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Compactness of the set of trajectories of the controllable system described by an affine integral equation
Applied Mathematics and Computation 219 (2013) 8416–8424
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Compactness of the set of trajectories of the controllable system described by an affine integral equation Nesir Huseyin ⇑, Anar Huseyin Anadolu University, Mathematics Department, Eskisehir 26470, Turkey
a r t i c l e
i n f o
Keywords: Affine integral equation Controllable system Integral constraint Set of trajectories
a b s t r a c t In this paper the control system with integral constraint on the controls is studied. It is assumed that the behavior of the system is described by an affine Volterra integral equation which is nonlinear with respect to the state vector and is linear with respect to the control vector. The closed ball of the space Lp ðp > 1Þ with radius l0 and centered at the origin, is chosen as the set of admissible control functions. It is proved that the set of trajectories is a bounded, closed, precompact and hence compact subset of the space of continuous functions. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction Control systems with integral constraint on the controls arise in various fields of the theory and applications of the applied mathematics. In general, they naturally arise as control problems with bounded Lp norms on the controls, control problems with prescribed bounded total energy and finance, and control problems with design uncertainties (see., e.g. [1–6]). For example, the motion of flying objects with variable mass, is described in the form of controllable system, where the control function has an integral constraint (see., e.g. [4–6]). Control system with integral constraint on the controls whose behavior is described by a differential equation is studied in [1–8]. In [7,8] various topological properties and numerical construction methods of the set of trajectories and attainable sets of the control system with integral constraint on the controls are studied where the behavior of the system is described by an affine differential equation. Integral equations appear in many problems of contemporary physics and mechanics (see., e.g. [9–17]). In this paper the control system with integral constraint on the controls whose behavior is described by a Volterra integral equation is considered. It is assumed that integral equation is affine, i.e. it is nonlinear with respect to the state vector and is linear with respect to the control vector. The closed ball of the space Lp ðp > 1Þ with radius l0 and centered at the origin, is chosen as the set of admissible control functions. The set of the trajectories of the system generated by all admissible control functions is studied. The paper is organized as follows: In Section 2 the conditions which satisfy the system are formulated (Conditions 2.A, 2.B and 2.C). In Section 3 the existence and uniqueness of the trajectory generated by a given admissible control function is shown (Theorem 3.1). In Section 4 the boundedness of the set of trajectories is proved (Theorem 4.1). In Section 5 it is proved that the set of trajectories is a precompact subset of the space of continuous functions (Theorem 5.1). In Section 6 the closedness of the set of trajectories is shown (Theorem 6.1).
⇑ Corresponding author. E-mail addresses: [email protected] (N. Huseyin), [email protected] (A. Huseyin). 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.03.005
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2. The system Consider controllable system the behavior of which is described by an integral equation
xðtÞ ¼ g ðt; xðt ÞÞ þ k
Z
t
½K 1 ðt; s; xðsÞÞ þ K 2 ðt; s; xðsÞÞuðsÞds;
ð2:1Þ
t0
where x 2 Rn is the state vector, u 2 Rm is the control vector, t 2 ½t 0 ; h. The system (2.1) is affine with respect to the control vector, but it is nonlinear with respect to the state vector. Therefore the system (2.1) will be called an affine control system. Let p > 1 and l0 > 0 be given numbers. The function uðÞ 2 Lp ð½t0 ; h; Rm Þ such that kuðÞkp 6 l0 is said to be an admissible R 1p h control function where kuðÞkp ¼ t0 kuðtÞkp dt . The set of all admissible control functions is denoted by symbol U p . Thus
n o U p ¼ uðÞ 2 Lp ð½t0 ; h; Rm Þ : kuðÞkp 6 l0 : It is obvious that the set of admissible control functions is the closed ball with radius l0 and centered at the origin in the space Lp ð½t 0 ; h; Rm Þ. It is assumed that the functions and a number k 2 R1 given in system (2.1) satisfy the following conditions: 2.A. the vector functions gð; Þ : ½t0 ; h Rn ! Rn ; K 1 ð; ; Þ : ½t 0 ; h ½t0 ; h Rn ! Rn and matrix function K 2 ð; ; Þ : ½t0 ; h ½t0 ; h Rn ! Rnm are continuous; 2.B. there exist L0 2 ½0; 1Þ; L1 P 0 and L2 P 0 such that
kgðt; x1 Þ gðt; x2 Þk 6 L0 kx1 x2 k; kK 1 ðt; s; x1 Þ K 1 ðt; s; x2 Þk 6 L1 kx1 x2 k; kK 2 ðt; s; x1 Þ K 2 ðt; s; x2 Þk 6 L2 kx1 x2 k n for every ðt;s; x1 Þ 2 ½t 0 ; h ½t0 ; h p1Rn ; ðt; s; x2 Þ 2 ½t0 ; h ½t 0 ; h R ; p 2.C. 0 6 k L1 ðh t0 Þ þ L2 ðh t0 Þ l0 < 1 L0 .
We denote
h i p1 LðkÞ ¼ L0 þ k L1 ðh t0 Þ þ L2 ðh t 0 Þ p l0 :
ð2:2Þ
According to the condition 2.C we obtain that LðkÞ < 1. Now, let us define a trajectory of the system (2.1) generated by an admissible control function. Let u ðÞ 2 U p . A continuous function x ðÞ : ½t0 ; h ! Rn satisfying the integral equation
x ðtÞ ¼ g ðt; x ðt ÞÞ þ k
Z
t
½K 1 ðt; s; x ðsÞÞ þ K 2 ðt; s; x ðsÞÞu ðsÞds
t0
for every t 2 ½t0 ; h is said to be a trajectory of the system (2.1) generated by the admissible control function u ðÞ 2 U p . The trajectory of the system (2.1) generated by the control function uðÞ 2 U p is denoted by xð; uðÞÞ and we set
Xp ¼ xð; uðÞÞ : uðÞ 2 U p : Xp is called the set of trajectories of the system (2.1). It is obvious that Xp C ð½t 0 ; h; Rn Þ where C ð½t 0 ; h; Rn Þ is the space of continuous functions xðÞ : ½t 0 ; h ! Rn with norm
kxðÞkC ¼ max fkxðtÞk : t 2 ½t 0 ; hg: For t 2 ½t 0 ; h we denote
Xp ðtÞ ¼ xðtÞ : xðÞ 2 Xp :
ð2:3Þ
The set Xp ðtÞ consists of points to which arrive the trajectories of the system at the instant of t. 3. Existence and uniqueness of trajectories In this section it is proved that under conditions 2.A – 2.C every admissible control function generates a unique trajectory. Theorem 3.1. Let u ðÞ 2 U p . Then the system (2.1) has unique x ðÞ ¼ xð; u ðÞÞ trajectory generated by the admissible control function u ðÞ. Proof. Let us define a map xðÞ ! AðxðÞÞ; xðÞ 2 C ð½t0 ; h; Rn Þ, setting
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AðxðÞÞjðtÞ ¼ g ðt; xðtÞÞ þ k
Z
t
½K 1 ðt; s; xðsÞÞ þ K 2 ðt; s; xðsÞÞu ðsÞds;
t 2 ½t 0 ; h:
ð3:1Þ
t0
Since u ðÞ 2 U p ; xðÞ 2 C ð½t 0 ; h; Rn Þ then by virtue of condition 2.A we get that the function t ! AðxðÞÞjðtÞ; t 2 ½t0 ; h, is continuous. Thus the map xðÞ ! AðxðÞÞ defined by (3.1) is in the form
AðÞ : C ð½t 0 ; h; Rn Þ ! C ð½t 0 ; h; Rn Þ: Let x1 ðÞ 2 C ð½t0 ; h; Rn Þ and x2 ðÞ 2 C ð½t 0 ; h; Rn Þ be arbitrarily chosen functions. Then applying condition 2.B we obtain
kAðx2 ðÞÞjðtÞ Aðx1 ðÞÞjðtÞk 6 L0 kx2 ðÞ x1 ðÞkC þ kL1
Z
t t0
kx2 ðÞ x1 ðÞkC ds þ kL2
Z
t
t0
kx2 ðÞ x1 ðÞkC ku ðsÞkds
Z t ¼ L0 þ kL1 ðh t 0 Þ þ kL2 ku ðsÞkds kx2 ðÞ x1 ðÞkC
ð3:2Þ
t0
for every t 2 ½t0 ; h. Since u ðÞ 2 U p then Hölder’s inequality implies
Z
t
ku ðsÞkds 6 ðt t 0 Þ
p1 p
Z
t0
t
p
ku ðsÞk ds
1p 6 ðh t 0 Þ
t0
p1 p
l0 :
From (2.2) and (3.2) it follows that the inequality
p1 kAðx2 ðÞÞjðtÞ Aðx1 ðÞÞjðtÞk 6 L0 þ kL1 ðh t0 Þ þ kL2 ðh t 0 Þ p l0 kx2 ðÞ x1 ðÞkC ¼ LðkÞkx2 ðÞ x1 ðÞkC
ð3:3Þ
holds for every t 2 ½t 0 ; h. Finally, (3.3) yields
kAðx2 ðÞÞjðÞ Aðx1 ðÞÞjðÞkC 6 LðkÞkx2 ðÞ x1 ðÞkC :
ð3:4Þ
According to the condition 2.C we have LðkÞ < 1. Then it follows from (3.4) that the map AðÞ : C ð½t0 ; h; Rn Þ ! C ð½t 0 ; h; Rn Þ defined by (3.1) is contractive and consequently it has a unique fixed point x ðÞ 2 C ð½t0 ; h; Rn Þ which is unique trajectory of the equation
x ðtÞ ¼ g ðt; x ðtÞÞ þ k
Z
t
½K 1 ðt; s; x ðsÞÞ þ K 2 ðt; s; x ðsÞÞu ðsÞds;
t 2 ½t0 ; h:
t0
h From condition 2.B it follows that for each fixed t 2 ½t0 ; h the function x ! gðt; xÞ is a contractive map and hence for each fixed t 2 ½t 0 ; h it has a unique fixed point. So there exists a unique sðtÞ 2 Rn ; t 2 ½t0 ; h, such that
sðtÞ ¼ gðt; sðtÞÞ;
t 2 ½t 0 ; h:
Let s0 ¼ sðt0 Þ. Then we have from (2.1) that xðt0 Þ ¼ s0 for every xðÞ 2 Xp . So we obtain the validity of the following proposition. Proposition 3.1. Xp ðt 0 Þ ¼ fs0 g.
