Vietnam J. Math (2015)43.417^37 DOI 10 1007/SI0013-015-0133-3
Stability and Asymptotic Behavior of Systems with Multi-time
Quoc-Phong Vu
Received: 31 December 2013 / Accepted 29 September 2014 / Published online 24 March 2015 0 Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore
Abstract Stability and asymptotic behavior of linear multi-time systems ^ w ( t )
•^i"(t) + / i (t) is considered, where A, are generators of commuting Co-semigroups on a Banach space. We formulate and prove versions ofthe results on exponential stability to homogeneous linear systems with multi-time. For non-homogeneous systems, we introduce the compatibility condition and obtain the variation-of-conslants formula. Using simulta neous solutions of Sylvester equations, we obtain results on asymptotic almost periodicity of solutions.
Keywords Lyapunov equation - Stability • Asymptotic stability • Exponential stability • Joint spectrum
Matiiematics Subject Classification (2010) Primary 34D20 37B25 • 47A62 47A10 • 47A13 . Secondary 15A24
1 Introduction
There is an extensive literature on relations between the stability, the theory of Sylvester- Lyapunov equations, and the asymptotic behavior of the linear evolution equation
u'it) ^AiiiD-l-fit), f > 0 . (1)
Dedicaied to Nguyen Khoa Son's 65th birthday.
Part of this work was done dunng the author's visii to the Vietnam Institute of Ad\anced Smdy ii Mathematics. Hanoi. Vietnam. June-July 2013.
Q.-RVutS)
Depanmenl of Mathematics, Ohio University. Athens, OH 45701. USA e-mail vu@ohio.edu
^ Sprii
where A is a linear operator on a Banacli space (see, e.g., [3,11, 17, 28, 35, 361). However, analogous questions for evolution equations with multi-time
^iiit) = AMt) + fiH), l=it,,...,tOfiKi (1 < > • < « , (2) at,
are less studied. In recent years, there has been a considerable interest in investiga- tions of some special classes of multi-time equations which occur namrally in integrable Hamiltonian systems (see, e.g., [5, 8, 26]), and control theory (see [31]). This has motivated us to consider to which extent the known results on the asymptotic behavior of solutions of (1) still remain valid for the multi-time equations (2).
In this paper, we study the stability and asymptotic behavior of solutions of (2) where A, are commuting, generally unbounded hnear operators on a Banach space £. In Section 2, we present some results on stability and asymptotic stability of the homogeneous system
^ H ( t ) - A , H ( t ) ( 1 < / < A ) (3) Section 3 is devoted to related systems of Sylvester equations. In Section 4, we extend the
vari ali on-of-con slants formula to multi-time non-homogeneous equations (2) and present some applications of the method of simultaneous Sylvester equations to the asymptonc behavior of the non-homogeneous equations (2).
This paper is dedicated to my friend and colleague, Nguyen Khoa Son, with whom I share memorable years of study and play at Kharkov Universiiy. We both were under strong influence of the Kharkov mathematical school and received encouragement and sup- port from Kharkov mathematicians. I would like also to acknowledge that my interest in differential equations with multi-time arose from my memories of old papers (and words) by A D. Myshkis on the subject.
Throughout the paper, K^ = |t ^ (ii, . . . , ; * ) : (, > OV/}, 0 = (0,. , 0 ) a n d l - ( 1 . . . . , I). IfT = ( T l , . . . , ? * ) e R^, then by [O.T] we denote the "time interval" in K^.:
namely (0. T] = (t ^ U], • •-'k) • 0 < I, < T, V i } . C * - {k = ih kk) : k, eC], C- = {k e C : Rek < 0) and C l - {X e C* ; ReA, < 0 V/}. The intenor of a set Q is denoted by Int(£^), so that, in particular, lnt(R5-) ^ (t = ( / | , . . . , r^) : t, > OVil.lf
^ = (Al Ak) and t = Ul /t), tiien we let ^ - 1 ^ Z L i ' i A , .
