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Nonstandard singular perturbation systems and higher index differential-algebraic systems
Applied Mathematics and Computation 134 (2003) 323–344 www.elsevier.com/locate/amc
Nonstandard singular perturbation systems and higher index differential-algebraic systems M. Etchechoury *, C. Muravchik Laboratorio de Electr onica Industrial, Control e Instrumentaci on (LEICI), Facultad de Ingenierıa, Universidad Nacional de La Plata, CC 91, 1900 La Plata, Argentina Departamento de Matem atica, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, CC 172, 1900 La Plata, Argentina
Abstract In order to obtain trajectory approximation results for a given singular perturbation system (SPS), two systems are derived from it: the slow and the fast one. Tikhonov’s theorem gives sufficient conditions on them to ensure a good approximation for a standard SPS, i.e., its corresponding slow system is a differential-algebraic system (DAS) of index 1. In this paper it is shown that a nonstandard SPS with the parameter set to zero can be seen as a DAS of higher index. This connection allows us to obtain a Tikhonov’s theorem when this DAS is of index 2. 2002 Elsevier Science Inc. All rights reserved. Keywords: Reduced system; Boundary layer system; Singular perturbation; Differential-algebraic equations; Index
1. Introduction Models of large scale systems typically involve interacting dynamic phenomena of widely differing time-scales. These type of models may be treated as two time-scale singular perturbation systems (SPS), where the perturbation parameter is the ratio of the time-scales of the slow and fast dynamics.
*
Corresponding author. E-mail address: [email protected] (M. Etchechoury).
0096-3003/02/$ - see front matter 2002 Elsevier Science Inc. All rights reserved. PII: S 0 0 9 6 - 3 0 0 3 ( 0 1 ) 0 0 2 8 8 - 0
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Indeed, it is well known that solving a singularly perturbed system: x_ ¼ f ðx; zÞ;
xðt0 Þ ¼ x0 ;
ð1Þ
_z ¼ gðx; zÞ;
zðt0 Þ ¼ z0
ð2Þ
is based on this time-scale decomposition; and then system (1) and (2) is replaced by reduced and boundary layer systems, see [1]. The two time-scale properties of the model give trajectory approximation results for the SPS (Tikhonov’s theorem, [2,3]). In (1) and (2) the positive constant is a small perturbation parameter; f and g are continuously differentiable for ðx; zÞ 2 D1 D2 , with D1 Rn and D2 Rm open connected sets. Setting ¼ 0, (1) and (2) becomes: x_ ¼ f ðx; zÞ;
ð3Þ
0 ¼ gðx; zÞ;
ð4Þ
i.e., a differential-algebraic system (DAS); see for instance [4–6]. In [7,8] McClamroch and Krischnan studied SPS with control that is not in the standard form. They characterized the reduced system and the boundary layer system by a set of DAS. In [9] another connection between SPS and DAS was explored: considering that the DAS (3) and (4) is of index 2 [11], the authors characterized the corresponding reduced system as a DAS and analyzed the existence and uniqueness of solution for this system in order to guarantee the existence and uniqueness of solution for (1) and (2). In this paper, assuming (3) and (4) of index 2, we study the asymptotic behavior for the trajectories of (1) and (2). Our analysis emphasizes in the corresponding boundary layer system, whose Jacobian is singular. As stability conditions for this system must be guaranteed, we invoke the center manifold theory in this particular case, see [10] and references therein. The paper is organized as follows. In Section 2 we review some notions and known results about DAS. In Section 3 we connect an SPS with its corresponding DAS, considering the index 1 case and the index 2 case; we also prove the stability of the equilibrium of the boundary layer system in the index 2 case. In Section 4 we set the main result of the paper, obtaining asymptotic approximations for the trajectories in the index 2 case. We discuss an example in Section 5, while we conclude in Section 6.
2. Differential-algebraic systems: some definitions and known results We shall give some definitions set by Reich [11,12], in order to review some facts about DAS. If M is a differentiable manifold, then TM denotes the tangent bundle of M, and Tx M is the tangent space of M at x 2 M.
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325
Definition 2.1. Given a real interval I, system (3) and (4) is said to be regular if there is a smooth submanifold M Rnþm and a vector field v : M ! TM such that a differentiable mapping s : I ! M is a solution of the vector field v iff the mapping c ¼ j s : I ! Rnþm is a solution of (3) and (4); with j : M ! Rnþm the canonical inclusion. Definition 2.2. The corresponding set for (3) and (4) is defined as H ¼ fðx; z; x_ ; z_ Þ 2 R2ðnþmÞ : x_ ¼ f ðx; zÞ ^ 0 ¼ gðx; zÞg: Remark 2.1. A curve c : I ! Rnþm is a solution for (3) and (4) iff ðcðtÞ; dc= dtðtÞÞ 2 H 8t 2 I. Moreover, define M1 ¼ p1 ðH Þ; where p1 : R2ðnþmÞ ! Rnþm is the canonical projection over the first n þ m components. Then, if M1 is a smooth manifold, c is a solution for (3) and (4) iff ðcðtÞ; dc=dtðtÞÞ 2 H \ TM1 8t 2 I. Define M2 ¼ p1 ðH \ TM1 Þ: Under the hypothesis that M2 is a smooth manifold, it can be proved that c is a solution for (3) and (4) iff ðcðtÞ; dc=dtðtÞÞ 2 H \ TM2 8t 2 I. We continue this process until Mi ¼ p1 ðH \ TMi1 Þ is a smooth manifold such that Miþ1 ¼ Mi . Definition 2.3. Let H be the corresponding set for (3) and (4), and define recursively the family of manifolds fMi gi¼0;...;s as follows: M0 ¼ Rnþm ; Miþ1 ¼ p1 ðH \ TMi Þ;
i ¼ 0; . . . ; s 1;
where s is the greatest integer so that Mi is a smooth manifold and Ms1 6¼ Ms . The family fMi gi¼0;...;s is called family of constrained manifolds and the integer s is the degree of the DAS. Definition 2.4. M ¼ Ms is called the configuration space of the DAS (3) and (4). Definition 2.5. If (3) and (4) is regular, the number s is called the index of the system. The following proposition, set in [13], gives a necessary and sufficient condition for (3) and (4) to be of index 1.
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Proposition 2.1. System (3) and (4) is of index 1 in M1 ¼ fðx; zÞ : gðx; zÞ ¼ 0g iff og rank ¼m ð5Þ oz for all ðx; zÞ 2 M1 . Remark 2.2. If (5) is valid in a neighborhood of ðx ; z Þ 2 M1 , then we say that (3) and (4) is of index 1 in this point. Remark 2.3. The notions of regularity and index set above have been extended in [14] to the special case of an n-dimensional dynamical system constrained to move on a m-dimensional manifold, i.e., w_ ¼ f~ðwÞ;
ð6Þ
0 ¼ g~ðwÞ
ð7Þ
with f~ : Rn ! Rn ; g~ : Rn ! Rm sufficiently smooth. It can be easily proved in s this case that the family of constrained manifolds fMi gi¼0 is recursively defined as Mi ¼ Mi1 \ fw 2 Rn : 0 ¼ Li1 g~ðwÞg 8i P 1; f~ where Lf~g~ðwÞ ¼
o~ g ~ f ðwÞ; ow
g~ÞðwÞ: Lif~g~ðwÞ ¼ Lf~ðLi1 f~
3. Reduced and boundary layer systems We assume that og rank ¼r oz in a neighborhood of ðx ; z Þ 2 M1 . 3.1. The ‘‘index one’’ case When r ¼ m, applying the implicit function theorem (IFT), there exist neighborhoods U and V of x and z , respectively, and a unique smooth function h : U ! V such that: fðx; zÞ 2 U V : gðx; zÞ ¼ 0g ¼ fðx; hðxÞÞ : x 2 U g;
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327
in particular, gðx; hðxÞÞ ¼ 0 8x 2 U . From system (3) and (4) we get the reduced system: x_ ¼ f ðx; hðxÞÞ;
xðt0 Þ ¼ x0 :
ð8Þ
With the coordinates change y ¼ z hðxÞ, system (1) and (2) becomes x_ ¼ f ðx; y þ hðxÞÞ;
xðt0 Þ ¼ x0 ;
y_ ¼ gðx; y þ hðxÞÞ
oh f ðx; y þ hðxÞÞ; ox
ð9Þ yðt0 Þ ¼ z0 hðx0 Þ:
ð10Þ
We set
dy dy ds ¼ ) ¼ 1= dt ds dt
and use s ¼ 0 as the initial value at t ¼ t0 ; so the new scale of time is s¼
t t0 :
In the time s, system (10) is dy oh ¼ gðx; y þ hðxÞÞ f ðx; y þ hðxÞÞ; ds ox
yð0Þ ¼ z0 hðx0 Þ:
ð11Þ
Setting ¼ 0, we obtain t ¼ t0 and x ¼ x0 which reduces system (11) to the autonomous system d^ y ¼ gðx0 ; y^ þ hðx0 ÞÞ; ds
y^ð0Þ ¼ z0 hðx0 Þ
ð12Þ
with an equilibrium at y ¼ 0, if we assume that x0 2 U . System (12) is called the boundary layer system. Remark. The existence and uniqueness of solution for (8) and the asymptotic stability of the equilibrium for (12) are basically the hypotheses of Tikhonov’s theorem. Then, existence and uniqueness of solution for (1) and (2) are guaranteed when (3) and (4) is of index 1 and asymptotic approximations of the trajectories are also valid. More precisely, if we assume that (8) admits a unique solution xr ðtÞ over an interval ½t0 ; t1 , then there exist positive constants l and such that for all kz0 hðx0 Þk < l, the SPS (1) and (2) has a unique solution ðxðt; Þ; zðt; ÞÞ, and xðt; Þ xr ðtÞ ¼ OðÞ;
ð13Þ
zðt; Þ hðxr ðtÞÞ y^ðsÞ ¼ OðÞ
ð14Þ
hold uniformly for t 2 ½t0 ; t1 and < , where y^ðsÞ is the solution of (12). Different proofs of this theorem can be found in [15,16].