4. Boundedness of the set of trajectories In this section the boundedness of the set of trajectories Xp is shown. Denote
c0 ¼ max fkgðt; 0Þk : t 2 ½t0 ; hg;
ð4:1Þ
c1 ¼ max fkK 1 ðt; s; 0Þk : ðt; sÞ 2 ½t 0 ; h ½t0 ; hg;
ð4:2Þ
c2 ¼ max fkK 2 ðt; s; 0Þk : ðt; sÞ 2 ½t 0 ; h ½t0 ; hg:
ð4:3Þ
Proposition 4.1. Let the functions gð; Þ : ½t 0 ; h Rn ! Rn ; K 1 ð; ; Þ : ½t 0 ; h ½t0 ; h Rn ! Rn and K 2 ð; ; Þ : ½t 0 ; h ½t0 ; h Rn ! Rnm satisfy the conditions 2.A and 2.B. Then
kgðt; xÞk 6 c0 þ L0 kxk;
kK 1 ðt; s; xÞk 6 c1 þ L1 kxk; n
kK 2 ðt; s; xÞk 6 c2 þ L2 kxk
for every ðt; s; xÞ 2 ½t 0 ; h ½t0 ; h R where the constants L0 ; L1 and L2 are given in condition 2.B.
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Proof. The proof of the proposition follows from Lipschitz continuity of the functions gð; Þ : ½t 0 ; h Rn ! Rn ; K 1 ð; ; Þ : ½t0 ; h ½t 0 ; h Rn ! Rn and K 2 ð; ; Þ : ½t0 ; h ½t 0 ; h Rn ! Rnm with respect to x and from definitions of the numbers c0 ; c1 and c2 , i.e. from (4.1)–(4.3). h Denote
r ¼
c0 þ kc1 ðh t0 Þ þ kc2 ðh t 0 Þ 1 L0
p1 p
l0
exp
LðkÞ L0 1 L0
ð4:4Þ
where L0 P 0 is defined in condition 2.B, LðkÞ is defined by (2.2), c0 P 0; c1 P 0 and c2 P 0 are defined by (4.1)–(4.3) respectively. Theorem 4.1. Let the functions gð; Þ : ½t0 ; h Rn ! Rn ; K 1 ð; ; Þ : ½t0 ; h ½t 0 ; h Rn ! Rn ; K 2 ð; ; Þ : ½t 0 ; h ½t 0 ; h Rn ! Rnm and the number k > 0 satisfy the conditions 2.A, 2.B and 2.C. Then for every xðÞ 2 Xp the inequality
kxðÞkC 6 r holds. Proof. Let xðÞ 2 Xp be an arbitrarily chosen trajectory. Then there exists uðÞ 2 U p such that
xðtÞ ¼ g ðt; xðt ÞÞ þ k
Z
t
½K 1 ðt; s; xðsÞÞ þ K 2 ðt; s; xðsÞÞuðsÞds
ð4:5Þ
t0
for every t 2 ½t0 ; h. From (4.5) and Proposition 4.1 it follows
kxðtÞk 6 kg ðt; xðt ÞÞk þ k
Z
t
kK 1 ðt; s; xðsÞÞkds þ k
Z
t0
6 c0 þ L0 kxðtÞk þ k
Z
t
t0 t
ðc1 þ L1 kxðsÞkÞds þ k
kK 2 ðt; s; xðsÞÞkkuðsÞkds Z
t0
t
ðc2 þ L2 kxðsÞkÞkuðsÞkds
t0 p1 p
6 L0 kxðtÞk þ c0 þ kc1 ðh t 0 Þ þ kc2 ðh t0 Þ
l0 þ k
Z
t
ðL1 þ L2 kuðsÞkÞkxðsÞkds
t0
for every t 2 ½t0 ; h. Since L0 2 ½0; 1Þ then from the last inequality we have
kxðtÞk 6
c0 þ kc1 ðh t 0 Þ þ kc2 ðh t 0 Þ 1 L0
p1 p
l0
þ
k 1 L0
Z
t
ðL1 þ L2 kuðsÞkÞkxðsÞkds
ð4:6Þ
t0
for every t 2 ½t0 ; h. (2.2), (4.4) and (4.6), Gronwall’s and Hölder’s inequalities yield
Z t k ðL1 þ L2 kuðsÞkÞds 1 L0 t 0 p1 p1 c0 þ kc1 ðh t 0 Þ þ kc2 ðh t 0 Þ p l0 k exp L1 ðh t 0 Þ þ L2 ðh t 0 Þ p l0 6 1 L0 1 L0 p1 c0 þ kc1 ðh t 0 Þ þ kc2 ðh t0 Þ p l0 LðkÞ L0 ¼ r exp ¼ 1 L0 1 L0
kxðtÞk 6
c0 þ kc1 ðh t 0 Þ þ kc2 ðh t 0 Þ 1 L0
for every t 2 ½t0 ; h. So, kxðÞkC 6 r .