The letters £,T,... denote a Banach or Hilbert space, Li£, T) is the space of all bounded linear operators from £ to J^ (we write Z.(£) instead of Li£, £)), A, B,T,..
denote single operators or one-parameter semigroups, and A, B, T for commuting /:-tuples of operators or multi-parameter semigroups. The domain of an operator A is denoted by DiA) The spectrum (the point spectrum, the approximate point spectrum) of a single operator A is denoted by o-(A) (resp. crpiA).G„iA)). while thejoini spectrum of a commut- ing k-niple A or the spectrum of representations T of semigroups is denoted by Sp(^) and Sp(T), respectively. Other notations are either standard and should not cause confusion, or will be introduced as needed
2 Stability of the Homogeneous Systems
A well-known theorem of Lyapunov states that for a complex matrix A (of order n xn),a necessary and sufficient condition that ail eigenvalues of A have negative real parts is that for each positive definite matrix H, the matrix equation
® Springer
Extension of Lyaponov Theorem
Extension of Lyaponov Theorem 419 has a unique positive defmite solution. Since all eigenvalues of A have negative real parts if
and only ifthe zero solution ofthe linear system;c'(f) ^ AxU)U > 0, .v(f) G R " ) is asymp- totically stable (which is also exponentially stable in this case), this theorem usually is referred to as a Lyapunov Stability Theorem. Note tiiat the Lyapunov Stability Theorem remams valid for case A isa bounded linear operator on a Hilbert space (see, e.g., [11]).
In this section, we present extensions of Ihe Lyapunov Stability Theorem to the case of (3). In order to precisely state the resuhs, let us introduce some definitions and terminology.
A strongly continuous homomorphism T from the semigroup R+ into Li£) is called a k-parameler Co-semigroup, In other words T(t) is a family of bounded linear operators on S such that:
(i) r ( 0 ) - / ;
(li) r(t + s)-r(t)r(s) (vt.se R*^);
(lli) The mapping t i-^ T(t)j: (from R^ to 5) is strongly continuous for every A: E £ If T(t) IS a fc-parameter Co-semigroup, then the formula 7", (r) :— Titi) (l > 0), where
\= iS,] S,k) iS,j is the Kronecker symbol), defines one-parameter semigroups, which arecommuting,i.e.7'Kj)r,(/) = Tj(r)7',(i')(V/, J > 0, I < i.j < it). Let A, denote the generators of the corresponding semigroup 7), with the domain DiAf) Then A, are closed, densely defined linear operators on £ (sec, e.g., [13, pp. 50-51]).
Proposition 1 The operators A, are commuting, i.e., AjAjx = AjA/X (VA: e DiA,Aj) fl DiAjA,)). their sum A = A] -f- - • + A^ ij closable, densely defined, and ils closure A is the generator of tlie semigroup Tit) — TiU)- • •T/.U)- Moreover, the linear manifold D2iA) := n'^^^^DiA,Aj) also is dense.
Proposition 1 is a well-known fact but since we could not find a reference in the available {to us) literature, we present a proof in the Appendix.
Consider tiie multi-time abstract Cauchy problem
iV,uit):=^uit) = A,uit) il ^i<k).
\uiO) = x, ' ^' where A, are closed, densely defined linear operators on £ such that n^^^ DiA,) is dense.
Generalizing the definition in [17], we call the problem (4) well-posed if: (i) for every X e n*_|D(A,}, there exists a unique classical solution u on R^, and (ii) the solution depends continuously on the initial condition, uniformly on each time interval [0, T]: if
«„(0) -^ uiO), tiien H„(t) - * H(t), uniformly on t e [0, T] (VT e R^.). It is well known that, for an one-time system, the Cauchy problem is well posed if and only if the cor- responding operator A is a generator of a one-parameter Co-semigroup (see, e g , [17]).