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3.2. The ‘‘index 2’’ case If r < m, we can reorder the components of z and g as follows: z g z¼ 1 ; g¼ 1 z2 g2 0
0
with z1 ¼ ½z1 z2 zr ; z2 ¼ ½zrþ1 zrþ2 zm ; and og1 rank ¼r oz1 in a neighborhood of ðx ; z Þ 2 M1 . Then, applying to g1 ðx; z1 ; z2 Þ the IFT, there exists a smooth function a1 ðx; w1 ; z2 Þ defined locally around ðx ; w1 ; z2 Þ such that g1 ðx; a1 ðx; w1 ; z2 Þ; z2 Þ ¼ w1 ;
ð15Þ
where w1 ¼ g1 ðx ; z1 ; z2 Þ. Moreover, ðx; w1 ; z2 Þ ¼ /ðx; z1 ; z2 Þ ¼ ðx; g1 ðx; z1 ; z2 Þ; z2 Þ
ð16Þ
is a C 1 function defined on an open neighborhood of ðx ; z1 ; z2 Þ, which is a diffeomorphism such that g1 /1 ðx; w1 ; z2 Þ ¼ w1 : So, applying the coordinate change /ðx; z1 ; z2 Þ, the system gðx; z1 ; z2 Þ ¼ 0 is equivalent to g /1 ðx; w1 ; z2 Þ ¼ 0; i.e., to w1 ¼ 0;
ð17Þ
g2 /1 ðx; w1 ; z2 Þ ¼ 0:
ð18Þ
Proposition 3.1. There exists a smooth function H, which depends only on x and is defined around x , such that, g /1 ðx; w1 ; z2 Þ ¼ 0 is equivalent to w1 ¼ 0;
ð19Þ
H ðxÞ ¼ 0:
ð20Þ
Proof. If we call g2 ðx; w1 ; z2 Þ ¼ g2 /1 ðx; w1 ; z2 Þ, considering x ¼ ðx1 ; . . . ; xn Þ; w1 ¼ ðw11 ; . . . ; wr1 Þ; z2 ¼ ðz11 ; . . . ; zmr 2 Þ; then the linearization of (17) and (18) in ðx; w1 ; z2 Þ is
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2
0 .. .
6 6 6 6 0 6 6 o 1 6 g2 6 1 6 ox 6 6 . 6 .. 6 4 o gmr 2
ox1
.. .
0 .. .
1 .. .
0 o g21 oxn .. . o g2mr oxn
0 o g21 ow11
.. .
o g2mr ow11
0 .. .
0 .. .
1 o g21 owr1 .. . o g2mr owr1
0 o g21 oz12 .. . o g2mr oz12
0 .. .
329
3
7 7 7 0 7 7 o g21 7 7 7; oz2mr 7 7 7 7 7 mr 5 o g 2
oz2mr
where o g2i =owl1 ; o g2i =ozj2 and o g2i =oxk , with 1 6 i; j 6 m r; 1 6 l 6 r; 1 6 k 6 n, are evaluated in ðx; w1 ; z2 Þ. Moreover, as rankðog=oðz1 ; z2 ÞÞ ¼ r, then oðg /1 Þ rank ¼ r; oðw1 ; z2 Þ which yields o g2i ðx; w1 ; z2 Þ ¼ 0 ozj2 for all 1 6 i; j 6 m r. So, we conclude that g2 ðx; w1 ; z2 Þ does not depend on z2 , i.e., g2 ðx; w1 ; z2 Þ ¼ g~2 ðx; w1 Þ: Then, (17) and (18) are equivalent to (19) and (20) with H ðxÞ ¼ g~2 ðx; 0Þ.
The following proposition [13] gives a sufficient condition for (3) and (4) to be of index 2 in ðx ; z Þ. Proposition 3.2. Assuming that oLf~H rank ¼ m r in ðx ; 0; z2 Þ; oz2 system (3) and (4) has index 2 in ðx ; z Þ; where Lf~H ðx; w1 ; z2 Þ ¼
dH ~ f ðx; w1 ; z2 Þ; dx
and f~ðx; w1 ; z2 Þ ¼ f ð/1 ðx; w1 ; z2 ÞÞ ¼ f ðx; a1 ðx; w1 ; z2 Þ; z2 Þ. We do not have yet an explicit form for z2 , so it is not possible to proceed as we did in the index 1 case. As the first step we can explore the reduced system corresponding to (1) and (2), with state ðx; z2 Þ 2 Rnþmr : x_ ¼ f ðx; z2 Þ;
ð21Þ
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0 ¼ H ðxÞ
ð22Þ
with f ðx; z2 Þ ¼ f~ðx; 0; z2 Þ. The following proposition [9] sets the index of (21) and (22). Proposition 3.3. If we assume that (3) and (4) is of index 2 in ðx ; z Þ, then (21) and (22) is a DAS of index 2 in ðx ; z2 Þ with configuration space: M ¼ fðx; z2 Þ 2 Rnþmr : H ðxÞ ¼ 0; Lf~H ðx; z2 Þ ¼ 0g: Remark 3.1. As a consequence of Proposition 3.3, if the initial condition ðx0 ; z20 Þ 2 M, then we can guarantee the existence and uniqueness of solution ðxðtÞ; z2 ðtÞÞ of (21) and (22) that also verifies: Lf H ðx; z2 Þ ¼ 0: Then, invoking Proposition 3.2 and IFT, we conclude that there is a smooth function a2 : U1 ! U2 , with U1 Rn and U2 Rmr neighborhoods of x and z2 , respectively, such that Lf H ðx; z2 Þ ¼ 0 iff z2 ¼ a2 ðxÞ. Finally, we define the reduced system corresponding to (1) and (2) as: x_ ¼ f ðx; a2 ðxÞÞ; xðt0 Þ ¼ x0 ;
ð23Þ
0 ¼ H ðxÞ:
ð24Þ
Note that (23) and (24) belongs to the class mentioned in Remark 2.3. The index of (23) and (24) was set in [9]. Proposition 3.4. (23) and (24) is a DAS of index 1 in x . Now, we set the corresponding boundary layer system. Applying the coordinate change (16), (1) and (2) becomes: x_ ¼ f~ðx; w1 ; z2 Þ; xðt0 Þ ¼ x0 ; ð25Þ og1 1 og1 1 ð/ ðx; w1 ; z2 ÞÞoxðxÞ f~ðx; w1 ; z2 Þ þ w_ 1 ¼ ð/ ðx; w1 ; z2 ÞÞw1 ox oz1 og1 1 þ ð/ ðx; w1 ; z2 ÞÞ~ g2 ðx; w1 Þ; w1 ðt0 Þ ¼ w10 ; ð26Þ oz2 _z2 ¼ g~2 ðx; w1 Þ;
z2 ðt0 Þ ¼ z20 ;
ð27Þ
with w1 ðt0 Þ ¼ g1 ðx0 ; z10 ; z20 Þ. In the time s ¼ ðt t0 Þ=, (25)–(27) is: dx ¼ f~ðx; w1 ; z2 Þ; ds
xð0Þ ¼ x0 ;
ð28Þ
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dw1 ¼ Cðx; w1 ; z2 Þf~ðx; w1 ; z2 Þ þ Aðx; w1 ; z2 Þw1 ds þ Bðx; w1 ; z2 Þ~ g2 ðx; w1 Þ; w1 ð0Þ ¼ w10 ; dz2 ¼ g~2 ðx; w1 Þ; ds
331
ð29Þ
z2 ð0Þ ¼ z20 ;
ð30Þ
where Aðx; w1 ; z2 Þ ¼
og1 1 ð/ ðx; w1 ; z2 ÞÞ; oz1
Bðx; w1 ; z2 Þ ¼
og1 1 ð/ ðx; w1 ; z2 ÞÞ; oz2
Cðx; w1 ; z2 Þ ¼
og1 1 ð/ ðx; w1 ; z2 ÞÞ: ox
Setting ¼ 0, (28)–(30) becomes: d^ x ¼ 0; ds
x^ð0Þ ¼ x0 ;
^1 dw ^ 1 ; ^z2 Þw ^ 1 þ Bð^ ^ 1 ; ^z2 Þ~ ^ 1 Þ; ¼ Að^ x; w x; w g2 ð^ x; w ds d^z2 ^ 1 Þ; ¼ g~2 ð^ x; w ds
ð31Þ ^ 1 ð0Þ ¼ w10 ; w
^z2 ð0Þ ¼ z20 :
ð32Þ ð33Þ
From (31) we obtain that x^ðsÞ ¼ x0 , then the system is equivalent to: ^1 dw ^ 1 ; ^z2 Þw ^ 1 þ Bðx0 ; w ^ 1 ; ^z2 Þ~ ^ 1 Þ; ¼ Aðx0 ; w g2 ðx0 ; w ds d^z2 ^ 1 Þ; ¼ g~2 ðx0 ; w ds
^z2 ð0Þ ¼ z20 :
^ 1 ð0Þ ¼ w10 ; w
ð34Þ ð35Þ
^ e1 ; ^ze2 Þ ¼ The boundary layer system (34) and (35) has an equilibrium in ðw ð0; a2 ðx0 ÞÞ. In order to investigate the stability of the equilibrium we evaluate its ^ 1 ðx0 ; 0Þ is null, it becomes Jacobian in this point. Assuming that o~ g2 =ow A1 0 J¼ 0 0 with A1 ¼ Aðx0 ; 0; a2 ðx0 ÞÞ an r r matrix that we will suppose Hurwitz, i.e., all its eigenvalues with negative real part. This Jacobian is singular, so we cannot conclude yet if the equilibrium has some kind of stability. As J has m r null eigenvalues, we