p1 p
l0
exp
h
5. Precompactness of the set of trajectories In this section precompactness of set of trajectories of the system (2.1) is prowed. Let us introduce some notations. Let D > 0 be a given number, r > 0 be defined by (4.4), Bn ðr Þ ¼ fx 2 Rn : kxk 6 r g. We set
D1 ¼ ½t0 ; h Bn ðr Þ;
ð5:1Þ
D2 ¼ ½t0 ; h ½t 0 ; h Bn ðr Þ;
ð5:2Þ
M1 ¼ max fkK 1 ðt; s; xÞk : ðt; s; xÞ 2 D2 g;
ð5:3Þ
M2 ¼ max fkK 2 ðt; s; xÞk : ðt; s; xÞ 2 D2 g;
ð5:4Þ
x0 ðDÞ ¼ max fkgðt2 ; xÞ gðt1 ; xÞk : jt2 t1 j 6 D; ðt1 ; xÞ 2 D1 ; ðt2 ; xÞ 2 D1 g;
ð5:5Þ
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x1 ðDÞ ¼ max fkK 1 ðt2 ; s; xÞ K 1 ðt1 ; s; xÞk : jt2 t1 j 6 D; ðt1 ; s; xÞ 2 D2 ; ðt2 ; s; xÞ 2 D2 g;
ð5:6Þ
x2 ðDÞ ¼ max fkK 2 ðt2 ; s; xÞ K 2 ðt1 ; s; xÞk : jt2 t1 j 6 D; ðt1 ; s; xÞ 2 D2 ; ðt2 ; s; xÞ 2 D2 g;
ð5:7Þ
i p1 p1 1 h x0 ðDÞ þ kx1 ðDÞðh t0 Þ þ kM1 D þ kx2 ðDÞðh t0 Þ p l0 þ kM2 D p l0 : 1 L0
ð5:8Þ
uðDÞ ¼
It is obvious that x0 ðDÞ ! 0þ ; x1 ðDÞ ! 0þ ; x2 ðDÞ ! 0þ ; uðDÞ ! 0þ as D ! 0þ and x0 ðD1 Þ 6 x0 ðD2 Þ; x1 ðD1 Þ 6 x1 ðD2 Þ; x2 ðD1 Þ 6 x2 ðD2 Þ; uðD1 Þ 6 uðD2 Þ if D1 < D2 . Proposition 5.1. For every xðÞ 2 Xp ; t 1 2 ½t 0 ; h; t 2 2 ½t 0 ; h the inequality
kxðt2 Þ xðt1 Þk 6 uðjt2 t 1 jÞ holds where uðÞ is defined by (5.8). Proof. Let us choose an arbitrary xðÞ 2 Xp and without less of generality let us assume that t 2 > t1 . Then there exists uðÞ 2 U p such that
xðtÞ ¼ g ðt; xðtÞÞ þ k
Z
t
½K 1 ðt; s; xðsÞÞ þ K 2 ðt; s; xðsÞÞuðsÞds
ð5:9Þ
t0
for every t 2 ½t0 ; h. Then it follows from (5.9)
kxðt2 Þ xðt1 Þk 6 kg ðt2 ; xðt2 ÞÞ g ðt1 ; xðt2 ÞÞk þ kg ðt1 ; xðt2 ÞÞ g ðt 1 ; xðt1 ÞÞk Z t2 Z t1 þk kK 1 ðt 2 ; s; xðsÞÞ K 1 ðt 1 ; s; xðsÞÞkds þ k kK 1 ðt 2 ; s; xðsÞÞkds t0
þk
Z
t1
t1
kK 2 ðt 2 ; s; xðsÞÞ K 2 ðt 1 ; s; xðsÞÞkkuðsÞkds þ k
t0
Z
t2
kK 2 ðt2 ; s; xðsÞÞkkuðsÞkds:
ð5:10Þ
t1
By virtue of condition 2.B
kg ðt1 ; xðt2 ÞÞ g ðt1 ; xðt1 ÞÞk 6 L0 kxðt 2 Þ xðt 1 Þk;
L0 2 ½0; 1Þ:
ð5:11Þ
Since xðÞ 2 Xp then Theorem 4.1 implies that
xðtÞ 2 Bn ðr Þ
ð5:12Þ
for every t 2 ½t0 ; h. (5.1), (5.5) and (5.12) imply
kg ðt2 ; xðt2 ÞÞ g ðt1 ; xðt2 ÞÞk 6 x0 ðjt 2 t1 jÞ:
ð5:13Þ
From (5.2), (5.6) and (5.12) it follows that
kK 1 ðt 2 ; s; xðsÞÞ K 1 ðt 1 ; s; xðsÞÞk 6 x1 ðjt 2 t1 jÞ for every s 2 ½t0 ; h and hence
Z
t1
kK 1 ðt 2 ; s; xðsÞÞ K 1 ðt 1 ; s; xðsÞÞkds 6 x1 ðjt 2 t 1 jÞðt 1 t 0 Þ 6 x1 ðjt 2 t1 jÞðh t 0 Þ:
ð5:14Þ
t0
(5.2), (5.7) and (5.12) yield
kK 2 ðt 2 ; s; xðsÞÞ K 2 ðt 1 ; s; xðsÞÞk 6 x2 ðjt 2 t1 jÞ
ð5:15Þ
for every s 2 ½t0 ; h. Since uðÞ 2 U p then we have from (5.15)
Z
t1
kK 2 ðt 2 ; s; xðsÞÞ K 2 ðt 1 ; s; xðsÞÞkkuðsÞkds 6
t0
Z
t1
t0
p1
x2 ðjt2 t1 jÞkuðsÞkds 6 x2 ðjt2 t1 jÞðh t0 Þ p l0 :
ð5:16Þ
From (5.2)–(5.4) and (5.12) it follows that
Z
t2
t1
kK 1 ðt 2 ; s; xðsÞÞkds 6 M 1 jt 2 t 1 j;
ð5:17Þ
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Z
t2
kK 2 ðt2 ; s; xðsÞÞkkuðsÞkds 6 M2
Z
t1
t2
kuðsÞkds 6 M 2 jt 2 t 1 j
p1 p
t1
l0 :
ð5:18Þ
From (5.10), (5.11), (5.13), (5.14), (5.16), (5.17) and (5.18) we obtain
kxðt2 Þ xðt 1 Þk 6 x0 ðjt 2 t 1 jÞ þ L0 kxðt2 Þ xðt1 Þk þ kx1 ðjt2 t1 jÞðh t 0 Þ þ kx2 ðjt2 t1 jÞðh t 0 Þ þ kM 1 jt 2 t 1 j þ kM 2 jt 2 t1 j
p1 p
p1 p
l0
l0 :
ð5:19Þ
Finally, (5.8) and (5.19) imply
kxðt2 Þ xðt1 Þk 6
i p1 p1 1 h x0 ðjt2 t1 jÞ þ kx1 ðjt2 t1 jÞðh t0 Þ þ kx2 ðjt2 t1 jÞðh t0 Þ p l0 þ kM1 jt2 t1 j þ kM2 jt2 t1 j p l0 1 L0
¼ uðjt 2 t 1 jÞ: h Proposition 5.2. The set of trajectories Xp C ð½t0 ; h; Rn Þ is a set of equicontinuous functions. Proof. Let us choose an inequality
e > 0. Since uðDÞ ! 0þ as D ! 0þ then there exists D ðeÞ > 0 such that for every D 2 ð0; D ðeÞ the
uðDÞ 6 e
ð5:20Þ
holds where uðÞ is defined by (5.8). Now let xðÞ 2 Xp be an arbitrarily chosen trajectory and jt2 t 1 j 6 D ðeÞ where t 1 2 ½t0 ; h; t 2 2 ½t 0 ; h. Then from (5.20) and Proposition 5.1 we have
kxðt2 Þ xðt 1 Þk 6 uðjt2 t1 jÞ 6 uðD ðeÞÞ 6 e and hence the set trajectories Xp C ð½t 0 ; h; Rn Þ is a set of equicontinuous functions.
h
From Theorem 4.1 and Proposition 5.2 we immediately obtain the validity of the following theorem. Theorem 5.1. The set of trajectories Xp is a precompact subset of the space C ð½t 0 ; h; Rn Þ. Note that from Proposition 5.1 it follows continuity of the set valued map t ! Xp ðtÞ; t 2 ½t 0 ; h. The Hausdorff distance between the sets F Rn and E Rn is denoted by hðF; EÞ and defined as
hðF; EÞ ¼ maxfsup dðx; EÞ; sup dðy; FÞg; x2F
y2E
where dðx; EÞ ¼ inf fkx yk : y 2 Eg. Proposition 5.3. For every t 1 2 ½t0 ; h and t 2 2 ½t 0 ; h the inequality
h Xp ðt 2 Þ; Xp ðt1 Þ 6 uðjt 2 t1 jÞ
is verified where the function uðÞ : ½0; 1Þ ! ½0; 1Þ is defined by (5.8), the sets Xp ðt 1 Þ and Xp ðt2 Þ are defined by (2.3). Since uðDÞ ! 0þ as D ! 0þ then we conclude the validity of the following corollary. Corollary 5.1. Set valued map t ! Xp ðtÞ; t 2 ½t 0 ; h, is continuous. 6. Closedness of the set of trajectories The following theorem characterizes closedness of the set of trajectories Xp . Theorem 6.1. The set of trajectories Xp is a closed subset of the space C ð½t0 ; h; Rn Þ. Proof. Let xk ðÞ 2 Xp for every k ¼ 1; 2; . . . and kxk ðÞ x0 ðÞkC ! 0 as k ! 1 where x0 ðÞ 2 C ð½t 0 ; h; Rn Þ. Let us show that x0 ðÞ 2 Xp . Since xk ðÞ 2 Xp then there exists uk ðÞ 2 U p such that
xk ðtÞ ¼ g ðt; xk ðt ÞÞ þ k
Z
t
t0
½K 1 ðt; s; xk ðsÞÞ þ K 2 ðt; s; xk ðsÞÞuk ðsÞds;
t 2 ½t0 ; h:
ð6:1Þ
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Since the set of admissible control functions U p is closed ball with radius l0 and centered at the origin of the space Lp ð½t0 ; h; Rn Þ ðp > 1Þ, then the set U p is weakly compact. Then without loss of generality, we can assume that the sequence n fuk ðÞg1 k¼1 weakly converges to a u ðÞ 2 U p . Let x ðÞ : ½t 0 ; h ! R be a trajectory of the system (2.1) generated by the admissible control function u ðÞ 2 U p . Then
x ðtÞ ¼ g ðt; x ðtÞÞ þ k
Z
t
½K 1 ðt; s; x ðsÞÞ þ K 2 ðt; s; x ðsÞÞu ðsÞds; t 2 ½t0 ; h:
ð6:2Þ
t0
From (6.1) , (6.2) and condition 2.B it follows
kxk ðt Þ x ðt Þk 6
k 1 L0
Z
t
ðL1 þ L2 kuk ðsÞkÞkxk ðsÞ x ðsÞkds þ
t0
Z t k K 2 ðt; s; x ðsÞÞðuk ðsÞ u ðsÞÞds 1 L0 t0
ð6:3Þ
for every t 2 ½t0 ; h. Let us denote zðt; sÞ ¼ K 2 ðt; s; x ðsÞÞ. Then the function zðt; sÞ : ½t0 ; h ½t0 ; h ! Rnm is continuous and the inequality (6.3) can be rewritten as
kxk ðt Þ x ðt Þk 6
k 1 L0
Z
t
ðL1 þ L2 kuk ðsÞkÞkxk ðsÞ x ðsÞkds þ
t0
Z t k zðt; sÞðuk ðsÞ u ðsÞÞds : 1 L0 t0
ð6:4Þ
n Since the sequence fuk ðÞg1 k¼1 weakly converges to u ðÞ 2 U p as k ! 1 in the space Lp ð½t 0 ; h; R Þ then, we have that for each nm fixed t 2 ½t 0 ; h and w ðÞ 2 Lq ð½t0 ; t; R Þ
Z t w ðsÞ½uk ðsÞ u ðsÞds ! 0 as k ! 1:
ð6:5Þ
t0
Since the function ðt; sÞ ! zðt; sÞ : ½t 0 ; h ½t 0 ; h ! Rnm is continuous then by virtue of (6.5) we have that
Z t zðt; sÞ½uk ðsÞ u ðsÞds ! 0 as k ! 1
ð6:6Þ
t0
for every fixed t 2 ½t 0 ; h. From (6.6) we conclude that for e > 0 and fixed t 2 ½t0 ; h there exists Kðe; tÞ > 0 such that for every k > Kðe; tÞ the inequality
Z t zðt; sÞ½uk ðsÞ u ðsÞds < e
ð6:7Þ
t0
holds. Now let us prove that for each t 2 ½t 0 ; h the inequality
e > 0 there exists kðeÞ > 0 (which does not depend on t) such that for every k > kðeÞ and
Z t zðt; sÞðuk ðsÞ u ðsÞÞds < e
ð6:8Þ
t0
holds. Let us assume the contrary, i.e. let there exist e > 0; ki > 0 and t i 2 ½t0 ; h ði ¼ 1; 2; . . .Þ such that ki ! 1 as i ! 1 and
Z
ti
t0
zðt i ; sÞ uki ðsÞ u ðsÞ ds P e :
ð6:9Þ
Since ti 2 ½t 0 ; h for every i ¼ 1; 2; . . . and ½t0 ; h is a compact set, then without loss of generality one can assume that t i ! t as i ! 1 and t 2 ½t 0 ; h. It is obvious that
Z
ti
t0
Z
zðt i ; sÞ uki ðsÞ u ðsÞ ds 6
t
t0
Z
zðti ; sÞ uki ðsÞ u ðsÞ ds þ
ti
t
zðti ; sÞ uki ðsÞ u ðsÞ ds
Z t Z
t
þ 6 ½ z ð t ; s Þ z ð t ; s Þ u ðsÞ u ðsÞ ds z ð t ; s Þ u ðsÞ u ðsÞ ds i k k i i t0 t0 Z ti
þ zðt i ; sÞ uki ðsÞ u ðsÞ ds
ð6:10Þ
t
for every i ¼ 1; 2; . . . (6.7) implies that for
Z
t
t0
e > 0 and t 2 ½t0 ; h there exists N1 > 0 such that for every i > N1 the inequality
e zðt ; sÞ uki ðsÞ u ðsÞ ds < 5
is verified. Continuity of the function zð; Þ : ½t0 ; h ½t 0 ; h ! Rnm yields that for given every i > N 2 and s 2 ½t0 ; h the inequality
ð6:11Þ
e 10ðht 0 Þ
p1 p
l0
there exists N 2 > 0 such that for
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kzðti ; sÞ zðt ; sÞk <
e 10ðh t 0 Þ
p1 p
ð6:12Þ
l0
holds. Since u ðÞ 2 U p ; uki ðÞ 2 U p for every i ¼ 1; 2; . . . ; 1q ¼ p1 then (6.12) and Hölder’s inequality implies that p
Z
t
t0
½zðti ; sÞ zðt ; sÞ uki ðsÞ u ðsÞ ds 6
Z
e 10ðh t0 Þ
62
p1 p
l0
e 10ðh t0 Þ
p1 p
l0
t
t0
uk ðsÞ þ ku ðsÞk ds i
ðh t 0 Þ
p1 p
l0 ¼
e 5
ð6:13Þ
for every i > N 2 . Since zðt; sÞ ¼ K 2 ðt; s; x ðsÞÞ; x ðÞ 2 Xp then we have from Theorem 4.1 and (5.4) that for every i ¼ 1; 2; . . . and s 2 ½t 0 ; h the inequality
kzðti ; sÞk 6 M2
ð6:14Þ
is satisfied. Without loss of generality let us assume that ti P t for every i ¼ 1; 2; . . . (Note that it can be assumed that ti 6 t for every i ¼ 1; 2; . . .). Since u ðÞ 2 U p ; uki ðÞ 2 U p for every i ¼ 1; 2; . . . ; 1q ¼ p1 then it follows from (6.14) and Hölder’s p inequality that
Z
ti
t
Z
6 M zðt i ; sÞ uki ðsÞ u ðsÞ ds 2
ti
t
uk ðsÞ ds þ i
Z
ti
t
p1 ku ðsÞkds 6 2M 2 ðt i t Þ p l0 :
ð6:15Þ
Since t i ! t as i ! 1 then we obtain from (6.15) that there exists N 3 > 0 such that for every i > N 3 the inequality
Z
ti
t
e zðt i ; sÞ uki ðsÞ u ðsÞ ds < 5
ð6:16Þ
holds. Now let N ¼ max fN 1 ; N 2 ; N 3 g. Then from (6.10), (6.11), (6.13) and (6.16) we have
Z
ti
t0
e e e 3e zðt i ; sÞ uki ðsÞ u ðsÞ ds < 5 þ 5 þ 5 ¼ 5 < e
ð6:17Þ
for every i > N . The inequalities (6.9) and (6.17) contradict and hence the inequality (6.8) is verified. Then from (6.4) and (6.8) we conclude that for every t 2 ½t0 ; h and k > kðeÞ the inequality
kxk ðt Þ x ðt Þk 6
ke k þ 1 L0 1 L0
Z
t
ðL1 þ L2 kuk ðsÞkÞkxk ðsÞ x ðsÞkds
ð6:18Þ
t0
is verified. Since uk ðÞ 2 U p for every k ¼ 1; 2; . . . then (2.2), (6.18) and Gronwall’s inequality yield that
Z h ke k exp ðL1 þ L2 kuk ðsÞkÞds 1 L0 t0 1 L0 p1 ke k ke LðkÞ L0 exp L1 ðh t 0 Þ þ L2 ðh t0 Þ p l0 ¼ exp 6 1 L0 1 L0 1 L0 1 L0
kxk ðt Þ x ðt Þk 6
ð6:19Þ
for every t 2 ½t 0 ; h and k > kðeÞ. Finally, we get from (6.19) that xk ðÞ ! x ðÞ as k ! 1. From uniqueness of the limit we have x0 ðÞ ¼ x ðÞ and consequently x0 ðÞ 2 Xp . h From Theorem 5.1 and Theorem 6.1 we obtain compactness of the set of trajectories. Theorem 6.2. The set of trajectories Xp is a compact subset of the space C ð½t0 ; h; Rn Þ.
References [1] M. Motta, C. Sartori, Minimum time with bounded energy, minimum energy with bounded time, SIAM J. Control Optim. 42 (2003) 789–809. [2] A.I. Subbotin, V.N. Ushakov, Alternative for an encounter-evasion differential game with integral constraints on the players controls, J. Appl. Math. Mech. 39 (1975) 367–375. [3] S.T. Zavalishchin, A.N. Sesekin, Dynamic impulse systems. Theory and application, Kluwer, 1997. [4] V.V. Beletskii, Studies of Motions of Celestial Bodies, Nauka, 1972 (in Russian). [5] N.N. Krasovskii, Theory of Control of Motion: Linear systems, Nauka, 1968 (in Russian). [6] V.I. Ukhobotov, One dimensional projection method in linear differential games with integral constraints, Chelyabinsk, 2005. (In Russian) [7] Kh.G. Guseinov, O. Ozer, E. Akyar, V.N. Ushakov, The approximation of reachable sets of control systems with integral constraint on controls, Nonlinear Differ. Equ. Appl. (NoDEA) 14 (2007) 57–73. [8] Kh.G. Guseinov, O. Ozer, E. Akyar, On the continuity properties of the attainable sets of control systems with integral constraints on control, Nonlinear Anal. TMA 56 (2004) 433–449. [9] J. Banas, A. Chlebowicz, On integrable solutions of a nonlinear Volterra integral equation under Carathéodory conditions, Bull. Lond. Math. Soc. 41 (2009) 1073–1084.