Using this fact, we obtain immediately that the multi-time Cauchy problem (4) is well posed if and only if A, are generators of some commuting Co-semigroups Tj (f) Therefore, assuming well posedness, one can identify (4) with commuting semigroups so that, for example, to say that the (deterministic) system (4) is stable (asymptotically stable) is the same as to say that the corresponding semigroup is stable (resp. asymptotically stable).
In order to define the long-time behavior for multi-systems, we must define direction of time (so that one can speak about "past" and 'Tuture"). There is a natural quasi order m M^, namely, we define t -; s if and only if s - t e R^. With this quasi order, R^ becomes a
^ Springer
directed set (for every t j , t i e R+ there exists s e M+ such that t | -; s. IT ^ s); hence, every function on R^. is a net, so that we can speak about convergence through R+. In the sequel, we denote this limit by limjji .
Note that if u is a (classical) solution of (4), then u also satisfies equation X>rM = iA-r)u, which descnbes the evolution of the state in the direction r. Intuitively speaking, evolu- tion from time to ^ (/[" tf^h to time ti ^ { / [ ' . . . . / " ' ) in R+ involves evolution in each time i, from t^^'' to r,''' independentiy, albeit with different speeds. The pairwise com- mutative property r,(f,)7"^(fj)j: = TjUj)T,U,)x means that the system is "deterministic"
(evolution of the "slate" x from time 0 to time t is independent from the particular lime direction from 0 to t).
Below, we give definitions of the stability notions, which are natural generalizations of the well-known stability notions for one-time systems, in the framework of the introduced time order in R^.
Definition 1 (i) The system (4) is called stable if the seimgroup Tit) is uniformly bounded, i.e.,
sup ||T(I)|| < oo,
(ll) The system (4) is called asymptotically stable if it is stable and lim^i, Tit)x = 0 for every x e £.
(iii) The system (4) is called exponentially stable if it is stable and lim^t ||T(t) || = 0.
In the sequel, uniformly bounded semigroups are simply called bounded semigroups.
Remark 1 It follows from the Uniform Boundedness Principle that if TU) is a one- parameter Co-semigroup and lim,^cc T(f)x exists for every x e £, then TU) is a bounded semigroup (i.e.. the corresponding one-time system is stable). For ft-parameter semi- groups with k >1 this is not necessarily valid For instance, if 7"i(r) is a nilpotent semi- group (i.e., Tl Uo) — 0 for some ^o > 0) and T2U) is an unbounded semigroup, then the two- parameter semigroup ri(/i)72{(2) converges to 0 through R ^ (it eventually vanishes).
On the other band, if 7~(t) can be extended to the whole group K* (in particular, if A, are bounded operators), then from convergence of Tit)x, it follows that Tit)x is bounded.
In fact, given limj^t Tit)x = y and a fixed e > 0, there exists to e K^ such that ||T(t)i - y|| < € whenever to -; t Thus, the set Q = [7"(t)j: : to -< t} is bounded, and so is {Tit)x:teR''+] = Ti-to)Q.
In view of Remark I, in the definitions of asymplolic stability and exponeniial stability of Tit), we require ihal the system (4) (5 stable (i.e.. semigroup Tit) is bounded).
Lemma 1 Suppose that TU) is a bounded k-parameter semigroup on a Banach space £, sup,gj(ii i|T(t)|| = A / < 00. r— (ri ru) is a fixed element in R^. and TrU) : = T(fr) (r > 0). Then, the following .statements hold:
(i) Tit) is asymptotically (exponentially) stable if and only if Trit) is asymptotically (resp. exponentially) stable for some (hence for all) r e InKRlj.);
^ Springer
Exiension of Lvauonov Theori^m
Extension of Lyaponov Theorem 421
(li) V TrgU) IS asymptotically (exponentially) stable for some rn 6 K ^ , then TU) is asymptotically (resp. exponentially) stable, hence Trit) is asymptotically (resp. expo- nentially) stable for all r e Int(R^).
