S(T,s)ds > 1.
Hence,
T
max min / Af(T, s, v(s))ds > 1. Aen(m) leA J l "
Therefore, there exists A € Q(m) such that the following inequalities hold for all l € A:
T
J A*(T, s, v(s))ds > 1.
(3.1)
max min / Af(t, s, v(s))ds > max / min Af(t, s,v(s))ds. (3.2)
en(m) leA J0 AeQ(m) Jo leA
This inequality and inequality (3.1) imply the validity of the lemma.
□
t
t
1
Let V be the set of all measurable functions v: [0, oo) ^ V. Let us define the number
t
T = inf{t > 0 : inf max minmin / Aj(t,s,v(s))ds > 1}. v(0eV Aen(m) leA jeJ 7 ljV 0
Consider the sets (i € I, j € J, v(-) € V)
t
Tij (v(-)) = {t > 0 : J Aj(T,s,v(s))ds > 1}. 0
Define the quantities (i € I, j € J, v(-) € V)
tij(v( )) = i'"f{t : t € Tij(v( ))) if Tij(v( ))= 0 (3.3)
j W' if Tij (v())= 0. ( '
Assumption 3. For any t € [0,T], v € V, l € I, and J0 C J, the selector Bi(T, t, v) = diag (0n(T, t, v),..., Afc(T, t, v)),
where
fljCT.T,v)4nAj'<T'T'v)' J J
[0, j € Jo,
satisfies the condition Bl (T, t, v) € Ml (T, t, v).
Theorem 1. Suppose that Assumptions 1, 2, and 3 hold and
t
lim / ¿(t, s)ds =
—J
t— 0
Then an m-fold capture occurs in the game G(n + 1).
Proof. By Lemma 1, T < Let v : [0, T] ^ V be an arbitrary measurable function and t € [0, T]. Let us introduce functions (^i1(T, t, v),... (T, t, v)) of the form
ftj(T,t,v) = iAij(T-T'v)- T € ^«'»! '
\0, t € (tij(v( )),T],
where t*j(v(-)) are defined by formula (3.3). Let B*(T, s,v) be a matrix of the form
/$1(T,s,v) 0 ... 0 \
B* (T,s,v) =
V 0 0 ... (T,s,v)/
Consider the multivalued maps (s € [0,T], v € V)
Ui(T, s, v) = {Ui € Ui : gi(T,s)(Ui - v) = -B* (T,s, v)fi(T)}.
0 ^*2(T,s,v) ... 0
By Assumption 3, B*(T, s, v) is a measurable selector of Mi(T, s, v). Therefore, the sets C/j(T, s, v) are nonempty for all i € I, s € [0, T], and v € V. Hence, by the measurable choice theorem [35, Theorem 20.6], there exists at least one measurable selector u*(T,s,v). We define the controls of pursuers Pi, i € I, assuming
Ui(T) = u* (T,r,v(r)).
By [12], the functions ui(■) are measurable. We show that these controls of the pursuers guarantee the m-fold capture of the evader. The solution of the Cauchy problem for system (2.1) has the form [9]:
t
zi(t) = fi(t) + |gi(t,s)(ui(s) - v(s))ds. 0
By the choice of controls of the pursuers, we obtain
T T
Zi (T) = fi(T) -J Bi(T,s,v(s))/i(T)ds = (e -J Bi(T,s,v(s))d^/i(T), 00
where E is an identity matrix. It follows from the definition of Bi(T, s,v(s)) that there exists A € Q(m) such that z(T) = 0 for all l € A. This proves the theorem. □
Assumption 4. The matrices Ai are diagonal matrices of the form
fan 0 ... 0 \
Ai = 0 ai2 . . . 0 with aij < 0 or all i € I, j € J.
U 0 ... aifc/
Let us introduce multivalued maps (v € V)
M0(v) = {Li : Li > 0, -Liz0 € (Li - v) }.
By Assumption 1, the sets M°(v) for all i € I and v £ V are nonempty and 0 € M°(v). Next, we define functions Ai(v) of the form
Ai(v) = sup min Aij- (v). (3.4)
Li&M°(v) j
Assumption 5. For all v € V, the supremum in (3.4) is attained. Assuming that the supremum in (3.4) is attained, we define the sets (v € V) M(v) = (Li(v) G MKv) : Mv) = mmXijiv)}.
Next, suppose that X*(v) is a measurable selector of Mi{v) and
5= inf max minA;*(v). vev Aen(m) ieA
Define ((t, t) € A)
a = max(—aij),
i,j
\a—1
9ij(t, s) = (t-s)a~lEi(al3(t-s)a,a), t^s,
Oi
g(t, s) = (t- s)a~lEi (-a(t -s)a,a), t / s,
Oi
gij (t,t) = g(t,t)=0.
