Sobolev, Vladimir A.

English

Sobolev Vladimir A.

University of Szeged

Acta Sci. Math., 51 (1987), 491—440 Integral manifolds, stability and decomposition of singularly perturbed systems in Banach space V. A. SOBOLEV*) 1. Introduction. This paper is dealing with the study of infinite dimensional singularly perturbed systems near an integral manifold. Consider the system x = f(t, x, y, e) (1.1) ey = Ay+sg(t, x,y,e) where x and y are elements of Banach spaces X and Y with norms || • ||, A is a con-stant linear bounded operator in Y, and / : 2 ? x X x r x [ 0 , e 0 ] - Z , g: RXXxYX[0,c0l - Y are continuous nonlinear operator functions. Using the method of integral mani-folds [1,2] we shall study the stability problem for (1.1) and the problem of decom-position of (1.1) by transforming it to the form (1.2) u = F(t, u, e), (1.3) sv = Av+sG(t, u, v, e). Then we shall apply this method for investigation of linear singularly perturbed systems. 2. Slow manifold. We first recall the definition of an integral manifold for the equation x=X(t, x) where x is an element of a Banach space. A set S is said to be an integral manifold if for (t0, x0)£S, the solution (/, x(t)), x(tQ)=x0 is in S for t£R. If (/, x(t))£S only at a finite interval, then we shall say that S is a local integral manifold. *) This research was completed while the author was visiting the Department of Mathematics at the Budapest University of Technology. Received August'2,1984,.andin revised form August 4,1986. 492- V. A. Sobolev Let Br={yiY, | | S r } , /^=[0, e0], Q = R X X x B r X l H - Assume that / and g are bounded and satisfy the Lipschitz condition in x, y on Q: (2.1) 1/0, x, y, 8)1 =S M, II g(/, y, 8)1 M, II f i t , X, y, e) - / ( / , X, y, e)|| =S/(lx-x||-H!>>-y||), (2.2) |g( ' , X, y, e)-g(t, x, y, 8)1 =5• J ( | * - x | + where M and / are positive constants. Assume that the spectrum a (A) of the linear bounded operator A satisfies the inequality Re a(A)^ — 2a<0. Then there exists a positive number K such that (2.3) S Ke—, t s 0. We shall say that the integral manifold of system (1.1) is a slow manifold if it can be represented of form y—h(t, x, e), where h is a continuous operator-func-tion. If e0 is sufficiently small then for each e€(0, e0) the system (1.1) has an integral manifold (slow manifold) represented of form y—eh(t, x, e) (see [1], p. 438). Here A is a continuous and bounded operator-function defined on Q ^ R X X X I ^ and satisfies the Lipschitz condition in x: (2.4) \\h(t,x,e)-h(t,x,e)\\^A\\x-x\\, A>0. Moreover, if / and g are continuously differentiate on Q to k order and their -derivatives are bounded and Lipschitzian in x, y then h is continuously differentiable on to k and its derivatives are bounded and Lipschitzian in x. In this case the operator-function h can be represented as asymptotic expansion eh=eh1(t, x) +... ...