Proof if) Suppose TH) is asymptotically stable and r e Int(M^). Let x G £ and £ > 0 Then, there exists to G R ^ such that if to -c t, tiien ||T(t)A:|| < j ^ Since r e Int(R^), there exists SQ > 0 such that to < íor; hence, ||T(íor)j:|| < ~. This implies that ||T(ir)j»:|| < e (Vr > ^o), ịẹ, rr(r) is asymptotically stable
Conversely, suppose that rr(() is asymptotically stable for some rGK+ and x e £,£ > 0 are given. Then, there exists ;o > 0 such dial \\Trito)x\\<-^. If for ^ L then t = ioi" + sfor some s e R+, which implies that \\Tii)x\\<e (Vfor -< t), ịẹ, TU) is asymptotically stablẹ
This proves (ii), as well as the " i f part of (i). The proof for the case of exponential stability
is analogous. D Remark 2 The converse statement of Lemma l-(ii) is not true; that is it may happen that
Tit) is asymptotically (exponentially) stable, but T^^U) is not asymptotically (resp, expo- nentially) stable for some r = (ri rk) e R ^ \ Int(R^). As an example, let T] it) be a bounded but not asymptotically stable Co-semigroup and let T2il) be a nilpotent Co- semigroup. Then, the two-parameter semigroup TU.t2) — T]iti)T2U2) is exponentially stable, while for ro = (1,0) the semigroup 7V(,(r) - T(r, 0) = TiU) is not asymptotically stablẹ
Note also that the assumption that Tit) is bounded is essential for the validity of Lemma l-(i): if, for example, Ai— 7, Aj — —/, and rj — (I, 2), r2 — (2, 1), then Tr^il) = sxpi-tl) U> 0) is exponentially stable, but TrjU) = exp(l/) (r > 0) and T(t) ^ ri(r|)r2((2) = exp[(r| ~ fi)?^] (t ^ il],t2) e Ế) are not exponentially stable (cf. also with the example in Remark 1).
In view of Lemma 1 and assuming that the underlying system is known to be stable, the asymptotic stability (exponential stability) of the multi-time system (4) is equivalent to the asymptotic stabihty (resp. exponential stability) of the one- parameter semigroup T^it) — Tili^) U > 0) for some r € Int{R^), with the generator Ar- Thus, Theorems 1 and 2 below are essentially an adaptation of the known results to the new class of multi-time systems. However, we present it in a form reflecting ađitional aspects of the multi-time case For this, we recall the notions of joint spectia of commuting operators.
Definition 2 (i) A point ị — (Ại, .. , A;-) e C* is called & joint eigenvalue of yl if there existsavectorjr G nf^|Z)(A,). ;c ^ 0, such that A,x — A,j:foraUi G {1 k).
The set Spp ( ^ ) consisting of all joint eigenvalues is called lh.& joint point spectrum of Ạ
(ii) A point k = ( A i , . . . , kk) G C is called an approximate joint eigenvalue of A if tiiere exists a sequence of vectors {jCnj^j C n^^j£)(A,-), ll^nll — 1, such that
\\AiX„ — kiX„\\ -* Oasn - * coforalli e { 1 , . . . , / : ) . T h e s e t Sp^(^) consistingof all joint approximate eigenvalues is called the joint approximate point spectrum of Ạ
Recall also tiiat a bounded linear operator P on a Hilbert space £ (with inner product (•, })is called positive semi-definite, denoted by P > 0 , if {FJ:, J;) > 0 (VJ: / O ) , and is called
Q Springer
positive definite, denoted by P > 0 , if there exists o O such that {Px,x}>c\\x\\^ (VJ: e £).
It is well known that P > Oif and only if P i s self-adjoint and cr(/') c R + , a n d P >Oifand only if P > 0 and is invertible
First, we consider the case when the operators A, are bounded, since the results in this case are more complete.