Lemma 2. Suppose that Assumptions 1, 4, and 5 hold, and 5 > 0 and aij < 0 for all i € I and j € J. Then there exists T > 0 such that, for every admissible function v(-), there is a set A € Q(m) such that the following inequalities hold for all l € A and j € J:
T
Ex(al3Ta, 1) - Jgl3(T,s)X3(v(s))ds < 0. 0
Proof. Let v(-) be an admissible function. Then 0 < —aij < a for all i and j. Therefore, the following inequalities hold [34] for all t > 0, s € [0, t], i € I, and i € J:
Ei(aij(t- s)a,a) > Ei(—a,(t — s)a, a).
a ex.
It follows from [37, Theorem 4.1.1] that Ei(z,^i) > 0 for all z € M1 and n € [a, +oo). Hence, the
Oi
inequalities
t t Jgij{t,s)Tij{v{s))ds > Jg{t,s)X*{v{s))ds 00
hold for all t > 0, i € I, and j € J. Next, we have
t t
max min / g(t, s)\*,(v(s))ds > max / g(t, s) min \Uv(s))d,s. (3.5)
Aen(m) leA J Aen(m) y leA
00
Using inequality (3.5), we obtain
t t
max / g{t,s)mm\*iv{s))ds > — / g{t,s) V min A¡{v{s))ds > Ven(m) J leA Cnm J leA
AeQ.(m)J leA Cm „
0 n 0 Aen(m)
t t I f -* S i
> —— / q(t, s) max min A; (v(s))ds > —— / q(t,s)d,s. ~ C™ J ^^ Asn(m) /SA lK ~ C™ J ^^ 00
By [13, Ch. 3, formula (1.15)],
t
J g{t,s)ds = taEi{-ata,a + 1).
Consider the functions (t € [0, to))
5
Mi) = Ei(aijta, 1) - —taEi(-ata, a + 1).
l
Since aij- < 0, - a < 0, it follows from [37, Theorem 1.2.1] that the following asymptotic representation holds as t ^ +to:
1] = -^rll-a) + ° ' +» = + '0 •
Therefore,
Consequently, lim hi?- (t) < 0 for all i € I and j € J. Hence, there exists T > 0 such that t—
hij(T) < 0 for all i € I and j € J. Next, let A € Q(m) be such that
T T
max min / g(T, s)XUv(s))ds = min / g(T, s)XUv(s))ds. Aen(m) leA J zeA J
00
Then, for all l € A, one has
T
jg(T, s)~X* (v(s))ds > ±-TaE^(-aTa,a + 1). o n
Therefore,
T T
- I gij(T,s)X^(v(s))ds < - I g{T, s)X*(v(s))ds < E±(-aTa ,a + 1).
00 Hence, the inequalities
T
sUi.inisl) < K, in^T" n -
a
Ei(aijTa, 1) - / gij(T, s)Xlj(v(s)) ds < Ei(aijTa, 1) - —TaEi(-aTa, a + 1) < 0
0
hold for all l € A and j € J. This proves the lemma. □
Lemma 3. Suppose that Assumptions 1, 4, and 5 hold, aij < 0, and 5 > 0. Then there exists T > 0 such that, for every admissible function v(-), there is a set A € Q(m) such that the following inequalities hold for all l € A and j € J:
T
Li(ayT°,l) - f gij(T, s]X*j(v(s)) ds < 0.
Proof. The proof is similar to the proof of Lemma 2.
□
Assumption 6. For all v € V, I € I, and J0 C J, the selector
Bi (v) = diag(Ai (v),...,Pik (v)),
where
* (v) = io'' (v)' 3J
[0, J/Jo,
satisfies the condition Bl(v) € M®(v).
Remark 1. Note that Assumption 6 does not always hold. Suppose that, in system (2.1), k = 2, n = 1, m = 1, z0 = (1,2), A1 is a zero matrix, and
Ui = V = {(ui,u2) : ui = U2, U2 € [-1,1]}.
Let v = 0. Then
i / A 0
Therefore,
Hence,
Mi(0)H(0 A/2), A€ [0,.
sup minAy = i. LeM°(o) j 2
M1<0> = {(0 1/°2
and the extremal selector is A1(0) = diag(1,1/2). However, the selector B1(0) = diag(1,0) / M1(0). Similarly, the selector B2(0) = diag(0,1/2) / M?(0).
Remark 2. If Assumption 1 holds, in particular, if the sets U have the form U = [ail,6il] x [ai2, bi2] x ... x [aik, bik] for all i, then Assumption 6 also holds.
Theorem 2. Suppose that Assumptions 1, 4, 5, and 6 hold and 5 > 0. Then an m-fold capture occurs in the game G(n + 1).
Proof. Define the number
t
T = infH>0 : sup min max max ( Ei(aijta, 1) — / gij(t, s)X*j(v(s))ds)< 0>.
u(.)gV Aen(m) /€A jeJ \ a I '■> J J
o
Then, by Lemma 3, T < Let v(-) be the admissible control of the evader. Consider the sets (i€l,j € J, v(-) €V)
Tij(v(-)) = {t ■ EiiauT01,!)- glj(T,s)\lj(v(s))ds<0}.
t
Next, let
* (( )) J inf{t : t € Tij(v(■ ))} if Tij(v(■ )) = 0, ß(.)=l Aj(v(t)), t € [0,tj(v(■ ))]
ij if Tij(v(■ ))= 0, P'jU \0, t € (tj(v(■ )),T
Bi(t) =diag(ßii(t),...,ßifc (t)).