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invoke the center manifold theory to investigate about the equilibrium’s stability. ^ 1 ¼ 0 is a center manifold for system (34) and (35) through the Proposition 3.5. w ^ e1 ; ^ze2 Þ. equilibrium ðw Proof. In general, a center manifold for (34) and (35) through an equilibrium ^ e1 ; ^ze2 Þ is a smooth function w ^ 1 ¼ cð^z2 Þ which verifies: ðw ^e1 . 1. cð^ze2 Þ ¼ w 2. oc=o^z2 ð^ze2 Þ ¼ 0. 3. The manifold is invariant for the system, i.e., ^1 dw dc ¼ Aðx0 ; cð^z2 Þ; ^z2 Þcð^z2 Þ þ Bðx0 ; cð^z2 Þ; ^z2 Þ~ g2 ðx0 ; cð^z2 ÞÞ ¼ g~ ðx0 ; cð^z2 ÞÞ: d^z2 2 ds ^ 1 ¼ 0, conditions 1 and 2 are trivially verified. In this case, for the manifold w ^ 1 ¼ 0 is: To prove 3 we note that the dynamics on the manifold w ^1 dw ¼ Bðx0 ; 0; ^z2 Þ~ g2 ðx0 ; 0Þ ¼ 0; ds ^ 1 ¼ 0 is a which yields the invariance for the manifold. Then we conclude that w center manifold for (34) and (35) through ð0; a2 ðx0 ÞÞ. Remark 3.2. By a known result of the theory of center manifold, the stability of the equilibrium of (34) and (35) is determined by the stability of ue ¼ a2 ðx0 Þ in the dynamics du ¼ g~2 ðx0 ; 0Þ: ds As in this case g~2 does not depend on u, and g~2 ðx0 ; 0Þ ¼ 0, then ð0; a2 ðx0 ÞÞ is a stable equilibrium for (34) and (35). By Proposition 3.5 and Remark 3.2 it can be proved that there is a constant q > 0 such that if s ! 1, then: ^ 1 ðsÞ ¼ Oðeqs Þ; w ^z2 ðsÞ a2 ðx0 Þ ¼ Oðeqs Þ: Then, we conclude: ^ e1 ; ^ze2 Þ ¼ ð0; a2 ðx0 ÞÞ of system (34) and (35) is Proposition 3.6. The equilibrium ðw asymptotically stable.
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333
In order to set the relation between the trajectories of systems (28)–(30) and (31)–(33), we invoke the perturbation theory for ordinary differential equations, but in systems which depend smoothly on its parameters, see [15]. Then, (28)–(30) is the perturbed system and (34) and (35) the nominal or unperturbed system. If we call 0
X ¼ ½x w1 z2 ;
^ 1 ^z2 0 ; X^ ¼ ½^ x w
F ðX ; Þ ¼ ½f~ðx; w1 ; z2 Þ Cðx; w1 ; z2 Þf~ðx; w1 ; z2 Þ þ Aðx; w1 ; z2 Þw1 0 þ Bðx; w1 ; z2 Þ~ g2 ðx; w1 Þ g~2 ðx; w1 Þ ;
^ 1 ; ^z2 Þw ^ 1 þ Bð^ ^ 1 ; ^z2 Þ~ ^ 1 Þ g~2 ð^ ^ 1 Þ0 ; F^ðX^ Þ ¼ ½0 Að^ x; w x; w g2 ð^ x; w x; w then we can state the following approximation result to be valid on the infinite interval (see [15]). Proposition 3.7. Suppose that • the function F is continuous and bounded and has continuous, bounded derivatives with respect to its arguments ðX ; Þ 2 D ½0; 0 , with D an open connected set. • X e ¼ ðx0 ; 0; a2 ðx0 ÞÞ is an asymptotically stable equilibrium of the unperturbed system. Then, X ðs; Þ X^ ðsÞ tends to zero as ! 0, uniformly in s, for all s P 0.
4. Main result The following result sets the existence and uniqueness of solution for (1) and (2), on the hypothesis that (3) and (4) is a DAS of index ‘‘2’’ in ðx ; z Þ. Asymptotic approximations for the trajectories are also obtained. By simplicity, we will work in coordinates ðx; w1 ; z2 Þ, then in these coordinates (3) and (4) becomes (25)–(27); and we will assume that the transformed initial condition ðx0 ; w10 ; z20 Þ satisfies (15). Theorem 4.1. We assume: 1. f~; g1 /1 ; g2 /1 and their first partial derivatives with respect to ðx; w1 ; z2 Þ are continuous and bounded for all ðx; w1 ; z2 Þ. The functions a1 ; a2 ; ðog1 /1 Þ=ox; ðog1 /1 Þ=ow1 and ðog1 /1 Þ=oz2 and their first partial derivatives are continuous and bounded. 2. The reduced system (23) and (24) has a unique solution xr ðtÞ defined on ½t0 ; t1 .
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3. Considering the boundary layer system (34) and (35), A1 is Hurwitz and ^ 1 ðx0 ; 0Þ ¼ 0. o~ g2 =ow Then, there is a positive constant l such that for all kw10 k < l the system (25)– (27) has a unique solution ðxðt; Þ; w1 ðt; Þ; z2 ðt; ÞÞ and xðt; Þ xr ðtÞ ! 0;
ð36Þ
t t t t 0 0 ^1 ðw1 ðt; Þ; z2 ðt; Þ a2 ðxr ðtÞÞÞ ! w ; ^z2 a2 ðx0 Þ ;
ð37Þ
when ! 0, uniformly in t 2 ½t0 ; t1 . Moreover, given tb > t0 , w1 ðt; Þ ! 0;
ð38Þ
when ! 0, uniformly in t 2 ½tb ; t1 . Proof. See Appendix A. 5. Example We consider the SPS: x_ 1 ¼ x22 þ z1 z2 ;
ð39Þ
x_ 2 ¼ x1 þ z1 ;
ð40Þ
_z1 ¼ z1 þ x1 x2 ;
ð41Þ
_z2 ¼ x1 þ x2 :
ð42Þ
Setting ¼ 0, we obtain the DAS: x_ 1 ¼ x22 þ z1 z2 ;
ð43Þ
x_ 2 ¼ x1 þ z1 ;
ð44Þ
0 ¼ z1 þ x1 x2 ;
ð45Þ
0 ¼ x1 þ x2 :
ð46Þ
From the algebraic restriction (45) we obtain z1 ¼ a1 ðx1 ; x2 Þ ¼ x1 x2 ; and from x_ 1 þ x_ 2 ¼ 0 the expression for z2 ¼ a2 ðx1 ; x2 Þ ¼ x22 þ 2x1 x2 þ x1 . Then, it yields the reduced system: x_ 1r ¼ x1r x2r x1r ;
ð47Þ
x_ 2r ¼ x1r x2r þ x1r ;
ð48Þ
0 ¼ x1r þ x2r ;
ð49Þ
which is a DAS of index 1.
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335
In the scale s ¼ t= the SPS becomes: dx1 ¼ ðx22 þ z1 z2 Þ; ds dx2 ¼ ðx1 þ z1 Þ; ds dz1 ¼ z1 þ x1 x2 ; ds dz2 ¼ x1 þ x2 : ds
ð50Þ ð51Þ ð52Þ ð53Þ
Assuming initial conditions ðx10 ; x20 ; z10 ; z20 Þ and setting ¼ 0, we obtain the corresponding boundary layer system: d^z1 ¼ ^z1 þ x10 x20 ; ds d^z2 ¼ x10 þ x20 : ds
ð54Þ ð55Þ
We know that the initial conditions must satisfy (55), then ð^ze1 ; ^ze2 Þ ¼ ðx10 x20 ; a2 ðx10 ; x20 ÞÞ is an equilibrium for (54) and (55). The corresponding center manifold is in this case ^z1 ¼ x10 x20 : In Figs. 1–11 we compare the trajectories of the SPS and those of its reduced system. Different values of the perturbation parameter were considered. In all cases the initial condition is ðx10 ; x20 ; z10 ; z20 Þ ¼ ð:5; :5; :2; :25Þ.