Theorem 1 Let A = ( A | , . . , Ak) be a commuting k-tuple of bounded linear operators on a Banach space £. Assume that the mulli-time syslem (4) generated by A is stable. Then A. The following conditions (i)-(iv) are equivalent
(i) The muhi-time system (4) is exponentially stable:
ill) Sp^iA) n (iR)* = 0,-
(til) For some {hence for all) T e InliR^), the one-lime system
u'iD^ArliU) ( f > 0 ) (5) IS exponentially stable:
(IV) For some (hence for all) r e Int(R*_), (j(Ar) fl iR - 0.
B. If £ is a Hilbert .••pace, then (i)-(iv) are equivalent to the following' (v) There exist self-adjoint operators H], .., Hk and P > 0 which salisjy the follow-
ing conditions:
(a)
// : - 5^ H, > 0. (6)
(b) The following identities hold:
A*Hj-\-HjA, =A)H,-\-H,Aj (VJ,> = I it). (7) (c) P satisfies simultaneously the Lyapunov equations
A ; P - | - P A , ^ - H , ( V i ^ l k). (8) (vi) For some ihence for all) r e InliR'^) the Lyapunov equation
A;P-i-PAr = -n (9) has a unique solution P > Ofor each / / > 0.
C Under the conditions (v) and i\\), ii) has a unique simultaneous solution P > Ofor each family H\, .., Hk of operators which satisfy (6)-(7).
u . lfdim£ < oo. then conditions ii)-i\i) are equivalent lo asymptotic stability of il).
Proof A. The equivalence (i) o (iii) is shown m Lemma 1, and the equivalence (iii) 4* (iv) is the well known fact of one-time systems. To show (ii) <? (iv) note that since A — Ai -h
• - • -i- Ai- generates a bounded semigroup, we have CT(A) C C _ , hence CT(A) n UM) = a a ( A ) n (iR). By the Spectral Mapping Theorem of Choi-Davis [9] (cf also [30]), aaiA] +
•••-\-Ai,) = {ki-\---l-kk:iki,...,kk) GSp^ (.4) I, hence condition S p „ ( ^ ) n ( m ) * = 0 implies OaiA) D IR = aiA) n ((R) = 0. Thus, the semigroup T ( ( l ) is exponentially stable. This imphes, by Lemma 1 that for all r G Int(R^), T(fr) is exponentially stable and, therefore, CT(Ar) 0 iR = 0. Conversely, let aiAr) n iR = id for some, r e Int(R+).
Then, the semigroup T ( / r ) is exponentially stable, hence, again by Lemma 1. T ( r l ) also is
Extension of Lyaponov Theorem 423
exponentially Stable (since 1 G Int(R*.)) ThereforeCTO(Ai-I h A i ) n (iR) = 0. and the Choi-Davis Spectral Mapping Theorem unpliesSpfl (.4.) n (i"R)* = 0.
B. Assume that f is a Hilbert space. The equivalence (iii) •«• (vi) is the Lyapunov the- orem for one-time systems. To see the equivalence (v) •*> (vi), note that if (9) is satisfied, dien the system (5), and hence (4), is exponentially stable. It follows that A — ^,=]Ai generates an exponentially stable semigroup. Let W > 0 be a positive definite operator.
Then, there exists a unique positive definite operator P such that A'P ^r PA — -H. Let Hi = -iAIP-VPA,) T h e n 7 / , , P s a t i s f y ( 7 ) a n d ( 8 ) , a n d ^ * ^ , W , ^ - ( A * P - | - P A ) - H.
Conversely, if //, and P satisfy (8) and H = E , = i ^ / > the" H- P are positive defi- nite and satisfy A*P -I- PA — —H. Therefore, the system (5) is exponentially stable for r = (1 1), hence, for all r / 0. This implies that (9) has a unique positive definite solution for each r ^ 0.