Define the controls of pursuers Pi, i € I, assuming
ui(t) = v(t) - Bi(t)z0.
The solution of the Cauchy problem for system (2.1) has the form [9]
t
Zi(t) = Ei(Aita, 1 )4> + /(i - s)a-lEi(Ai(t - s)a-\a)(Ui(s) - v(s))ds.
J 0
Therefore,
T
zl3(T) = (e^T", 1) - / gij{T, s)Bij(s)ds^z®j
a
0
= /' gl3{T,s)XMs))ds]z"r
0
It follows from the assumptions of the theorem and the definition of Bi(t) (i € /, t € [0, to)) that there exists A € Q(m) such that Zj(T) = 0 for all l € A and j € J, which implies that an m-fold capture occurs in the game G(n + 1). This proves the theorem. □
Example 1. Suppose that, in system (2.1), k = 2, n = 1, m = 1, z0 = (1,2), A is a zero matrix, V = {0}, and
Li = {(u, u2) : u=0, u2€[-1,1]} U {(u, u2) : u2=0, u€[-1,1]} U {(u, u2) : ui=u2€[-1,1]}. Then
M0(0) = {(0 A) .A € [0,1/2]}u{(A S).A € [0,1]}u{(A a02) .A € [0,1]}.
Hence,
Consequently,
sup min A1j = 1/2.
LeM?(0) j
mi<0>HI'0 i02
and the extremal selector is A1 (0) = diag (1,1/2). Therefore, T = (2oT(a))i/a, and the control of the pursuer Pi has the form
i(-i,-1), t € [A,Ti] , iW | (A,-1), t € (ri,r],
where Ti = T — (ar(a))1/a. Then [9]
T
Zl(T) = + _L y(T _ s)»-\il{s)ds_ 0
Therefore,
Ti T
zn(T) = z°n - -L J(T- sY~lds = 0, z12(T) = z°12 - ^ J(T - sr-'ds = 0. 00
Note that the use of scalar resolving functions, i.e., functions of the form
'A 0N
L '.0 A
does not allow one to get the capture since, in this case, the condition -Lz0 € U1 — v is satisfied only for the zero matrix L.
We now present conditions on the game parameters under which the capture is guaranteed when scalar resolving functions are used.
Assumption 7. In system (2.1), the matrices Ai have the form Ai = aiE, ai < 0, i € I, E is an identity matrix, and
¿0 = inf max min uAv) > 0, v&V Aen(m) leA
where ^i(v) = sup{^ > 0 : — € Ul — v}.
Theorem 3. Suppose that Assumptions 1 and 7 hold. Then an m-fold capture occurs in the game G(n + 1).
Proof. It follows from the conditions of the theorem that the following equations hold for all i € I, j € J:
gij(t,s) = (t-s)a~lEi (ai(t-s)a,a)=gi(t,s), t^s, gij(t,t) = 0,
Oi
Ei(aijta, 1) = Ei(aita,l).
Oi Oi
Therefore, it follows from Lemma 3 that there exists a time T > 0 such that, for every admissible function v(-) € V, there is a set A € Q(m) such that the inequalities
T
E^aiT", 1) - Jgi(T,s)^(v(s))ds < 0
hold for all l € A. Define the number
t
To = inf-u>0 : sup min max) Ei(aita, 1) — / gi(t, s)^i(v(s))ds)< 0 L i,(.) Aen(m) /€A V a J )
Next, let v( ■ ) be the admissible control of the evader:
t
n = infji > 0 : ExicnTS1,1) - Jgi(T0,s)m(v(s))ds < o}.
0
It follows from the above proof that there exists a set A0 € Q(m) such that the inequalities T < T0 hold for all l € A0. Define the controls of pursuers Pj, i € I, assuming
u(t) = i v(t) - ^j(v(t))z°, t € [0,Tj], j \v(t), t € [Tj,T0].
The solution of the Cauchy problem for system (2.1) has the form [9]
To
zi(T0) = (E±_{aiT£,l)- jgi{T0,s)^{v{s))ds)zl
0
This equation and the definition of A0 imply that z (T0) = 0 for all l € A0. This proves the theorem. □
Corollary 1. Suppose that, in system (2.1), the matrices A have the form A = fl,E, a < 0, i € /, E is an identity matrix, U = V for all i € /, V is a strictly convex compact set with a smooth boundary, and
0 € p| Intco {z0,l € A}, (3.6)
Aen(ra-m+1)
where Int A and co A denote the interior and the convex hull of the set A, respectively. Then an m-fold capture occurs in the game G(n + 1).
Indeed, in this case, condition (3.6) implies that ¿0 > 0 [30].
4. Conclusion
We obtained new sufficient conditions for multiple capture of the evader in the group pursuit problem with fractional derivatives. To solve the problem, we introduced matrix resolving functions.
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