+^hk(t, x)+hk+1(t, x,e) where hk+l=0(ek+1). The coefficients ht of this expansion can be found from the equation „ dh dh . (2.5) x> Bh>£) = A h + g ( i . *> eft. e)-For finite dimensional systems this method of approximating slow manifolds was essentially used in [3]. The method of approximation used in [4] can .be gen-eralized to infinite dimensional problems in an obvious way. The flow on a slow manifold is governed by the reduced equation (1.2), where F(t, u, e) = / ( / , u, eh(t, u, e), e). It is well-known for finite dimensional spaces X that the condition f(t, 0,0, e) =0, g(t, 0,0, e)=0 implies h(t, 0, e)=0 and if the zero solution of (1.2) is stable (asymp-totically stable, unstable) then the zero solution of (1.1) is stable (asymptotically stable, unstable). We shall prove below this statement for infinite dimensional X. 3. Integral manifold for auxiliary system. Let us suppose that / and g are con-tinuously differentiable on Q and their derivatives are bounded and Lipschitzian Singularly perturbed systems in Banach space 493. in x, y and introduce new variables u, z and xx by the formulae z=y~eh(t, x, e), xx=x—u where u satisfies (1.2). Consider the following auxiliary differential system u = F(t, u, e) ez = Az+eZ(t, u, xx, z, e), where fx =f(t, u+xx, z+eh(t, u+xx,e), e ) - F ( t , u,e), Z = g(t, m+XI, z+eh(t, u+xx, e), e ) - g ( i , u+xx, eh(t, «H-^, e), e ) -- e | j ( i , u+xi, e)[ / ( i , u+xx, z+sh(t, u+xx, e), e)-/(/, u+xx, eh(t, u+xx, e), e)]. By means of our assumptions it is easy to show that there exists a constant 0 such that fx and Z satisfy the following inequalities =S + ll^ll + ||z||] [ | |z-z| | + (1 +11*11) И * - хЛ ч-СИхжВ + Ы ) 1И-ЙЩ, (3.7) \\Z(t,u,xx,z,e)-Z(t,ü,xx,z,e)\\^N[\\z-n + \\zmu-ü^ + lxx-xxm, where t£R, и, й£Х, xx,xx£X, z,z£Bri, 0<r1Sr. We shall show that the system (3.1) has an integral manifold represented of form xx = sH(t, u, z, e), where H is an operator-function defined and continuous on Q2 = R x X x B e X l e , 0 < e S £ 0 l and H satisfies the inequalities: 1 K. (3.8) \H(t,u,z,z)\\ =M|z[l, (3.9) IIH(t, u, z, e ) - H ( t , u, z, e)| S b\z-z\\, (3.10) WHO, u, z, e)—H(t, й, z, e)|l c||z|| • | |ы-й||, with a,b,c>0 for t£R, и,й£Х, z,z£Bt, e€/£ l . The flow on this manifold is governed by reduced equations (1.2), (1.3), where F=f(t, u, eh(j, и, с), e), G=Z{t, u, sH(t, u, v, e), v, e). (3.1) = / i ( í , U, xx, z, e) (3.2) (3.3) (3.4) (3.5) (3.6) ll/lO» "> Xx, Z, 8)|| S iVdl*! +I|z||), \Z(t,u,xx,z,B)\\^N\z\\, 1 / l C , U, XX, Z, ¿ ) - f x ( t , U, XX,Z, 8)11 ^ ( l ^ - x j + llz-zl), \\Z(t, U, XX, Z, 8) -Z(t, u, xx, z, e)|| á N(\\xx - x j + | |z-z| | ) , \\fi(t, U, xx, z, e ) - f x ( t , й, xx, z, 8)11 S 494- V. A. Sobolev Moreover, every solution of (1.1) with ^y(Q-zh{ t 0 , л(/0), е) | |ёо can be represented of form x = u+eH(t, u, v, e), (3.11) у = v+eh(t, x, e) = v+eh(t, u+sH(t, u, v, e), s), where u, v is the corresponding solution of (1.2), (1.3). Our proof of this statements is modelled on KELLEY [5]. Let 5 be the set of operator-functions sH: Q2-*X such that H is continuous and satisfies (3.8)—(3.10). Let d be a metric on 5" defined by d(eH, EH) = sup{-|i | -ei#(r, u, z, e)-H(t, u, z, s)|], t£R, u£X, z€£ e } for each E€(0, e j , EH, EH£S and note that S is a complete metric space with respect to d. For each SH£ S, we consider the system (3.12) й — F(t, u, e), (3.13) ez = Az+sZ(t, u, sH(t, u, z, s), z, e), with solutions denoted by и—Ф(1, i0, u0, S), z=W(t, t0, u0, z0, E\H) where «К'О.'