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[10] T.A. Burton, J.R. Haddock, Qualitative properties of solutions of integral equations, Nonlinear Anal. TMA 71 (2009) 5712–5723. [11] A.I. Guseinov, H.S. Muhtarov, Introduction to the Theory of Nonlinear Singular Integral Equations, Nauka, 1980 (in Russian). [12] A.M. Lyapunov, Sur les figures d’equilibre peu differentes des ellipoides d’une masse liquide homogene douee d’un mouvement de rotation, Premiere partle. Etude generale du probleme, Zapiski Akademii Nauk, St.Petersburg, 1925, pp. 1–25. [13] T. Malolepszy, M. Niedziela, A note on blow-up solutions to some nonlinear Volterra integral equations, Appl. Math. Comput. (217) (2011) 5279–5284. [14] A.D. Polyanin, A.V. Manzhurov, Handbook of Integral Equation, CRC Press, 1998. [15] D. O’Regan, Applications of linking to Hammerstein integral equations, Appl. Math. Comput. (218) (2012) 6401–6406. [16] P.S. Urysohn, Works on topology and other fields of mathematics, I, Gosudarstv. Izdat. Tehn.-Teoret. Lit., 1951. (In Russian) [17] M. Väth, Volterra and Integral Equations of Vector Functions, M. Deccer Inc., 2000.
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Compactness of the set of trajectories of the controllable system described by an affine integral equation Nesir Huseyin ⇑, Anar Huseyin Anadolu University, Mathematics Department, Eskisehir 26470, Turkey
a r t i c l e
i n f o
Keywords: Affine integral equation Controllable system Integral constraint Set of trajectories
a b s t r a c t In this paper the control system with integral constraint on the controls is studied. It is assumed that the behavior of the system is described by an affine Volterra integral equation which is nonlinear with respect to the state vector and is linear with respect to the control vector. The closed ball of the space Lp ðp > 1Þ with radius l0 and centered at the origin, is chosen as the set of admissible control functions. It is proved that the set of trajectories is a bounded, closed, precompact and hence compact subset of the space of continuous functions. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction Control systems with integral constraint on the controls arise in various fields of the theory and applications of the applied mathematics. In general, they naturally arise as control problems with bounded Lp norms on the controls, control problems with prescribed bounded total energy and finance, and control problems with design uncertainties (see., e.g. [1–6]). For example, the motion of flying objects with variable mass, is described in the form of controllable system, where the control function has an integral constraint (see., e.g. [4–6]). Control system with integral constraint on the controls whose behavior is described by a differential equation is studied in [1–8]. In [7,8] various topological properties and numerical construction methods of the set of trajectories and attainable sets of the control system with integral constraint on the controls are studied where the behavior of the system is described by an affine differential equation. Integral equations appear in many problems of contemporary physics and mechanics (see., e.g. [9–17]). In this paper the control system with integral constraint on the controls whose behavior is described by a Volterra integral equation is considered. It is assumed that integral equation is affine, i.e. it is nonlinear with respect to the state vector and is linear with respect to the control vector. The closed ball of the space Lp ðp > 1Þ with radius l0 and centered at the origin, is chosen as the set of admissible control functions. The set of the trajectories of the system generated by all admissible control functions is studied. The paper is organized as follows: In Section 2 the conditions which satisfy the system are formulated (Conditions 2.A, 2.B and 2.C). In Section 3 the existence and uniqueness of the trajectory generated by a given admissible control function is shown (Theorem 3.1). In Section 4 the boundedness of the set of trajectories is proved (Theorem 4.1). In Section 5 it is proved that the set of trajectories is a precompact subset of the space of continuous functions (Theorem 5.1). In Section 6 the closedness of the set of trajectories is shown (Theorem 6.1).
⇑ Corresponding author. E-mail addresses: [email protected] (N. Huseyin), [email protected] (A. Huseyin). 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.03.005
N. Huseyin, A. Huseyin / Applied Mathematics and Computation 219 (2013) 8416–8424
8417
2. The system Consider controllable system the behavior of which is described by an integral equation
xðtÞ ¼ g ðt; xðt ÞÞ þ k
Z
t
½K 1 ðt; s; xðsÞÞ þ K 2 ðt; s; xðsÞÞuðsÞds;
ð2:1Þ
t0
where x 2 Rn is the state vector, u 2 Rm is the control vector, t 2 ½t 0 ; h. The system (2.1) is affine with respect to the control vector, but it is nonlinear with respect to the state vector. Therefore the system (2.1) will be called an affine control system. Let p > 1 and l0 > 0 be given numbers. The function uðÞ 2 Lp ð½t0 ; h; Rm Þ such that kuðÞkp 6 l0 is said to be an admissible R 1p h control function where kuðÞkp ¼ t0 kuðtÞkp dt . The set of all admissible control functions is denoted by symbol U p . Thus
n o U p ¼ uðÞ 2 Lp ð½t0 ; h; Rm Þ : kuðÞkp 6 l0 : It is obvious that the set of admissible control functions is the closed ball with radius l0 and centered at the origin in the space Lp ð½t 0 ; h; Rm Þ. It is assumed that the functions and a number k 2 R1 given in system (2.1) satisfy the following conditions: 2.A. the vector functions gð; Þ : ½t0 ; h Rn ! Rn ; K 1 ð; ; Þ : ½t 0 ; h ½t0 ; h Rn ! Rn and matrix function K 2 ð; ; Þ : ½t0 ; h ½t0 ; h Rn ! Rnm are continuous; 2.B. there exist L0 2 ½0; 1Þ; L1 P 0 and L2 P 0 such that
kgðt; x1 Þ gðt; x2 Þk 6 L0 kx1 x2 k; kK 1 ðt; s; x1 Þ K 1 ðt; s; x2 Þk 6 L1 kx1 x2 k; kK 2 ðt; s; x1 Þ K 2 ðt; s; x2 Þk 6 L2 kx1 x2 k n for every ðt;s; x1 Þ 2 ½t 0 ; h ½t0 ; h p1Rn ; ðt; s; x2 Þ 2 ½t0 ; h ½t 0 ; h R ; p 2.C. 0 6 k L1 ðh t0 Þ þ L2 ðh t0 Þ l0 < 1 L0 .