Fig. 1. ¼ :0005; x1 ðtÞ and x1r ðtÞ.
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Fig. 2. ¼ :0005; x2 ðtÞ and x2r ðtÞ.
Fig. 3. ¼ :0005; z1 ðtÞ and z1a ðtÞ.
In particular, we compare the fast variables z1 and z2 with their corresponding approximations z1a ¼ a1 ðx1r ; x2r Þ ¼ x1r x2r and z2a ¼ a2 ðx1r ; x2r Þ ¼ x22r þ 2x1r x2r þ x1r , respectively. Moreover, in order to observe the boundary layer system we simulate z1 and z1a in the interval [0, .1].
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337
Fig. 4. ¼ :0005; z2 ðtÞ and z2a ðtÞ.
Fig. 5. ¼ :0005; z1 ðtÞ and z1a ðtÞ in ½0; :1.
6. Conclusions We have seen that an SPS becomes a DAS when the perturbation parameter is set to zero. Under the hypothesis that the DAS is of index 2, we have approximated the trajectories of the SPS using the trajectories of the DAS. We invoked the center manifold theory to guarantee some stability condition needed to prove the main result of the paper.
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Fig. 6. ¼ :0001; z1 ðtÞ and z1a ðtÞ.
Fig. 7. ¼ :0001; z2 ðtÞ and z2a ðtÞ.
We are currently studying the connection between a non-standard SPS with control and its corresponding DAS.
Appendix A Proof of Theorem 4.1. By Hypothesis 1, we can ensure, for each > 0, the existence and uniqueness of solution for (1) and (2).
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339
Fig. 8. ¼ :0001; z1 ðtÞ and z1a ðtÞ in ½0; :1.
Error for x: If we define ex ðt; Þ ¼ xðt; Þ xr ðtÞ, then Z t ex ðt; Þ ¼ ½f~ðxðs; Þ; w1 ðs; Þ; z2 ðs; ÞÞ f~ðxr ÞðsÞ; 0; a2 ðxr ðsÞÞ ds t0
¼
Z
t
½f~ðxðs; Þ; w1 ðs; Þ; z2 ðs; ÞÞ f~ðxr ðsÞ; w1 ðs; Þ; z2 ðs; ÞÞ ds
t0
þ
Z
t
½f~ðxr ðsÞ; w1 ðs; Þ; z2 ðs; ÞÞ f~ðxr ðsÞ; 0; a2 ðxðs; ÞÞÞ ds
t0
þ
Z
t
½f~ðxr ðsÞ; 0; a2 ðxðs; ÞÞÞ f~ðxr ðsÞ; 0; a2 ðxr ðsÞÞÞ ds
t0
and kex ðt; Þk 6
Z
t
k1 kxðs; Þ xr ðsÞk ds þ t0
a2 ðxðs; ÞÞk ds þ
Z
Z
t
k2 kw1 ðs; Þ; z2 ðs; Þ
t0 t
k3 kxðs; Þ xr ðsÞk ds
t0
using that f~; g and a2 are Lipschitz in their components, which is true by Hypothesis 1. To obtain a bound for ðw1 ðs; Þ; z2 ðs; Þ a2 ðxðs; ÞÞÞ
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Fig. 9. ¼ :00005; z1 ðtÞ and z1a ðtÞ.
Fig. 10. ¼ :00005; z2 ðtÞ and z2a ðtÞ.
we work, for simplicity, in the time s, then ^ 1 ðsÞ; z2 ðs; Þ ^z2 ðsÞÞk kðw1 ðs; Þ; z2 ðs; Þ a2 ðxðs; ÞÞÞk 6 kðw1 ðs; Þ w þ kðc w1 ðsÞ; ^z2 ðsÞ a2 ðx0 ÞÞk þ ka2 ðxðs; ÞÞ a2 ðx0 ÞÞk: By Proposition 3.7, given d1 > 0, there exists 1 > 0 such that
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341
Fig. 11. ¼ :00005; z1 ðtÞ and z1a ðtÞ in ½0; :1.
^1 ðsÞ; z2 ðs; Þ ^z2 ðsÞk < kðw1 ðs; Þ w
d1 3
8s P 0:
By Remark 3.2, there exists a constant l > 0 such that if kw10 k < l, then ^ 1 ðsÞ; ^z2 ðsÞ a2 ðx0 ÞÞk < kðw
d1 3
8s P 0:
Note that in this case the initial condition z20 is exactly a2 ðx0 Þ by Hypothesis 2. Moreover, there is a constant K > 0, such that ka2 ðxðs; ÞÞ a2 ðx0 ÞÞk 6 Kkxðs; Þ x0 k; where we have used that a2 in Lipschitz in x. So, given d2 ¼ d1 =3K, there exists 2 > 0 such that kxðs; Þ x0 k < d2 for all < 2 . Then, we conclude that ka2 ðxðs; ÞÞ a2 ðx0 Þk <
d1 3
8 < 2 ; 8s P 0:
Finally, if we choose ¼ minð1 ; 2 Þ, we obtain kðw1 ðs; Þ; z2 ðs; Þ a2 ðxðs; ÞÞÞk < d1
8 < ; 8s P 0
or equivalently, kðw1 ðs; Þ; z2 ðs; Þ a2 ðxðs; ÞÞÞk < d1
8 < ; 8s 2 ½t0 ; t1 :
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Now, given d > 0, we have Z t kex ðt; Þk 6 K1 kex ðs; Þk ds þ k2 d1 ðt1 t0 Þ t0
with K1 ¼ minðk1 ; k3 Þ. By Gronwall’s inequality, we conclude kex ðt; Þk 6 k2 d1 ðt1 t0 ÞeK1 ðt1 t0 Þ
8t 2 ½t0 ; t1 :
So, if we choose d1 ¼
d k2 ðt1 t0 ÞeK1 ðt1 t0 Þ
it yields, kex ðt; Þk 6 d
8 < ¼ minð1 ; 2 Þ
uniformly in t 2 ½t0 ; t1 ; which proves (36). Error for ðw1 ; z2 Þ: To prove error (37) we work again in the time s ^ 1 ðsÞ; ^z2 a2 ðx0 ÞÞ ðw1 ðs; Þ; z2 ðs; Þ a2 ðxr ðsÞÞÞ ðw ^ 1 ðsÞ; z2 ðs; Þ ^z2 ðsÞÞ þ ð0; a2 ðxðs; ÞÞ a2 ðxr ðsÞÞÞ ¼ ðw1 ðs; Þ w þ ð0; a2 ðx0 Þ a2 ðxðs; ÞÞÞ; which tends to zero for sufficiently small, and uniformly in s P 0; where we have used Proposition 3.7, (36), and that a2 is Lipschitz in x. Finally, given tb > t0 and d > 0, there exists 3 > 0 such that t t 0 ^1 w1 ðt; Þ w < d=2 for all < 3 and all t 2 ½t0 ; t1 . Then t t 0 ^1 kw1 ðt; Þk < d=2 þ w and invoking again the theory of center manifold, if kw10 k is sufficiently small, there exist positive constants c1 and b such that t t 0 ^1 ð56Þ w 6 c1 ebððtt0 Þ=Þ kw10 k: Since ebððtt0 Þ=Þ 6 8bðt t0 Þ P ln
1 ;
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343
then (56) will be OðÞ uniformly on ½tb ; t1 if is small enough to satisfy 1 ln 6 bðtb t0 Þ: More precisely, there exist constants k > 0 and 4 > 0 such that t t 0 ^1 w 6 k for all < 4 , uniformly in t 2 ½tb ; t1 . Now we can choose 5 6 4 such that k < d=2, for all < 5 ; then t t 0 ^1 w 6 d=2 for all < 5 , uniformly in t 2 ½tb ; t1 . Finally, if ¼ minð3 ; 5 Þ, then kw1 ðt; Þk < d
8 <
uniformly in t 2 ½tb ; t1 ; which proves (38).
References [1] P. Kokotovic, H.K. Khalil, J. O’Reilly, Singular Perturbation Methods in Control: Analysis and Design, Academic Press, Harcourt, Brace and Jovanovich, New York, 1986. [2] V.F. Butuzov, A.B. Vasil’eva, M.V. Fedoryuk, Asymptotic methods in the theory of ordinary differential equations, in: R.V. Gamkrelidze (Ed.), Progress in Mathematics, Plenum Press, New York, 1970. [3] F. Hoppensteadt, Properties of solutions of ordinary differential equations with small parameters, Comm. Pure Appl. Math. 24 (1971) 807–840. [4] W. Rheinboldt, Differential algebraic systems as differential equations on manifolds, Math. Comput. 43 (1984) 473–482. [5] A. Szatkowski, Geometric characterization of singular differential algebraic equations, Int. J. Syst. Sci. 23 (1992) 167–186. [6] K. Brenan, S. Campbell, L. Petzold, Numerical Solution of Initial-Value Problem in Differential-Algebraic Equations, SIAM, Philadelphia, PA, 1996. [7] H. Krischnan, N. McClamroch, On the connection between nonlinear differential-algebraic equations and singularly perturbed control systems in nonstandard form, IEEE Trans. Automat. Control 39 (1993) 1079–1084. [8] N. McClamroch, H. Krischnan, Non-standard singularly perturbed control systems and differential-algebraic systems, Int. J. Control 55 (1995) 1239–1253. [9] M. Etchechoury, C. Muravchik, Singular perturbation models and differential algebraic systems, Lat. Am. Appl. Res. 29 (1999) 205–212. [10] J. Carr, Applications of Centre Manifold Theory, Springer, New York, 1981. [11] S. Reich, On a geometrical interpretation of differential-algebraic equations, Circ. Syst. Sign. Proc. 9 (1990) 367–382. [12] S. Reich, On an existence and uniqueness theory for nonlinear differential algebraic equations, Circ. Syst. Sign. Proc. 10 (1991) 343–359.