C. Suppose conditions (v)-(vi) hold and H, satisfy (7). Let H Yl\^]^i- •^ —
^,=]A,. From (7) we have
A^H-\-HA, = Y1{A'HJ-1- HjA,)
= ^ ( A * / / , + /y,A^) =A'Hi-^H,A (Vf = 1 , . . . . ^ ) . (10) Since the one-time system
u'it) ^ Auit) U > 0)
is exponentially stable, there exists a unique positive definite operator P such that
A*P -I- PA ^ - H. ( I I ) We show that P IS the unique simultaneous solution of (8) From (10), we have
A*(A,*P-I-PA, +H,) + {AlP+ PA, +H,)A
^ A'^A'P -1-A*PA, -h A ; ' P A - | - PA,A-\-A'H, -1- H,A
= A* (A*P -I- PA) -F (A'P + PA) A, -+- A*H -f HA,
= A^i-H) -F i-H)A, + A*H + HA, = 0,
i e . , A ; P - l - P A , - l - / f , isasolutionofA*P-t-PA - O.Tberefore. A ; P - | - P A , - | - W , = 0 for all i = 1 it. I.e., P is a simultaneous solution of (8). The uniqueness of the simultaneous solution of (8) follows from the uniqueness ofthe solution of (11).
D. It follows from Lemma 1 that if the multi-time system (4) is asymptotically stable, then so is the one-time system (5), hence cjpiAr) fl iR = 0. If dim£' < oo, then (J,,iAr) = cr(Ar). hence (T(Ar) C (C_ \ iR), so thai (5) and (4) are exponentially stable. D
Note that conditions (7) are compatibitiry condilions for (8). If we introduce operators A, on Li£) by
A , X = : A*X-lXA, U = \. ..k.X e L{£)).
Ihen A, are pairwise commuting if and only if A, are pairwise commuting. The system (7) can be rewritten in the following form
^ Spni
424 Q -P Vu
Therefore, if P is a simultaneous solution of (8), then H, must satisfy the conditions A/ Hj =
\jHt(Vi / y", 1 < (. j < k). i.e., H, must satisfy (7).
Before slating a version of Theorem I for unbounded operators A, let us give a definition of a solution of Lyapunov equations (8) m this case
Definition 3 Let A, be possibly unbounded closed hnear operators on a Hilbert space £ such that Dl (4.) :— nf^, D(A,) is dense in f, and l e t / / , be self-adjoint operators on £ such that D]iA) C DiH,) V/. An operator P G Li£) is called a simultaneous solution of
A ; P - | - P A , =-H, ( / ^ l . . . . , i )
ifforeveryA-.y G D, we have ( P J : , A,y) 4- (PA,j:.y) - {-H,x.y)iVi = 1 *).
In particular, P is a solution of equation
A ' P - I - P A ^ - / / (12) if for every ,i:,y G D(A) we have {PJ:, Ay) -H (PAjr.y) = -{Hx,y).
Note also that, for the case of unbounded A,, the operator A^ is not closed, in general (see, e.g , [4]), but is densely defined and closable, audits closure Ar is the generator ofthe one-parameter semigroup 7"r(/). Below for generators A, of commuting Co-semigroups on
£ ' , w e l e t D | ^ nf^j D(A;) and D2(^) := nf^^, D(A,Aj). Then Di(.A) andD2(4) are dense in £ (see Appendix).
Lemma 2 Suppose TU) is an exponentially stable Co-semigroup on a Hilbert space £, with generaior A, Z c £ is dense and Tit)Z c Z c D(A)(Vf > 0), 5 G Li£) and (Sx, Ay) -H iSAx. y) = 0 (V.v, y G Z). Then S ^ 0.
Proof From {Sx. Ay)-f-{5Aj:, y) =^ 0 (Vjr, y e Z) wehave iSTU)x, ATil)y}-i-iSATit)x, Tii)y) ^ 0 (V:c, y e Z). Therefore
0 = / l{STis)x.ATis)y) + {SATis)x.Tis)y){dt Jl)
-[{STis)x,Tis)y)-\ds ) ds
- {STil)x.Tit)y)-{Sx,y) (V;r, y G Z).