We denote
h i p1 LðkÞ ¼ L0 þ k L1 ðh t0 Þ þ L2 ðh t 0 Þ p l0 :
ð2:2Þ
According to the condition 2.C we obtain that LðkÞ < 1. Now, let us define a trajectory of the system (2.1) generated by an admissible control function. Let u ðÞ 2 U p . A continuous function x ðÞ : ½t0 ; h ! Rn satisfying the integral equation
x ðtÞ ¼ g ðt; x ðt ÞÞ þ k
Z
t
½K 1 ðt; s; x ðsÞÞ þ K 2 ðt; s; x ðsÞÞu ðsÞds
t0
for every t 2 ½t0 ; h is said to be a trajectory of the system (2.1) generated by the admissible control function u ðÞ 2 U p . The trajectory of the system (2.1) generated by the control function uðÞ 2 U p is denoted by xð; uðÞÞ and we set
Xp ¼ xð; uðÞÞ : uðÞ 2 U p : Xp is called the set of trajectories of the system (2.1). It is obvious that Xp C ð½t 0 ; h; Rn Þ where C ð½t 0 ; h; Rn Þ is the space of continuous functions xðÞ : ½t 0 ; h ! Rn with norm
kxðÞkC ¼ max fkxðtÞk : t 2 ½t 0 ; hg: For t 2 ½t 0 ; h we denote
Xp ðtÞ ¼ xðtÞ : xðÞ 2 Xp :
ð2:3Þ
The set Xp ðtÞ consists of points to which arrive the trajectories of the system at the instant of t. 3. Existence and uniqueness of trajectories In this section it is proved that under conditions 2.A – 2.C every admissible control function generates a unique trajectory. Theorem 3.1. Let u ðÞ 2 U p . Then the system (2.1) has unique x ðÞ ¼ xð; u ðÞÞ trajectory generated by the admissible control function u ðÞ. Proof. Let us define a map xðÞ ! AðxðÞÞ; xðÞ 2 C ð½t0 ; h; Rn Þ, setting
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AðxðÞÞjðtÞ ¼ g ðt; xðtÞÞ þ k
Z
t
½K 1 ðt; s; xðsÞÞ þ K 2 ðt; s; xðsÞÞu ðsÞds;
t 2 ½t 0 ; h:
ð3:1Þ
t0
Since u ðÞ 2 U p ; xðÞ 2 C ð½t 0 ; h; Rn Þ then by virtue of condition 2.A we get that the function t ! AðxðÞÞjðtÞ; t 2 ½t0 ; h, is continuous. Thus the map xðÞ ! AðxðÞÞ defined by (3.1) is in the form
AðÞ : C ð½t 0 ; h; Rn Þ ! C ð½t 0 ; h; Rn Þ: Let x1 ðÞ 2 C ð½t0 ; h; Rn Þ and x2 ðÞ 2 C ð½t 0 ; h; Rn Þ be arbitrarily chosen functions. Then applying condition 2.B we obtain
kAðx2 ðÞÞjðtÞ Aðx1 ðÞÞjðtÞk 6 L0 kx2 ðÞ x1 ðÞkC þ kL1
Z
t t0
kx2 ðÞ x1 ðÞkC ds þ kL2
Z
t
t0
kx2 ðÞ x1 ðÞkC ku ðsÞkds
Z t ¼ L0 þ kL1 ðh t 0 Þ þ kL2 ku ðsÞkds kx2 ðÞ x1 ðÞkC
ð3:2Þ
t0
for every t 2 ½t0 ; h. Since u ðÞ 2 U p then Hölder’s inequality implies
Z
t
ku ðsÞkds 6 ðt t 0 Þ
p1 p
Z
t0
t
p
ku ðsÞk ds
1p 6 ðh t 0 Þ
t0
p1 p
l0 :
From (2.2) and (3.2) it follows that the inequality
p1 kAðx2 ðÞÞjðtÞ Aðx1 ðÞÞjðtÞk 6 L0 þ kL1 ðh t0 Þ þ kL2 ðh t 0 Þ p l0 kx2 ðÞ x1 ðÞkC ¼ LðkÞkx2 ðÞ x1 ðÞkC
ð3:3Þ
holds for every t 2 ½t 0 ; h. Finally, (3.3) yields
kAðx2 ðÞÞjðÞ Aðx1 ðÞÞjðÞkC 6 LðkÞkx2 ðÞ x1 ðÞkC :
ð3:4Þ
According to the condition 2.C we have LðkÞ < 1. Then it follows from (3.4) that the map AðÞ : C ð½t0 ; h; Rn Þ ! C ð½t 0 ; h; Rn Þ defined by (3.1) is contractive and consequently it has a unique fixed point x ðÞ 2 C ð½t0 ; h; Rn Þ which is unique trajectory of the equation
x ðtÞ ¼ g ðt; x ðtÞÞ þ k
Z
t
½K 1 ðt; s; x ðsÞÞ þ K 2 ðt; s; x ðsÞÞu ðsÞds;
t 2 ½t0 ; h:
t0
h From condition 2.B it follows that for each fixed t 2 ½t0 ; h the function x ! gðt; xÞ is a contractive map and hence for each fixed t 2 ½t 0 ; h it has a unique fixed point. So there exists a unique sðtÞ 2 Rn ; t 2 ½t0 ; h, such that
sðtÞ ¼ gðt; sðtÞÞ;
t 2 ½t 0 ; h:
Let s0 ¼ sðt0 Þ. Then we have from (2.1) that xðt0 Þ ¼ s0 for every xðÞ 2 Xp . So we obtain the validity of the following proposition. Proposition 3.1. Xp ðt 0 Þ ¼ fs0 g.
4. Boundedness of the set of trajectories In this section the boundedness of the set of trajectories Xp is shown. Denote
c0 ¼ max fkgðt; 0Þk : t 2 ½t0 ; hg;
ð4:1Þ
c1 ¼ max fkK 1 ðt; s; 0Þk : ðt; sÞ 2 ½t 0 ; h ½t0 ; hg;
ð4:2Þ
c2 ¼ max fkK 2 ðt; s; 0Þk : ðt; sÞ 2 ½t 0 ; h ½t0 ; hg:
ð4:3Þ
Proposition 4.1. Let the functions gð; Þ : ½t 0 ; h Rn ! Rn ; K 1 ð; ; Þ : ½t 0 ; h ½t0 ; h Rn ! Rn and K 2 ð; ; Þ : ½t 0 ; h ½t0 ; h Rn ! Rnm satisfy the conditions 2.A and 2.B. Then
kgðt; xÞk 6 c0 þ L0 kxk;
kK 1 ðt; s; xÞk 6 c1 þ L1 kxk; n
kK 2 ðt; s; xÞk 6 c2 þ L2 kxk
for every ðt; s; xÞ 2 ½t 0 ; h ½t0 ; h R where the constants L0 ; L1 and L2 are given in condition 2.B.
N. Huseyin, A. Huseyin / Applied Mathematics and Computation 219 (2013) 8416–8424
8419
Proof. The proof of the proposition follows from Lipschitz continuity of the functions gð; Þ : ½t 0 ; h Rn ! Rn ; K 1 ð; ; Þ : ½t0 ; h ½t 0 ; h Rn ! Rn and K 2 ð; ; Þ : ½t0 ; h ½t 0 ; h Rn ! Rnm with respect to x and from definitions of the numbers c0 ; c1 and c2 , i.e. from (4.1)–(4.3). h Denote
r ¼
c0 þ kc1 ðh t0 Þ þ kc2 ðh t 0 Þ 1 L0
p1 p
l0
exp
LðkÞ L0 1 L0
ð4:4Þ
where L0 P 0 is defined in condition 2.B, LðkÞ is defined by (2.2), c0 P 0; c1 P 0 and c2 P 0 are defined by (4.1)–(4.3) respectively. Theorem 4.1. Let the functions gð; Þ : ½t0 ; h Rn ! Rn ; K 1 ð; ; Þ : ½t0 ; h ½t 0 ; h Rn ! Rn ; K 2 ð; ; Þ : ½t 0 ; h ½t 0 ; h Rn ! Rnm and the number k > 0 satisfy the conditions 2.A, 2.B and 2.C. Then for every xðÞ 2 Xp the inequality
kxðÞkC 6 r holds. Proof. Let xðÞ 2 Xp be an arbitrarily chosen trajectory. Then there exists uðÞ 2 U p such that
xðtÞ ¼ g ðt; xðt ÞÞ þ k
Z
t
½K 1 ðt; s; xðsÞÞ þ K 2 ðt; s; xðsÞÞuðsÞds
ð4:5Þ
t0
for every t 2 ½t0 ; h. From (4.5) and Proposition 4.1 it follows
kxðtÞk 6 kg ðt; xðt ÞÞk þ k
Z
t
kK 1 ðt; s; xðsÞÞkds þ k
Z
t0
6 c0 þ L0 kxðtÞk þ k
Z
t
t0 t
ðc1 þ L1 kxðsÞkÞds þ k
kK 2 ðt; s; xðsÞÞkkuðsÞkds Z
t0
t
ðc2 þ L2 kxðsÞkÞkuðsÞkds
t0 p1 p
6 L0 kxðtÞk þ c0 þ kc1 ðh t 0 Þ þ kc2 ðh t0 Þ
l0 þ k
Z
t
ðL1 þ L2 kuðsÞkÞkxðsÞkds
t0
for every t 2 ½t0 ; h. Since L0 2 ½0; 1Þ then from the last inequality we have
kxðtÞk 6
c0 þ kc1 ðh t 0 Þ þ kc2 ðh t 0 Þ 1 L0
p1 p
l0
þ
k 1 L0
Z
t
ðL1 þ L2 kuðsÞkÞkxðsÞkds
ð4:6Þ
t0
for every t 2 ½t0 ; h. (2.2), (4.4) and (4.6), Gronwall’s and Hölder’s inequalities yield
Z t k ðL1 þ L2 kuðsÞkÞds 1 L0 t 0 p1 p1 c0 þ kc1 ðh t 0 Þ þ kc2 ðh t 0 Þ p l0 k exp L1 ðh t 0 Þ þ L2 ðh t 0 Þ p l0 6 1 L0 1 L0 p1 c0 þ kc1 ðh t 0 Þ þ kc2 ðh t0 Þ p l0 LðkÞ L0 ¼ r exp ¼ 1 L0 1 L0
kxðtÞk 6
c0 þ kc1 ðh t 0 Þ þ kc2 ðh t 0 Þ 1 L0
for every t 2 ½t0 ; h. So, kxðÞkC 6 r .