О. "О. Е)=«О> У(h, 'о, "О. zo> s\H)=zQ. The operator-functions F(t,u,e), Z(t, u, EH(t, u, z, e), z, г) are uniformly bounded on their domains, hence, any solution of (3.12), (3.13) is defined for all t. As usually, (see [1,2,5]) the equality Xi=eH(t, u, z, e) describes an integral manifold for (3.1) if and only if the operator-function sH is a solution of the equation ЕЕ (3.14) еН(т, и, z, E) — — J Л(/, ФИ, т, и, е), £#(/, Ф(/, т, и, Е), т У( t , т, и, z, Е\Н), е), ¥(t, т, и, z, е |Я), E)dt. Let </;(?)=Ф(г, т, и, е), — т, и, z, г\Н) then by the "variation of constants" formula t ф(0 = e(im(t-r)z+ J e( i /£)A(,-s)Z(S j ^ ^ ^ d s t By (2.3), (3.3) and (3.8) there holds for all - < ~ < т | | z | | s e , e€(0, e j : t №(01 - [jzl + / tfe-(a/£)('-s>iV|]«Ks)|| ds. X Singularly perturbed systems in Banach space 495. Therefore, by Gronwall's Lemma, we obtain (3.15) - » < t S / < » , where ct l=a—EKN^y^'0 for sufficiently small ea. Mow define an operator T on S by setting oo (3.16) T(H)(r, u,z,e) = - I A(t, 4>(/), eH(t, (p(t), (i), e), Ф(t), e) dt. T The improper integral here converges by virtue of (3.2), (3.8) and (3.15). It is clear that T[H) as defined in (3.16) is continuous on Q2. Also, by (3.2), (3.8) and (3.15) we obtain 00 NK \\T(H)(T, U, Z, e)|| ^ f N( 1 +ea)Ke-(*>||z[| dt = e — ( 1 + s a ) ||z||, / «1 NK and therefore T(H) satisfies the boundedness condition required by (3.8) if < 1 «i NK If NK\ and as= /11 —e 1. «1 / v ; To prove that T(H) satisfies the conditions, required by (3.9), (3.10) we reason as follows. Let u f X , z,z€Be, ф^Ч'О, т, и, z, е|Я). Then, by (3.5), (3.9), (2.3) a:nd by the "variations of constants" formula we have t Н(1)~ФЛ')1 / Ke~^l^-s)N(l + eb)j|ф(s)-фг(s)|j ds. t Therefore, by Gronwall's Lemma, we obtain (3.17) UiO-MOW K e - ^ W - ^ W z - z l , — t § / < a2 = tx-eKN(l+eb). Then, by (3.4), (3.9) and (3.17) oo |Г(Я)(т, и, z, e) — T(H) (T, u,z,e)|| S / N(l +eb) H^(0-^xC0il dt t S e ( l+£b) | |z -z | | a 2 It is clear that for sufficiently small % a constant b can be choosen such that oc2>y K2N and (1+е г£)^6. From this inequality it follows that T(H) satisfies the «2 Lipschitz condition required by (3.9). In exactly the same way by the inequality I Ф(/, т, и, Е) — Ф ( / , т, w,.e)|| S — »< т Ё/<ОО 496- V. A. Sobolev and (3.10), (3.7) and (3.6) it is easy to show that for some c > 0 and sufficiently small Ej the operator-function T(H) satisfies the condition (3,10). Now, let eH, eH£ S, ij/z(t) = 4>(t, т, u, z, e\H). Then by (3.4) and (3.9) (3.18) | |Г(Я)(r, u, z, e)—T(H)(x, u, z, e)| S CO ^ / iV[(l +eb) \\Ф0)—ФЖ)\\ +e | | # ( i , <р(г), Mt), e)-H(t, <p(t), фг(t), e)||] dt s X oo S / iV[(l +eb) U(t) - ф М \ + Ке-Ы*'-* || z|| d(EH, EH)] dt. Z Using (3.5) and (3.9) we find that 11^(0-^2(011 ^ t 25 / Ke~(ot/£){'~z)N[(l +eb)\\ijj(s)—ij/2(s)l +Ke~^t-3)\z\d(EH, eH)]dt. t Substitution of this into (3.18) yields y L ЦГ(Я)(т, и, z, e) - Г ( Я ) ( г , и, z, e)|| S £ — f ( l +ЕЬ) l l d(sH, EE). ||z|| у I a2-y J From this last inequality it easily follows that T is a contraction mapping if ег is sufficiently small. Thus, Г is a contraction mapping of 5 into itself and so, by the known Banach Contraction Principle, T must have a unique fixed point EH^S. The operator-function eH is a solution of (3.14) and, therefore, the equality xx=EH(t, и, z, e) represents an integral manifold for (3.1). The flow on this manifold is governed by (1.2), (1.3) where F=f(t, u, eh(t, U,E), E), G = Z(t, u, eH(t, u, v, E), v, e). 4. Decomposition and stability. Our next object is to obtain the representation (3.11). Let x-x(t), y=y(t) be a solution of (1.1) with X ( / 0 ) = A ' 0 , y(to)=y<» | |^0-£/i(i0 , xQ, e)|| ̂ Q. We shall show that there exists a solution u=u(t), u(t0)=uQ, v=v(t), v(tQ) = v0 of (1.2), (1.3) such that x(t) = u(t)+sH{t, u(t), v(t), E), (4.1) y(t) = v(t)+eh{t,x(t), £). It is sufficient to show that (4.1) holds for t=t0. Substitution t=ta into (4.1) yields = u0+EH(t0, u0, v0, £), y0 = v0+eh(ta, x0, E) Singularly perturbed systems in Banach space 497. and, therefore, v0=y0—eh(t0, x0, e). For u0 we obtain the equation (4-2) x„ = u0+eH(t0, u0,y0-eh(t0,x0,e), e). This last equation can be represented of form "о = P(u0 , e) = х 0 - е Я ( / 0 , m0, y0-eh{t0, *o>£), «)• From (3.10) it is easy to obtain that for each £6(0, EJ and fixed x0, y0 such that ||y0—£Й(?0,ХО5£)11 — » P is a contraction mapping of X into itself and so, SyC by the Banach Contraction Mapping Principle, P must have a unique fixed point u0€X which is the required solution of (4.2). Now, we consider the stability problem for (1.1). Using (4.1) we obtain that every solution x=x(t), y=y(t) with |]>-0—е/г(/0, x0, s)|| SQ can be represented as x(t) = uiO+cp^t), (4-3) y(t) = eh(t,u(t),e) + (p2(t), where (u( t ) , eh(t, u(t), s)) is a solution lying in the manifold y=eh(t, x, e); <pL= = sH{t, m(0, v(t), e),cp2 = v(t)+eh(t, u(t)+eH(t, u(t),v(t),e), s)-eh(t, u(t), e). This and (2.4), (3.8) and (3.15) allow us to write 1^(01 еаКе-ЫМ-^Ы, (4.4) 11 (̂011 s (1 +е*аА)Ке-Ы*'-'о>\\ио1 «€(0,«!], ts=t0, v0 = y0-eh(t0,x0, E). Assume that f(t, 0,0, £)=0, g(t, 0,0, E)—0; then h(t, 0, E)=0 and F(t, 0, E)=0. By (4.3) and (4.4) we obtain | |x(0|| 'S |]и(0|| ЦЯ011 Si eA ||И(0|| +(1 +£2аЛ)*е-<^<'- 'о>к| | , i S i 0 . From this last inequalities it easily follows that if the zero solution of (1.2) is stable (asymptotically stable) then the zero solution of (1.1) is stable (asymptotically stable). It is obvious that the instability of the zero solution of (1.2) implies the instability of the zero solution of (1.1). Now, we can summarize our results in the following T h e o r e m 4.1. Let f and g in (1.1) be continuous, bounded and satisfy (2.1), (2.2) w j RXXxBrXl^: let us assume that the spectrum of the linear bounded oper-ator A satisfies Re a{A)^ — 2 ж 0 . Then theie exist numbers sx and Qx such that the