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[13] M. Etchechoury, C. Muravchik, Regularidad de sistemas diferenciales algebraicos semiexplıcitos, in: Proceedings de la VI Reuni on en Procesamiento de la Informaci on y Control RPIC’97, Universidad Nacional de San Juan, Argentina, vol. 1, 1997, pp. 57–63. [14] M. Etchechoury, Sistemas diferenciales ordinarios con restricciones algebraicas. Internal Report LEICI, Facultad de Ingenierıa, Universidad Nacional de La Plata, Argentina, 1998. [15] H. Khalil, Nonlinear Systems, Macmillan, New York, 1992. [16] M. Corless, L. Glielmo, On the exponential stability of singularly perturbed systems, SIAM J. Control Opt. 30 (1992) 1338–1360.
Nonstandard singular perturbation systems and higher index differential-algebraic systems M. Etchechoury *, C. Muravchik Laboratorio de Electr onica Industrial, Control e Instrumentaci on (LEICI), Facultad de Ingenierıa, Universidad Nacional de La Plata, CC 91, 1900 La Plata, Argentina Departamento de Matem atica, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, CC 172, 1900 La Plata, Argentina
Abstract In order to obtain trajectory approximation results for a given singular perturbation system (SPS), two systems are derived from it: the slow and the fast one. Tikhonov’s theorem gives sufficient conditions on them to ensure a good approximation for a standard SPS, i.e., its corresponding slow system is a differential-algebraic system (DAS) of index 1. In this paper it is shown that a nonstandard SPS with the parameter set to zero can be seen as a DAS of higher index. This connection allows us to obtain a Tikhonov’s theorem when this DAS is of index 2. 2002 Elsevier Science Inc. All rights reserved. Keywords: Reduced system; Boundary layer system; Singular perturbation; Differential-algebraic equations; Index
1. Introduction Models of large scale systems typically involve interacting dynamic phenomena of widely differing time-scales. These type of models may be treated as two time-scale singular perturbation systems (SPS), where the perturbation parameter is the ratio of the time-scales of the slow and fast dynamics.
*
Corresponding author. E-mail address: [email protected] (M. Etchechoury).
0096-3003/02/$ - see front matter 2002 Elsevier Science Inc. All rights reserved. PII: S 0 0 9 6 - 3 0 0 3 ( 0 1 ) 0 0 2 8 8 - 0
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Indeed, it is well known that solving a singularly perturbed system: x_ ¼ f ðx; zÞ;
xðt0 Þ ¼ x0 ;
ð1Þ
_z ¼ gðx; zÞ;
zðt0 Þ ¼ z0
ð2Þ
is based on this time-scale decomposition; and then system (1) and (2) is replaced by reduced and boundary layer systems, see [1]. The two time-scale properties of the model give trajectory approximation results for the SPS (Tikhonov’s theorem, [2,3]). In (1) and (2) the positive constant is a small perturbation parameter; f and g are continuously differentiable for ðx; zÞ 2 D1 D2 , with D1 Rn and D2 Rm open connected sets. Setting ¼ 0, (1) and (2) becomes: x_ ¼ f ðx; zÞ;
ð3Þ
0 ¼ gðx; zÞ;
ð4Þ
i.e., a differential-algebraic system (DAS); see for instance [4–6]. In [7,8] McClamroch and Krischnan studied SPS with control that is not in the standard form. They characterized the reduced system and the boundary layer system by a set of DAS. In [9] another connection between SPS and DAS was explored: considering that the DAS (3) and (4) is of index 2 [11], the authors characterized the corresponding reduced system as a DAS and analyzed the existence and uniqueness of solution for this system in order to guarantee the existence and uniqueness of solution for (1) and (2). In this paper, assuming (3) and (4) of index 2, we study the asymptotic behavior for the trajectories of (1) and (2). Our analysis emphasizes in the corresponding boundary layer system, whose Jacobian is singular. As stability conditions for this system must be guaranteed, we invoke the center manifold theory in this particular case, see [10] and references therein. The paper is organized as follows. In Section 2 we review some notions and known results about DAS. In Section 3 we connect an SPS with its corresponding DAS, considering the index 1 case and the index 2 case; we also prove the stability of the equilibrium of the boundary layer system in the index 2 case. In Section 4 we set the main result of the paper, obtaining asymptotic approximations for the trajectories in the index 2 case. We discuss an example in Section 5, while we conclude in Section 6.
2. Differential-algebraic systems: some definitions and known results We shall give some definitions set by Reich [11,12], in order to review some facts about DAS. If M is a differentiable manifold, then TM denotes the tangent bundle of M, and Tx M is the tangent space of M at x 2 M.
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325
Definition 2.1. Given a real interval I, system (3) and (4) is said to be regular if there is a smooth submanifold M Rnþm and a vector field v : M ! TM such that a differentiable mapping s : I ! M is a solution of the vector field v iff the mapping c ¼ j s : I ! Rnþm is a solution of (3) and (4); with j : M ! Rnþm the canonical inclusion. Definition 2.2. The corresponding set for (3) and (4) is defined as H ¼ fðx; z; x_ ; z_ Þ 2 R2ðnþmÞ : x_ ¼ f ðx; zÞ ^ 0 ¼ gðx; zÞg: Remark 2.1. A curve c : I ! Rnþm is a solution for (3) and (4) iff ðcðtÞ; dc= dtðtÞÞ 2 H 8t 2 I. Moreover, define M1 ¼ p1 ðH Þ; where p1 : R2ðnþmÞ ! Rnþm is the canonical projection over the first n þ m components. Then, if M1 is a smooth manifold, c is a solution for (3) and (4) iff ðcðtÞ; dc=dtðtÞÞ 2 H \ TM1 8t 2 I. Define M2 ¼ p1 ðH \ TM1 Þ: Under the hypothesis that M2 is a smooth manifold, it can be proved that c is a solution for (3) and (4) iff ðcðtÞ; dc=dtðtÞÞ 2 H \ TM2 8t 2 I. We continue this process until Mi ¼ p1 ðH \ TMi1 Þ is a smooth manifold such that Miþ1 ¼ Mi . Definition 2.3. Let H be the corresponding set for (3) and (4), and define recursively the family of manifolds fMi gi¼0;...;s as follows: M0 ¼ Rnþm ; Miþ1 ¼ p1 ðH \ TMi Þ;
i ¼ 0; . . . ; s 1;
where s is the greatest integer so that Mi is a smooth manifold and Ms1 6¼ Ms . The family fMi gi¼0;...;s is called family of constrained manifolds and the integer s is the degree of the DAS. Definition 2.4. M ¼ Ms is called the configuration space of the DAS (3) and (4). Definition 2.5. If (3) and (4) is regular, the number s is called the index of the system. The following proposition, set in [13], gives a necessary and sufficient condition for (3) and (4) to be of index 1.