p1 p
l0
exp
h
5. Precompactness of the set of trajectories In this section precompactness of set of trajectories of the system (2.1) is prowed. Let us introduce some notations. Let D > 0 be a given number, r > 0 be defined by (4.4), Bn ðr Þ ¼ fx 2 Rn : kxk 6 r g. We set
D1 ¼ ½t0 ; h Bn ðr Þ;
ð5:1Þ
D2 ¼ ½t0 ; h ½t 0 ; h Bn ðr Þ;
ð5:2Þ
M1 ¼ max fkK 1 ðt; s; xÞk : ðt; s; xÞ 2 D2 g;
ð5:3Þ
M2 ¼ max fkK 2 ðt; s; xÞk : ðt; s; xÞ 2 D2 g;
ð5:4Þ
x0 ðDÞ ¼ max fkgðt2 ; xÞ gðt1 ; xÞk : jt2 t1 j 6 D; ðt1 ; xÞ 2 D1 ; ðt2 ; xÞ 2 D1 g;
ð5:5Þ
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x1 ðDÞ ¼ max fkK 1 ðt2 ; s; xÞ K 1 ðt1 ; s; xÞk : jt2 t1 j 6 D; ðt1 ; s; xÞ 2 D2 ; ðt2 ; s; xÞ 2 D2 g;
ð5:6Þ
x2 ðDÞ ¼ max fkK 2 ðt2 ; s; xÞ K 2 ðt1 ; s; xÞk : jt2 t1 j 6 D; ðt1 ; s; xÞ 2 D2 ; ðt2 ; s; xÞ 2 D2 g;
ð5:7Þ
i p1 p1 1 h x0 ðDÞ þ kx1 ðDÞðh t0 Þ þ kM1 D þ kx2 ðDÞðh t0 Þ p l0 þ kM2 D p l0 : 1 L0
ð5:8Þ
uðDÞ ¼
It is obvious that x0 ðDÞ ! 0þ ; x1 ðDÞ ! 0þ ; x2 ðDÞ ! 0þ ; uðDÞ ! 0þ as D ! 0þ and x0 ðD1 Þ 6 x0 ðD2 Þ; x1 ðD1 Þ 6 x1 ðD2 Þ; x2 ðD1 Þ 6 x2 ðD2 Þ; uðD1 Þ 6 uðD2 Þ if D1 < D2 . Proposition 5.1. For every xðÞ 2 Xp ; t 1 2 ½t 0 ; h; t 2 2 ½t 0 ; h the inequality
kxðt2 Þ xðt1 Þk 6 uðjt2 t 1 jÞ holds where uðÞ is defined by (5.8). Proof. Let us choose an arbitrary xðÞ 2 Xp and without less of generality let us assume that t 2 > t1 . Then there exists uðÞ 2 U p such that
xðtÞ ¼ g ðt; xðtÞÞ þ k
Z
t
½K 1 ðt; s; xðsÞÞ þ K 2 ðt; s; xðsÞÞuðsÞds
ð5:9Þ
t0
for every t 2 ½t0 ; h. Then it follows from (5.9)
kxðt2 Þ xðt1 Þk 6 kg ðt2 ; xðt2 ÞÞ g ðt1 ; xðt2 ÞÞk þ kg ðt1 ; xðt2 ÞÞ g ðt 1 ; xðt1 ÞÞk Z t2 Z t1 þk kK 1 ðt 2 ; s; xðsÞÞ K 1 ðt 1 ; s; xðsÞÞkds þ k kK 1 ðt 2 ; s; xðsÞÞkds t0
þk
Z
t1
t1
kK 2 ðt 2 ; s; xðsÞÞ K 2 ðt 1 ; s; xðsÞÞkkuðsÞkds þ k
t0
Z
t2
kK 2 ðt2 ; s; xðsÞÞkkuðsÞkds:
ð5:10Þ
t1
By virtue of condition 2.B
kg ðt1 ; xðt2 ÞÞ g ðt1 ; xðt1 ÞÞk 6 L0 kxðt 2 Þ xðt 1 Þk;
L0 2 ½0; 1Þ:
ð5:11Þ
Since xðÞ 2 Xp then Theorem 4.1 implies that
xðtÞ 2 Bn ðr Þ
ð5:12Þ
for every t 2 ½t0 ; h. (5.1), (5.5) and (5.12) imply
kg ðt2 ; xðt2 ÞÞ g ðt1 ; xðt2 ÞÞk 6 x0 ðjt 2 t1 jÞ:
ð5:13Þ
From (5.2), (5.6) and (5.12) it follows that
kK 1 ðt 2 ; s; xðsÞÞ K 1 ðt 1 ; s; xðsÞÞk 6 x1 ðjt 2 t1 jÞ for every s 2 ½t0 ; h and hence
Z
t1
kK 1 ðt 2 ; s; xðsÞÞ K 1 ðt 1 ; s; xðsÞÞkds 6 x1 ðjt 2 t 1 jÞðt 1 t 0 Þ 6 x1 ðjt 2 t1 jÞðh t 0 Þ:
ð5:14Þ
t0
(5.2), (5.7) and (5.12) yield
kK 2 ðt 2 ; s; xðsÞÞ K 2 ðt 1 ; s; xðsÞÞk 6 x2 ðjt 2 t1 jÞ
ð5:15Þ
for every s 2 ½t0 ; h. Since uðÞ 2 U p then we have from (5.15)
Z
t1
kK 2 ðt 2 ; s; xðsÞÞ K 2 ðt 1 ; s; xðsÞÞkkuðsÞkds 6
t0
Z
t1
t0
p1
x2 ðjt2 t1 jÞkuðsÞkds 6 x2 ðjt2 t1 jÞðh t0 Þ p l0 :
ð5:16Þ
From (5.2)–(5.4) and (5.12) it follows that
Z
t2
t1
kK 1 ðt 2 ; s; xðsÞÞkds 6 M 1 jt 2 t 1 j;
ð5:17Þ
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Z
t2
kK 2 ðt2 ; s; xðsÞÞkkuðsÞkds 6 M2
Z
t1
t2
kuðsÞkds 6 M 2 jt 2 t 1 j
p1 p
t1
l0 :
ð5:18Þ
From (5.10), (5.11), (5.13), (5.14), (5.16), (5.17) and (5.18) we obtain
kxðt2 Þ xðt 1 Þk 6 x0 ðjt 2 t 1 jÞ þ L0 kxðt2 Þ xðt1 Þk þ kx1 ðjt2 t1 jÞðh t 0 Þ þ kx2 ðjt2 t1 jÞðh t 0 Þ þ kM 1 jt 2 t 1 j þ kM 2 jt 2 t1 j
p1 p
p1 p
l0
l0 :
ð5:19Þ
Finally, (5.8) and (5.19) imply
kxðt2 Þ xðt1 Þk 6
i p1 p1 1 h x0 ðjt2 t1 jÞ þ kx1 ðjt2 t1 jÞðh t0 Þ þ kx2 ðjt2 t1 jÞðh t0 Þ p l0 þ kM1 jt2 t1 j þ kM2 jt2 t1 j p l0 1 L0
¼ uðjt 2 t 1 jÞ: h Proposition 5.2. The set of trajectories Xp C ð½t0 ; h; Rn Þ is a set of equicontinuous functions. Proof. Let us choose an inequality
e > 0. Since uðDÞ ! 0þ as D ! 0þ then there exists D ðeÞ > 0 such that for every D 2 ð0; D ðeÞ the
uðDÞ 6 e
ð5:20Þ
holds where uðÞ is defined by (5.8). Now let xðÞ 2 Xp be an arbitrarily chosen trajectory and jt2 t 1 j 6 D ðeÞ where t 1 2 ½t0 ; h; t 2 2 ½t 0 ; h. Then from (5.20) and Proposition 5.1 we have
kxðt2 Þ xðt 1 Þk 6 uðjt2 t1 jÞ 6 uðD ðeÞÞ 6 e and hence the set trajectories Xp C ð½t 0 ; h; Rn Þ is a set of equicontinuous functions.