following assert at ions are true: 498- V. A. Sobolev (i) For each c€(0, e j , g£(0, gx) and t0 there exists for (3.1) an integral manifold represented by an equation of form xx=eH(t, u, z, e) where H is an operator-function defined and continuous on RXXXBeXlCi and, moreover, H satisfies (3.8)—(3.10). (ii) Every solution x=x(t), y=y(t) of (1.1) with x(t0)=x0,y(t0)=y0, li^o-£Й('о> ;r0, can be represented of form (3.11), where u = u(t), u(t0)=u0 is a solution of (1.2), u0is asolution of (4.2); v—v(t) is a solution of'(1.3) with u = u(t), v(t0)=v0=y0-Eh(t0,x0,e). (iii) If f(t, 0, 0, e)=0, g(t, 0, 0, £)=0 and the zero solution of (1.2) is stable (asymptotically stable, unstable), then the zero solution of (1.1) is stable (asymp-totically stable, unstable). Note, that in the proof of this theorem we did not use the boundedness of A. So, Theorem 4.1 can be extended onto the system (1.1) with an unbounded operator A, if A is the generator of a strongly continuous linear semigroup S(t) such that | |S'(/)||^*e-M, t s 0 . It should be observed that similar problems for systems with unbounded oper-ators were studied in [2, 6]. The next result shows that, in principle, the operator-function H can be approxi-ЪН mated to any degree of accuracy with respect to e. Let D(eH)=e—— at дН &H Fit, u, e)+—{Av+eZ(t, u, sH, v, i))-fx(t, u, cH, v, e). If Z ) (еЯ) -= 0 ( e * + 1 ) then \\H-H\\ = 0(t*). The^idea of the proof of this statement is very simple. Let us introduce a new variable^x2 =x1 — sH(t, u, z, e); then for u, x2, z we obtain the following system " = f0, u, £ ) , *2 =fi(t,U,X2,Z,E), EZ = Az+eZ(t, u, X2+EH, Z, E), dH where fx=fx{t, u, x2+EH, z, £)-fx(t, и, EH, Z, £ ) -£ — - [Z(t, u, X2+EH, Z, £ ) -oz —Z(t,u,eH,z,e)], EH=eH(t,u,z,E). This last system has an integral manifold x2=eHk+x(t, it, z, s) such that Ht+J=0(£). It means that the system (3.1) has the integral manifold xx=EH(t, u, z, ¿)=eH{t, u, z, £)+0(e*+1). In many problems, Я can be found as asymptotic expansion £ Я = £Я1(/, и, v) + ... +Е*Я*(/, U, v) + 0(Ek+1) from the equation £>(еЯ)=0. Note, that u0 can be found as asymptotic expansion "o = "o(e) = »o+^o+.- .+^Ko+O^" 1" 1 ) from (4.2). It is easy to see that u\= — Hx(t0, xa, y0). Singularly perturbed systems in Banach space 499. 5. Linear systems. Consider the following system *1 = ^11*1+^12*2+/l, (5.1) £X2 — /421*1+^22*2-1"^» where Xi, fi=fi(i,B) vary in the Banach space Xt, and Au=Ai}(t,e) are operator-functions Au: Xj—Xiii, J =1,2). Assume Au and ft to have high order continuous and bounded derivatives with respect to t and e, for t£R, e£[0, e0]. Therefore, they can be represented as asymptotic expansions AtJ = Ai? (t)+eAtf (/) + ... +ekAtf (/)+0(ek+1), ft = fim 0)+e/i(1) (0+• • • + ekfi(k) (0+O (ek+1) with smooth and bounded coefficients. Let us suppose that the family Aj$(t), t£R, is compact, the spectrum c(A<$) of A(22(t) satisfies the inequality (5.2) Re e(A$) ^ - 2 a < 0, t£R and there exists bounded operator t ^ ] - 1 . Under such assumptions there exists a transformation Xi = u+eH(t, e)v, x2 = v+L(t, e)x1 + l(t, e) = (/+sLH) v+Lu + l(t, e), analogous to (3.11) for the linear case. The new variables u, v satisfy the equations (5.3) u = (A11+A12L)u+f1+A12l, (5.4) ev = (A22-eLA12) V. The operator-functions L, H and the function I can be found from the equations (5.5) £l+£Z,(^u+/41 2L) = A21+A2SL, (5.6) sH+H(A22-eLA12) = e(A11+A12L)H+Ali, (5.7) el+eLf1 = (A22-eLA12)l+f2 as asymptotic expansions L=Lm(t)+eLm(t) + ... +ekLik)(t) + 0(ek+1), H = Hw(t)+eH™(i) + ...+£k-1H(-k-i'>(t)+0(f!1), I = /«» (t)+ei(1) (0+• • • + e*/(k) (t)+0 (e*+1). It is a straightforward computation to obtain expressions for L(0, H{i), /(i) from (5.5)—(5.7). Note that L is a bounded solution of the Riccati equation (5.5) on R and, there-fore, satisfies the integral equation 2 * L(t, e) = — f U(t, s, e)[^2l(s, e)-eL(s, e)(/4u(s, e) + Aia(s, e)L(s, £))] ds, 500 V. A. Sobolev: Singularly perturbed systems in Banach space where U is the evolutional operator of the equation EX2—A 22 x2. Using (5.2) and the compactness of A$( t ) we obtain (5.8) || U(t, s, e)|| 2= Ke-Wt)U~s\ - <*=< s t < For H and I we have the exact expressions j ~ • H = f V(t,s,e)A12(s,e)JV(s,t,e)ds, e J 1 ' l = — J W(U S, S ) [ / 2 ( S , E) - eL(s, еШз, £)] ds, — 0 0 where V is the evolutional operator of the equation x1=(A11 + A12L)Xl and W is the one of the equation ЕХ2=(А22—ЕЬА12)хг. The improper integrals here con-verge by virtue of (5.8). As earlier, the stability of (5.3) is equivalent to the stability of (5.1). In conclusion it should be noted that the stability and decomposition problems for finite dimensional systems were considered in [7]. References [1] Ю. А. Митропольский, О. Б. Лыкова, Интегральные многообразия в нелинейной механике, Наука (Москва, 1973). [2] D. HENRY, Geometric theory of semilinear parabolic equations, Lecture Notes in Math., 840, Springer-Verlag (Berlin—New York, 1981). [3] В. В. Стрыгин, В. А. Соболев, Влияние геометрических и кинетических параметров и диссипации энергии на устойчивость ориентации спутников с двойным вращением, Космические Исследования, 3 (1976), 366—371. [4] В. А. Соболев, К теории интегральных многообразий одного класса систем сингулярно возмущенных уравнений, Приближенные методы исследования дифференциальных уравнений и их приложения, Куйбышевский Университет (1980), 124—147. [5] A. KELLEY, The stable, center-stable, center, center-unstable and unstable manifolds, J. Dif-ferential Equations, 3 (1967), 546—570. [6] О. Б. Лыкова, Принцип'сведения в банаховом пространстве, Укр. Матем. Ж.. 4 (1971), 464—471. [7] V. A. SOBOLEV, Integral manifolds and decomposition of singularly perturbed systems, Systems Control Lett., 5(1984), 169—179. DEPARTMENT OF DIFFERENTIAL EQUATIONS KUIBYSHEV STATE UNIVERSITY KUIBYSHEV, U.S.S.R. 

Integral manifolds, stability and decomposition of singularly perturbed systems in Banach space

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Integral manifolds, stability and decomposition of singularly perturbed systems in Banach space

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