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Proposition 2.1. System (3) and (4) is of index 1 in M1 ¼ fðx; zÞ : gðx; zÞ ¼ 0g iff og rank ¼m ð5Þ oz for all ðx; zÞ 2 M1 . Remark 2.2. If (5) is valid in a neighborhood of ðx ; z Þ 2 M1 , then we say that (3) and (4) is of index 1 in this point. Remark 2.3. The notions of regularity and index set above have been extended in [14] to the special case of an n-dimensional dynamical system constrained to move on a m-dimensional manifold, i.e., w_ ¼ f~ðwÞ;
ð6Þ
0 ¼ g~ðwÞ
ð7Þ
with f~ : Rn ! Rn ; g~ : Rn ! Rm sufficiently smooth. It can be easily proved in s this case that the family of constrained manifolds fMi gi¼0 is recursively defined as Mi ¼ Mi1 \ fw 2 Rn : 0 ¼ Li1 g~ðwÞg 8i P 1; f~ where Lf~g~ðwÞ ¼
o~ g ~ f ðwÞ; ow
g~ÞðwÞ: Lif~g~ðwÞ ¼ Lf~ðLi1 f~
3. Reduced and boundary layer systems We assume that og rank ¼r oz in a neighborhood of ðx ; z Þ 2 M1 . 3.1. The ‘‘index one’’ case When r ¼ m, applying the implicit function theorem (IFT), there exist neighborhoods U and V of x and z , respectively, and a unique smooth function h : U ! V such that: fðx; zÞ 2 U V : gðx; zÞ ¼ 0g ¼ fðx; hðxÞÞ : x 2 U g;
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327
in particular, gðx; hðxÞÞ ¼ 0 8x 2 U . From system (3) and (4) we get the reduced system: x_ ¼ f ðx; hðxÞÞ;
xðt0 Þ ¼ x0 :
ð8Þ
With the coordinates change y ¼ z hðxÞ, system (1) and (2) becomes x_ ¼ f ðx; y þ hðxÞÞ;
xðt0 Þ ¼ x0 ;
y_ ¼ gðx; y þ hðxÞÞ
oh f ðx; y þ hðxÞÞ; ox
ð9Þ yðt0 Þ ¼ z0 hðx0 Þ:
ð10Þ
We set
dy dy ds ¼ ) ¼ 1= dt ds dt
and use s ¼ 0 as the initial value at t ¼ t0 ; so the new scale of time is s¼
t t0 :
In the time s, system (10) is dy oh ¼ gðx; y þ hðxÞÞ f ðx; y þ hðxÞÞ; ds ox
yð0Þ ¼ z0 hðx0 Þ:
ð11Þ
Setting ¼ 0, we obtain t ¼ t0 and x ¼ x0 which reduces system (11) to the autonomous system d^ y ¼ gðx0 ; y^ þ hðx0 ÞÞ; ds
y^ð0Þ ¼ z0 hðx0 Þ
ð12Þ
with an equilibrium at y ¼ 0, if we assume that x0 2 U . System (12) is called the boundary layer system. Remark. The existence and uniqueness of solution for (8) and the asymptotic stability of the equilibrium for (12) are basically the hypotheses of Tikhonov’s theorem. Then, existence and uniqueness of solution for (1) and (2) are guaranteed when (3) and (4) is of index 1 and asymptotic approximations of the trajectories are also valid. More precisely, if we assume that (8) admits a unique solution xr ðtÞ over an interval ½t0 ; t1 , then there exist positive constants l and such that for all kz0 hðx0 Þk < l, the SPS (1) and (2) has a unique solution ðxðt; Þ; zðt; ÞÞ, and xðt; Þ xr ðtÞ ¼ OðÞ;
ð13Þ
zðt; Þ hðxr ðtÞÞ y^ðsÞ ¼ OðÞ
ð14Þ
hold uniformly for t 2 ½t0 ; t1 and < , where y^ðsÞ is the solution of (12). Different proofs of this theorem can be found in [15,16].
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3.2. The ‘‘index 2’’ case If r < m, we can reorder the components of z and g as follows: z g z¼ 1 ; g¼ 1 z2 g2 0
0
with z1 ¼ ½z1 z2 zr ; z2 ¼ ½zrþ1 zrþ2 zm ; and og1 rank ¼r oz1 in a neighborhood of ðx ; z Þ 2 M1 . Then, applying to g1 ðx; z1 ; z2 Þ the IFT, there exists a smooth function a1 ðx; w1 ; z2 Þ defined locally around ðx ; w1 ; z2 Þ such that g1 ðx; a1 ðx; w1 ; z2 Þ; z2 Þ ¼ w1 ;
ð15Þ
where w1 ¼ g1 ðx ; z1 ; z2 Þ. Moreover, ðx; w1 ; z2 Þ ¼ /ðx; z1 ; z2 Þ ¼ ðx; g1 ðx; z1 ; z2 Þ; z2 Þ
ð16Þ
is a C 1 function defined on an open neighborhood of ðx ; z1 ; z2 Þ, which is a diffeomorphism such that g1 /1 ðx; w1 ; z2 Þ ¼ w1 : So, applying the coordinate change /ðx; z1 ; z2 Þ, the system gðx; z1 ; z2 Þ ¼ 0 is equivalent to g /1 ðx; w1 ; z2 Þ ¼ 0; i.e., to w1 ¼ 0;
ð17Þ
g2 /1 ðx; w1 ; z2 Þ ¼ 0:
ð18Þ
Proposition 3.1. There exists a smooth function H, which depends only on x and is defined around x , such that, g /1 ðx; w1 ; z2 Þ ¼ 0 is equivalent to w1 ¼ 0;
ð19Þ
H ðxÞ ¼ 0:
ð20Þ
Proof. If we call g2 ðx; w1 ; z2 Þ ¼ g2 /1 ðx; w1 ; z2 Þ, considering x ¼ ðx1 ; . . . ; xn Þ; w1 ¼ ðw11 ; . . . ; wr1 Þ; z2 ¼ ðz11 ; . . . ; zmr 2 Þ; then the linearization of (17) and (18) in ðx; w1 ; z2 Þ is
M. Etchechoury, C. Muravchik / Appl. Math. Comput. 134 (2003) 323–344
2
0 .. .
6 6 6 6 0 6 6 o 1 6 g2 6 1 6 ox 6 6 . 6 .. 6 4 o gmr 2
ox1
.. .
0 .. .
1 .. .
0 o g21 oxn .. . o g2mr oxn
0 o g21 ow11
.. .
o g2mr ow11
0 .. .
0 .. .
1 o g21 owr1 .. . o g2mr owr1
0 o g21 oz12 .. . o g2mr oz12
0 .. .
329
3
7 7 7 0 7 7 o g21 7 7 7; oz2mr 7 7 7 7 7 mr 5 o g 2
oz2mr
where o g2i =owl1 ; o g2i =ozj2 and o g2i =oxk , with 1 6 i; j 6 m r; 1 6 l 6 r; 1 6 k 6 n, are evaluated in ðx; w1 ; z2 Þ. Moreover, as rankðog=oðz1 ; z2 ÞÞ ¼ r, then oðg /1 Þ rank ¼ r; oðw1 ; z2 Þ which yields o g2i ðx; w1 ; z2 Þ ¼ 0 ozj2 for all 1 6 i; j 6 m r. So, we conclude that g2 ðx; w1 ; z2 Þ does not depend on z2 , i.e., g2 ðx; w1 ; z2 Þ ¼ g~2 ðx; w1 Þ: Then, (17) and (18) are equivalent to (19) and (20) with H ðxÞ ¼ g~2 ðx; 0Þ.
The following proposition [13] gives a sufficient condition for (3) and (4) to be of index 2 in ðx ; z Þ. Proposition 3.2. Assuming that oLf~H rank ¼ m r in ðx ; 0; z2 Þ; oz2 system (3) and (4) has index 2 in ðx ; z Þ; where Lf~H ðx; w1 ; z2 Þ ¼
dH ~ f ðx; w1 ; z2 Þ; dx
and f~ðx; w1 ; z2 Þ ¼ f ð/1 ðx; w1 ; z2 ÞÞ ¼ f ðx; a1 ðx; w1 ; z2 Þ; z2 Þ. We do not have yet an explicit form for z2 , so it is not possible to proceed as we did in the index 1 case. As the first step we can explore the reduced system corresponding to (1) and (2), with state ðx; z2 Þ 2 Rnþmr : x_ ¼ f ðx; z2 Þ;
ð21Þ
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0 ¼ H ðxÞ
ð22Þ
with f ðx; z2 Þ ¼ f~ðx; 0; z2 Þ. The following proposition [9] sets the index of (21) and (22). Proposition 3.3. If we assume that (3) and (4) is of index 2 in ðx ; z Þ, then (21) and (22) is a DAS of index 2 in ðx ; z2 Þ with configuration space: M ¼ fðx; z2 Þ 2 Rnþmr : H ðxÞ ¼ 0; Lf~H ðx; z2 Þ ¼ 0g: Remark 3.1. As a consequence of Proposition 3.3, if the initial condition ðx0 ; z20 Þ 2 M, then we can guarantee the existence and uniqueness of solution ðxðtÞ; z2 ðtÞÞ of (21) and (22) that also verifies: Lf H ðx; z2 Þ ¼ 0: Then, invoking Proposition 3.2 and IFT, we conclude that there is a smooth function a2 : U1 ! U2 , with U1 Rn and U2 Rmr neighborhoods of x and z2 , respectively, such that Lf H ðx; z2 Þ ¼ 0 iff z2 ¼ a2 ðxÞ. Finally, we define the reduced system corresponding to (1) and (2) as: x_ ¼ f ðx; a2 ðxÞÞ; xðt0 Þ ¼ x0 ;
ð23Þ
0 ¼ H ðxÞ:
ð24Þ