h
From Theorem 4.1 and Proposition 5.2 we immediately obtain the validity of the following theorem. Theorem 5.1. The set of trajectories Xp is a precompact subset of the space C ð½t 0 ; h; Rn Þ. Note that from Proposition 5.1 it follows continuity of the set valued map t ! Xp ðtÞ; t 2 ½t 0 ; h. The Hausdorff distance between the sets F Rn and E Rn is denoted by hðF; EÞ and defined as
hðF; EÞ ¼ maxfsup dðx; EÞ; sup dðy; FÞg; x2F
y2E
where dðx; EÞ ¼ inf fkx yk : y 2 Eg. Proposition 5.3. For every t 1 2 ½t0 ; h and t 2 2 ½t 0 ; h the inequality
h Xp ðt 2 Þ; Xp ðt1 Þ 6 uðjt 2 t1 jÞ
is verified where the function uðÞ : ½0; 1Þ ! ½0; 1Þ is defined by (5.8), the sets Xp ðt 1 Þ and Xp ðt2 Þ are defined by (2.3). Since uðDÞ ! 0þ as D ! 0þ then we conclude the validity of the following corollary. Corollary 5.1. Set valued map t ! Xp ðtÞ; t 2 ½t 0 ; h, is continuous. 6. Closedness of the set of trajectories The following theorem characterizes closedness of the set of trajectories Xp . Theorem 6.1. The set of trajectories Xp is a closed subset of the space C ð½t0 ; h; Rn Þ. Proof. Let xk ðÞ 2 Xp for every k ¼ 1; 2; . . . and kxk ðÞ x0 ðÞkC ! 0 as k ! 1 where x0 ðÞ 2 C ð½t 0 ; h; Rn Þ. Let us show that x0 ðÞ 2 Xp . Since xk ðÞ 2 Xp then there exists uk ðÞ 2 U p such that
xk ðtÞ ¼ g ðt; xk ðt ÞÞ þ k
Z
t
t0
½K 1 ðt; s; xk ðsÞÞ þ K 2 ðt; s; xk ðsÞÞuk ðsÞds;
t 2 ½t0 ; h:
ð6:1Þ
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Since the set of admissible control functions U p is closed ball with radius l0 and centered at the origin of the space Lp ð½t0 ; h; Rn Þ ðp > 1Þ, then the set U p is weakly compact. Then without loss of generality, we can assume that the sequence n fuk ðÞg1 k¼1 weakly converges to a u ðÞ 2 U p . Let x ðÞ : ½t 0 ; h ! R be a trajectory of the system (2.1) generated by the admissible control function u ðÞ 2 U p . Then
x ðtÞ ¼ g ðt; x ðtÞÞ þ k
Z
t
½K 1 ðt; s; x ðsÞÞ þ K 2 ðt; s; x ðsÞÞu ðsÞds; t 2 ½t0 ; h:
ð6:2Þ
t0
From (6.1) , (6.2) and condition 2.B it follows
kxk ðt Þ x ðt Þk 6
k 1 L0
Z
t
ðL1 þ L2 kuk ðsÞkÞkxk ðsÞ x ðsÞkds þ
t0
Z t k K 2 ðt; s; x ðsÞÞðuk ðsÞ u ðsÞÞds 1 L0 t0
ð6:3Þ
for every t 2 ½t0 ; h. Let us denote zðt; sÞ ¼ K 2 ðt; s; x ðsÞÞ. Then the function zðt; sÞ : ½t0 ; h ½t0 ; h ! Rnm is continuous and the inequality (6.3) can be rewritten as
kxk ðt Þ x ðt Þk 6
k 1 L0
Z
t
ðL1 þ L2 kuk ðsÞkÞkxk ðsÞ x ðsÞkds þ
t0
Z t k zðt; sÞðuk ðsÞ u ðsÞÞds : 1 L0 t0
ð6:4Þ
n Since the sequence fuk ðÞg1 k¼1 weakly converges to u ðÞ 2 U p as k ! 1 in the space Lp ð½t 0 ; h; R Þ then, we have that for each nm fixed t 2 ½t 0 ; h and w ðÞ 2 Lq ð½t0 ; t; R Þ
Z t w ðsÞ½uk ðsÞ u ðsÞds ! 0 as k ! 1:
ð6:5Þ
t0
Since the function ðt; sÞ ! zðt; sÞ : ½t 0 ; h ½t 0 ; h ! Rnm is continuous then by virtue of (6.5) we have that
Z t zðt; sÞ½uk ðsÞ u ðsÞds ! 0 as k ! 1
ð6:6Þ
t0
for every fixed t 2 ½t 0 ; h. From (6.6) we conclude that for e > 0 and fixed t 2 ½t0 ; h there exists Kðe; tÞ > 0 such that for every k > Kðe; tÞ the inequality
Z t zðt; sÞ½uk ðsÞ u ðsÞds < e
ð6:7Þ
t0
holds. Now let us prove that for each t 2 ½t 0 ; h the inequality
e > 0 there exists kðeÞ > 0 (which does not depend on t) such that for every k > kðeÞ and
Z t zðt; sÞðuk ðsÞ u ðsÞÞds < e
ð6:8Þ
t0
holds. Let us assume the contrary, i.e. let there exist e > 0; ki > 0 and t i 2 ½t0 ; h ði ¼ 1; 2; . . .Þ such that ki ! 1 as i ! 1 and
Z
ti
t0
zðt i ; sÞ uki ðsÞ u ðsÞ ds P e :
ð6:9Þ
Since ti 2 ½t 0 ; h for every i ¼ 1; 2; . . . and ½t0 ; h is a compact set, then without loss of generality one can assume that t i ! t as i ! 1 and t 2 ½t 0 ; h. It is obvious that
Z
ti
t0
Z
zðt i ; sÞ uki ðsÞ u ðsÞ ds 6
t
t0
Z
zðti ; sÞ uki ðsÞ u ðsÞ ds þ
ti
t
zðti ; sÞ uki ðsÞ u ðsÞ ds
Z t Z
t
þ 6 ½ z ð t ; s Þ z ð t ; s Þ u ðsÞ u ðsÞ ds z ð t ; s Þ u ðsÞ u ðsÞ ds i k k i i t0 t0 Z ti
þ zðt i ; sÞ uki ðsÞ u ðsÞ ds
ð6:10Þ
t
for every i ¼ 1; 2; . . . (6.7) implies that for
Z
t
t0
e > 0 and t 2 ½t0 ; h there exists N1 > 0 such that for every i > N1 the inequality
e zðt ; sÞ uki ðsÞ u ðsÞ ds < 5
is verified. Continuity of the function zð; Þ : ½t0 ; h ½t 0 ; h ! Rnm yields that for given every i > N 2 and s 2 ½t0 ; h the inequality
ð6:11Þ
e 10ðht 0 Þ
p1 p
l0
there exists N 2 > 0 such that for
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kzðti ; sÞ zðt ; sÞk <
e 10ðh t 0 Þ
p1 p
ð6:12Þ
l0
holds. Since u ðÞ 2 U p ; uki ðÞ 2 U p for every i ¼ 1; 2; . . . ; 1q ¼ p1 then (6.12) and Hölder’s inequality implies that p
Z
t
t0
½zðti ; sÞ zðt ; sÞ uki ðsÞ u ðsÞ ds 6
Z
e 10ðh t0 Þ
62
p1 p
l0
e 10ðh t0 Þ
p1 p
l0
t
t0
uk ðsÞ þ ku ðsÞk ds i
ðh t 0 Þ
p1 p
l0 ¼
e 5
ð6:13Þ
for every i > N 2 . Since zðt; sÞ ¼ K 2 ðt; s; x ðsÞÞ; x ðÞ 2 Xp then we have from Theorem 4.1 and (5.4) that for every i ¼ 1; 2; . . . and s 2 ½t 0 ; h the inequality
kzðti ; sÞk 6 M2
ð6:14Þ
is satisfied. Without loss of generality let us assume that ti P t for every i ¼ 1; 2; . . . (Note that it can be assumed that ti 6 t for every i ¼ 1; 2; . . .). Since u ðÞ 2 U p ; uki ðÞ 2 U p for every i ¼ 1; 2; . . . ; 1q ¼ p1 then it follows from (6.14) and Hölder’s p inequality that
Z
ti
t
Z
6 M zðt i ; sÞ uki ðsÞ u ðsÞ ds 2
ti
t
uk ðsÞ ds þ i
Z
ti
t
p1 ku ðsÞkds 6 2M 2 ðt i t Þ p l0 :
ð6:15Þ
Since t i ! t as i ! 1 then we obtain from (6.15) that there exists N 3 > 0 such that for every i > N 3 the inequality
Z
ti
t
e zðt i ; sÞ uki ðsÞ u ðsÞ ds < 5
ð6:16Þ
holds. Now let N ¼ max fN 1 ; N 2 ; N 3 g. Then from (6.10), (6.11), (6.13) and (6.16) we have
Z
ti
t0
e e e 3e zðt i ; sÞ uki ðsÞ u ðsÞ ds < 5 þ 5 þ 5 ¼ 5 < e
ð6:17Þ
for every i > N . The inequalities (6.9) and (6.17) contradict and hence the inequality (6.8) is verified. Then from (6.4) and (6.8) we conclude that for every t 2 ½t0 ; h and k > kðeÞ the inequality
kxk ðt Þ x ðt Þk 6
ke k þ 1 L0 1 L0
Z
t
ðL1 þ L2 kuk ðsÞkÞkxk ðsÞ x ðsÞkds
ð6:18Þ
t0
is verified. Since uk ðÞ 2 U p for every k ¼ 1; 2; . . . then (2.2), (6.18) and Gronwall’s inequality yield that
Z h ke k exp ðL1 þ L2 kuk ðsÞkÞds 1 L0 t0 1 L0 p1 ke k ke LðkÞ L0 exp L1 ðh t 0 Þ þ L2 ðh t0 Þ p l0 ¼ exp 6 1 L0 1 L0 1 L0 1 L0
kxk ðt Þ x ðt Þk 6
ð6:19Þ
for every t 2 ½t 0 ; h and k > kðeÞ. Finally, we get from (6.19) that xk ðÞ ! x ðÞ as k ! 1. From uniqueness of the limit we have x0 ðÞ ¼ x ðÞ and consequently x0 ðÞ 2 Xp . h From Theorem 5.1 and Theorem 6.1 we obtain compactness of the set of trajectories. Theorem 6.2. The set of trajectories Xp is a compact subset of the space C ð½t0 ; h; Rn Þ.
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