Note that (23) and (24) belongs to the class mentioned in Remark 2.3. The index of (23) and (24) was set in [9]. Proposition 3.4. (23) and (24) is a DAS of index 1 in x . Now, we set the corresponding boundary layer system. Applying the coordinate change (16), (1) and (2) becomes: x_ ¼ f~ðx; w1 ; z2 Þ; xðt0 Þ ¼ x0 ; ð25Þ og1 1 og1 1 ð/ ðx; w1 ; z2 ÞÞoxðxÞ f~ðx; w1 ; z2 Þ þ w_ 1 ¼ ð/ ðx; w1 ; z2 ÞÞw1 ox oz1 og1 1 þ ð/ ðx; w1 ; z2 ÞÞ~ g2 ðx; w1 Þ; w1 ðt0 Þ ¼ w10 ; ð26Þ oz2 _z2 ¼ g~2 ðx; w1 Þ;
z2 ðt0 Þ ¼ z20 ;
ð27Þ
with w1 ðt0 Þ ¼ g1 ðx0 ; z10 ; z20 Þ. In the time s ¼ ðt t0 Þ=, (25)–(27) is: dx ¼ f~ðx; w1 ; z2 Þ; ds
xð0Þ ¼ x0 ;
ð28Þ
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dw1 ¼ Cðx; w1 ; z2 Þf~ðx; w1 ; z2 Þ þ Aðx; w1 ; z2 Þw1 ds þ Bðx; w1 ; z2 Þ~ g2 ðx; w1 Þ; w1 ð0Þ ¼ w10 ; dz2 ¼ g~2 ðx; w1 Þ; ds
331
ð29Þ
z2 ð0Þ ¼ z20 ;
ð30Þ
where Aðx; w1 ; z2 Þ ¼
og1 1 ð/ ðx; w1 ; z2 ÞÞ; oz1
Bðx; w1 ; z2 Þ ¼
og1 1 ð/ ðx; w1 ; z2 ÞÞ; oz2
Cðx; w1 ; z2 Þ ¼
og1 1 ð/ ðx; w1 ; z2 ÞÞ: ox
Setting ¼ 0, (28)–(30) becomes: d^ x ¼ 0; ds
x^ð0Þ ¼ x0 ;
^1 dw ^ 1 ; ^z2 Þw ^ 1 þ Bð^ ^ 1 ; ^z2 Þ~ ^ 1 Þ; ¼ Að^ x; w x; w g2 ð^ x; w ds d^z2 ^ 1 Þ; ¼ g~2 ð^ x; w ds
ð31Þ ^ 1 ð0Þ ¼ w10 ; w
^z2 ð0Þ ¼ z20 :
ð32Þ ð33Þ
From (31) we obtain that x^ðsÞ ¼ x0 , then the system is equivalent to: ^1 dw ^ 1 ; ^z2 Þw ^ 1 þ Bðx0 ; w ^ 1 ; ^z2 Þ~ ^ 1 Þ; ¼ Aðx0 ; w g2 ðx0 ; w ds d^z2 ^ 1 Þ; ¼ g~2 ðx0 ; w ds
^z2 ð0Þ ¼ z20 :
^ 1 ð0Þ ¼ w10 ; w
ð34Þ ð35Þ
^ e1 ; ^ze2 Þ ¼ The boundary layer system (34) and (35) has an equilibrium in ðw ð0; a2 ðx0 ÞÞ. In order to investigate the stability of the equilibrium we evaluate its ^ 1 ðx0 ; 0Þ is null, it becomes Jacobian in this point. Assuming that o~ g2 =ow A1 0 J¼ 0 0 with A1 ¼ Aðx0 ; 0; a2 ðx0 ÞÞ an r r matrix that we will suppose Hurwitz, i.e., all its eigenvalues with negative real part. This Jacobian is singular, so we cannot conclude yet if the equilibrium has some kind of stability. As J has m r null eigenvalues, we
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invoke the center manifold theory to investigate about the equilibrium’s stability. ^ 1 ¼ 0 is a center manifold for system (34) and (35) through the Proposition 3.5. w ^ e1 ; ^ze2 Þ. equilibrium ðw Proof. In general, a center manifold for (34) and (35) through an equilibrium ^ e1 ; ^ze2 Þ is a smooth function w ^ 1 ¼ cð^z2 Þ which verifies: ðw ^e1 . 1. cð^ze2 Þ ¼ w 2. oc=o^z2 ð^ze2 Þ ¼ 0. 3. The manifold is invariant for the system, i.e., ^1 dw dc ¼ Aðx0 ; cð^z2 Þ; ^z2 Þcð^z2 Þ þ Bðx0 ; cð^z2 Þ; ^z2 Þ~ g2 ðx0 ; cð^z2 ÞÞ ¼ g~ ðx0 ; cð^z2 ÞÞ: d^z2 2 ds ^ 1 ¼ 0, conditions 1 and 2 are trivially verified. In this case, for the manifold w ^ 1 ¼ 0 is: To prove 3 we note that the dynamics on the manifold w ^1 dw ¼ Bðx0 ; 0; ^z2 Þ~ g2 ðx0 ; 0Þ ¼ 0; ds ^ 1 ¼ 0 is a which yields the invariance for the manifold. Then we conclude that w center manifold for (34) and (35) through ð0; a2 ðx0 ÞÞ. Remark 3.2. By a known result of the theory of center manifold, the stability of the equilibrium of (34) and (35) is determined by the stability of ue ¼ a2 ðx0 Þ in the dynamics du ¼ g~2 ðx0 ; 0Þ: ds As in this case g~2 does not depend on u, and g~2 ðx0 ; 0Þ ¼ 0, then ð0; a2 ðx0 ÞÞ is a stable equilibrium for (34) and (35). By Proposition 3.5 and Remark 3.2 it can be proved that there is a constant q > 0 such that if s ! 1, then: ^ 1 ðsÞ ¼ Oðeqs Þ; w ^z2 ðsÞ a2 ðx0 Þ ¼ Oðeqs Þ: Then, we conclude: ^ e1 ; ^ze2 Þ ¼ ð0; a2 ðx0 ÞÞ of system (34) and (35) is Proposition 3.6. The equilibrium ðw asymptotically stable.
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333
In order to set the relation between the trajectories of systems (28)–(30) and (31)–(33), we invoke the perturbation theory for ordinary differential equations, but in systems which depend smoothly on its parameters, see [15]. Then, (28)–(30) is the perturbed system and (34) and (35) the nominal or unperturbed system. If we call 0
X ¼ ½x w1 z2 ;
^ 1 ^z2 0 ; X^ ¼ ½^ x w
F ðX ; Þ ¼ ½f~ðx; w1 ; z2 Þ Cðx; w1 ; z2 Þf~ðx; w1 ; z2 Þ þ Aðx; w1 ; z2 Þw1 0 þ Bðx; w1 ; z2 Þ~ g2 ðx; w1 Þ g~2 ðx; w1 Þ ;
^ 1 ; ^z2 Þw ^ 1 þ Bð^ ^ 1 ; ^z2 Þ~ ^ 1 Þ g~2 ð^ ^ 1 Þ0 ; F^ðX^ Þ ¼ ½0 Að^ x; w x; w g2 ð^ x; w x; w then we can state the following approximation result to be valid on the infinite interval (see [15]). Proposition 3.7. Suppose that • the function F is continuous and bounded and has continuous, bounded derivatives with respect to its arguments ðX ; Þ 2 D ½0; 0 , with D an open connected set. • X e ¼ ðx0 ; 0; a2 ðx0 ÞÞ is an asymptotically stable equilibrium of the unperturbed system. Then, X ðs; Þ X^ ðsÞ tends to zero as ! 0, uniformly in s, for all s P 0.
4. Main result The following result sets the existence and uniqueness of solution for (1) and (2), on the hypothesis that (3) and (4) is a DAS of index ‘‘2’’ in ðx ; z Þ. Asymptotic approximations for the trajectories are also obtained. By simplicity, we will work in coordinates ðx; w1 ; z2 Þ, then in these coordinates (3) and (4) becomes (25)–(27); and we will assume that the transformed initial condition ðx0 ; w10 ; z20 Þ satisfies (15). Theorem 4.1. We assume: 1. f~; g1 /1 ; g2 /1 and their first partial derivatives with respect to ðx; w1 ; z2 Þ are continuous and bounded for all ðx; w1 ; z2 Þ. The functions a1 ; a2 ; ðog1 /1 Þ=ox; ðog1 /1 Þ=ow1 and ðog1 /1 Þ=oz2 and their first partial derivatives are continuous and bounded. 2. The reduced system (23) and (24) has a unique solution xr ðtÞ defined on ½t0 ; t1 .
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3. Considering the boundary layer system (34) and (35), A1 is Hurwitz and ^ 1 ðx0 ; 0Þ ¼ 0. o~ g2 =ow Then, there is a positive constant l such that for all kw10 k < l the system (25)– (27) has a unique solution ðxðt; Þ; w1 ðt; Þ; z2 ðt; ÞÞ and xðt; Þ xr ðtÞ ! 0;
ð36Þ
t t t t 0 0 ^1 ðw1 ðt; Þ; z2 ðt; Þ a2 ðxr ðtÞÞÞ ! w ; ^z2 a2 ðx0 Þ ;
ð37Þ
when ! 0, uniformly in t 2 ½t0 ; t1 . Moreover, given tb > t0 , w1 ðt; Þ ! 0;
ð38Þ
when ! 0, uniformly in t 2 ½tb ; t1 . Proof. See Appendix A. 5. Example We consider the SPS: x_ 1 ¼ x22 þ z1 z2 ;
ð39Þ
x_ 2 ¼ x1 þ z1 ;
ð40Þ
_z1 ¼ z1 þ x1 x2 ;
ð41Þ
_z2 ¼ x1 þ x2 :
ð42Þ
Setting ¼ 0, we obtain the DAS: x_ 1 ¼ x22 þ z1 z2 ;
ð43Þ
x_ 2 ¼ x1 þ z1 ;
ð44Þ
0 ¼ z1 þ x1 x2 ;
ð45Þ
0 ¼ x1 þ x2 :
ð46Þ
From the algebraic restriction (45) we obtain z1 ¼ a1 ðx1 ; x2 Þ ¼ x1 x2 ; and from x_ 1 þ x_ 2 ¼ 0 the expression for z2 ¼ a2 ðx1 ; x2 Þ ¼ x22 þ 2x1 x2 þ x1 . Then, it yields the reduced system: x_ 1r ¼ x1r x2r x1r ;
ð47Þ
x_ 2r ¼ x1r x2r þ x1r ;
ð48Þ
0 ¼ x1r þ x2r ;
ð49Þ
which is a DAS of index 1.
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335
In the scale s ¼ t= the SPS becomes: dx1 ¼ ðx22 þ z1 z2 Þ; ds dx2 ¼ ðx1 þ z1 Þ; ds dz1 ¼ z1 þ x1 x2 ; ds dz2 ¼ x1 þ x2 : ds
ð50Þ ð51Þ ð52Þ ð53Þ
Assuming initial conditions ðx10 ; x20 ; z10 ; z20 Þ and setting ¼ 0, we obtain the corresponding boundary layer system: d^z1 ¼ ^z1 þ x10 x20 ; ds d^z2 ¼ x10 þ x20 : ds
ð54Þ ð55Þ
We know that the initial conditions must satisfy (55), then ð^ze1 ; ^ze2 Þ ¼ ðx10 x20 ; a2 ðx10 ; x20 ÞÞ is an equilibrium for (54) and (55). The corresponding center manifold is in this case ^z1 ¼ x10 x20 : In Figs. 1–11 we compare the trajectories of the SPS and those of its reduced system. Different values of the perturbation parameter were considered. In all cases the initial condition is ðx10 ; x20 ; z10 ; z20 Þ ¼ ð:5; :5; :2; :25Þ.
Fig. 1. ¼ :0005; x1 ðtÞ and x1r ðtÞ.
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Fig. 2. ¼ :0005; x2 ðtÞ and x2r ðtÞ.
Fig. 3. ¼ :0005; z1 ðtÞ and z1a ðtÞ.
In particular, we compare the fast variables z1 and z2 with their corresponding approximations z1a ¼ a1 ðx1r ; x2r Þ ¼ x1r x2r and z2a ¼ a2 ðx1r ; x2r Þ ¼ x22r þ 2x1r x2r þ x1r , respectively. Moreover, in order to observe the boundary layer system we simulate z1 and z1a in the interval [0, .1].
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337
Fig. 4. ¼ :0005; z2 ðtÞ and z2a ðtÞ.
Fig. 5. ¼ :0005; z1 ðtÞ and z1a ðtÞ in ½0; :1.
6. Conclusions We have seen that an SPS becomes a DAS when the perturbation parameter is set to zero. Under the hypothesis that the DAS is of index 2, we have approximated the trajectories of the SPS using the trajectories of the DAS. We invoked the center manifold theory to guarantee some stability condition needed to prove the main result of the paper.
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Fig. 6. ¼ :0001; z1 ðtÞ and z1a ðtÞ.
Fig. 7. ¼ :0001; z2 ðtÞ and z2a ðtÞ.
We are currently studying the connection between a non-standard SPS with control and its corresponding DAS.
Appendix A Proof of Theorem 4.1. By Hypothesis 1, we can ensure, for each > 0, the existence and uniqueness of solution for (1) and (2).
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339
Fig. 8. ¼ :0001; z1 ðtÞ and z1a ðtÞ in ½0; :1.
Error for x: If we define ex ðt; Þ ¼ xðt; Þ xr ðtÞ, then Z t ex ðt; Þ ¼ ½f~ðxðs; Þ; w1 ðs; Þ; z2 ðs; ÞÞ f~ðxr ÞðsÞ; 0; a2 ðxr ðsÞÞ ds t0
¼
Z
t
½f~ðxðs; Þ; w1 ðs; Þ; z2 ðs; ÞÞ f~ðxr ðsÞ; w1 ðs; Þ; z2 ðs; ÞÞ ds
t0
þ
Z
t
½f~ðxr ðsÞ; w1 ðs; Þ; z2 ðs; ÞÞ f~ðxr ðsÞ; 0; a2 ðxðs; ÞÞÞ ds
t0
þ
Z
t
½f~ðxr ðsÞ; 0; a2 ðxðs; ÞÞÞ f~ðxr ðsÞ; 0; a2 ðxr ðsÞÞÞ ds
t0
and kex ðt; Þk 6
Z
t
k1 kxðs; Þ xr ðsÞk ds þ t0
a2 ðxðs; ÞÞk ds þ
Z
Z
t
k2 kw1 ðs; Þ; z2 ðs; Þ
t0 t
k3 kxðs; Þ xr ðsÞk ds
t0
using that f~; g and a2 are Lipschitz in their components, which is true by Hypothesis 1. To obtain a bound for ðw1 ðs; Þ; z2 ðs; Þ a2 ðxðs; ÞÞÞ
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Fig. 9. ¼ :00005; z1 ðtÞ and z1a ðtÞ.
Fig. 10. ¼ :00005; z2 ðtÞ and z2a ðtÞ.
we work, for simplicity, in the time s, then ^ 1 ðsÞ; z2 ðs; Þ ^z2 ðsÞÞk kðw1 ðs; Þ; z2 ðs; Þ a2 ðxðs; ÞÞÞk 6 kðw1 ðs; Þ w þ kðc w1 ðsÞ; ^z2 ðsÞ a2 ðx0 ÞÞk þ ka2 ðxðs; ÞÞ a2 ðx0 ÞÞk: By Proposition 3.7, given d1 > 0, there exists 1 > 0 such that
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341
Fig. 11. ¼ :00005; z1 ðtÞ and z1a ðtÞ in ½0; :1.
^1 ðsÞ; z2 ðs; Þ ^z2 ðsÞk < kðw1 ðs; Þ w
d1 3
8s P 0:
By Remark 3.2, there exists a constant l > 0 such that if kw10 k < l, then ^ 1 ðsÞ; ^z2 ðsÞ a2 ðx0 ÞÞk < kðw
d1 3
8s P 0:
Note that in this case the initial condition z20 is exactly a2 ðx0 Þ by Hypothesis 2. Moreover, there is a constant K > 0, such that ka2 ðxðs; ÞÞ a2 ðx0 ÞÞk 6 Kkxðs; Þ x0 k; where we have used that a2 in Lipschitz in x. So, given d2 ¼ d1 =3K, there exists 2 > 0 such that kxðs; Þ x0 k < d2 for all < 2 . Then, we conclude that ka2 ðxðs; ÞÞ a2 ðx0 Þk <
d1 3
8 < 2 ; 8s P 0:
Finally, if we choose ¼ minð1 ; 2 Þ, we obtain kðw1 ðs; Þ; z2 ðs; Þ a2 ðxðs; ÞÞÞk < d1
8 < ; 8s P 0
or equivalently, kðw1 ðs; Þ; z2 ðs; Þ a2 ðxðs; ÞÞÞk < d1
8 < ; 8s 2 ½t0 ; t1 :
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Now, given d > 0, we have Z t kex ðt; Þk 6 K1 kex ðs; Þk ds þ k2 d1 ðt1 t0 Þ t0
with K1 ¼ minðk1 ; k3 Þ. By Gronwall’s inequality, we conclude kex ðt; Þk 6 k2 d1 ðt1 t0 ÞeK1 ðt1 t0 Þ
8t 2 ½t0 ; t1 :
So, if we choose d1 ¼
d k2 ðt1 t0 ÞeK1 ðt1 t0 Þ
it yields, kex ðt; Þk 6 d
8 < ¼ minð1 ; 2 Þ
uniformly in t 2 ½t0 ; t1 ; which proves (36). Error for ðw1 ; z2 Þ: To prove error (37) we work again in the time s ^ 1 ðsÞ; ^z2 a2 ðx0 ÞÞ ðw1 ðs; Þ; z2 ðs; Þ a2 ðxr ðsÞÞÞ ðw ^ 1 ðsÞ; z2 ðs; Þ ^z2 ðsÞÞ þ ð0; a2 ðxðs; ÞÞ a2 ðxr ðsÞÞÞ ¼ ðw1 ðs; Þ w þ ð0; a2 ðx0 Þ a2 ðxðs; ÞÞÞ; which tends to zero for sufficiently small, and uniformly in s P 0; where we have used Proposition 3.7, (36), and that a2 is Lipschitz in x. Finally, given tb > t0 and d > 0, there exists 3 > 0 such that t t 0 ^1 w1 ðt; Þ w < d=2 for all < 3 and all t 2 ½t0 ; t1 . Then t t 0 ^1 kw1 ðt; Þk < d=2 þ w and invoking again the theory of center manifold, if kw10 k is sufficiently small, there exist positive constants c1 and b such that t t 0 ^1 ð56Þ w 6 c1 ebððtt0 Þ=Þ kw10 k: Since ebððtt0 Þ=Þ 6 8bðt t0 Þ P ln
1 ;
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343
then (56) will be OðÞ uniformly on ½tb ; t1 if is small enough to satisfy 1 ln 6 bðtb t0 Þ: More precisely, there exist constants k > 0 and 4 > 0 such that t t 0 ^1 w 6 k for all < 4 , uniformly in t 2 ½tb ; t1 . Now we can choose 5 6 4 such that k < d=2, for all < 5 ; then t t 0 ^1 w 6 d=2 for all < 5 , uniformly in t 2 ½tb ; t1 . Finally, if ¼ minð3 ; 5 Þ, then kw1 ðt; Þk < d
8 <
uniformly in t 2 ½tb ; t1 ; which proves (38).
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