By letting i ^ oo, we get {Sx. y) = 0 (Vj:, y e Z), which implies S = 0. D Theorem 2 Let A := i A ],..., A^) be a k-tuple of (generally unbounded) generators of commuting Co-semigroups T,it) U = 1, ...,k) on a Banach space £. Assume that the mulli-time system (4) generated by A is stable. Then the following conditions (i)-{iv) are equivalent
(i) The mulli-time syslem (4) is exponentially stable;
(ll) For some Ihence for all) r e InliR^) the one-lime system
H ' ( 0 - A r H ( r ) (f > 0 ) (13) is exponentially .liable;
(iii) For some (hence for all) r e InliR''^) and 0 < p < oo I \\Trit)x\\Pdt < CC (VJ: G £).
Jo
^ Springer
I:
Extension of Lyaponov Theorem
If£ is a Hilbert space, then (i)-(iii) are equivalent to the following:
(iv) There exist self-adjoint operators / / , , . . , / / * and P > 0 such thai H ^ Y.';_^^H, is bounded, positive definite, n f ^ , D ( A J C DiH,). H,x G n*^,/)(A*)(Vj: e D]iA)), and
A';HJX + HjA,x ^ A*H,x + H,AjX (Vx € D]iA)), (14) A ; P J : + P A , X - - H , J : i^x e D]iA)); (15) (v) For some I hence for all )r e JnliR'\_) the Lyapunov (9) has a unique solution P > 0
for each H > Q,
Proof The equivalence (i) 4^ (ii) is shown in Lemma 1. the equivalences (ii) -s- (in) -^ (v) are known as the Datko-Pazy theorem (see [10, 25]). To show that (iv) o (v), we note that i f j : e n f ^ i D ( A * ) t b e n j : e D(A*), and ^^f^, Afj: = T x , where A - E L I A , .
Let//, a n d P satisfy (]4)-(15),i.e,Vx G n f ^ | D ( A , ) , Px e nl^^DiA*) and A* Px + PA,x ^ -HjxiVi = l,...,k). Then, as noted abov_e, Px e_D(A'*) and A * P J : + P A J : = T!l^]iA^Px+PA,x) - - E f = i ^ i - ^ = ~Hx,i.e.,A*P-\-PA = -//.ByDatko'stheorem [10], the system (13) with r ^ (1, . , , 1) is exponentially stable, hence the multi-time system (4) is exponentially stable, and so is every one-time system (13) with r e Int(IK^), by Lemma 1 Apply Datko theorem again we get (v). Conversely, if (v) holds, then (13) is exponentially stable, hence (4) is exponentially stable, which in turn implies that the one- time system (13), with r = ( I , . . . , 1) is exponentially stable. Thus, there exist P > 0, H > OsuchlhatA*P -i- PA = - / / , where A = E f = i ' ^ ' - L e t / / ; = AfP -\- PA, Then it is straightforward that H, and P satisfy condition (14)-(I5).
Finally, suppose that (iv) and (v) hold. Let Tit) = T]it)- - -T^il). Then TU) is a Co- semigroup with generator A, where A — J2i=]^i- Since Tit) is exponentially stable, there exists a unique solution P > Oof (12).Define,foreach t e R^, bounded hnear operators Qit) and Rit) by:
-iQit)x.y) = f\H]Tiis)x.T]is)y)ds-l- [ ^{H2T2is)T]Ui)x.T2is)Tiit])y)ds + • • • Jo Jo
^r (HiJi(s)j|(r,)..,rt_i(ft„i)j-.ri(5)ri(/,).,.ri_,(fi-_,)j)<ii
ix.ye D,iA)),
iRit)x, y) = iPTit)x, Tit)y) - {Px, y) ix, y e D,iA)).
It is straightforward lo verify that
-^{QIX»!,y) = {H,Tmx,Tit)y),
— iRil)x,y) = iPAiTit)x,TiOy) + {FTit)x, AiTHiy) ix.y € D,iA)).
Using the identity {Hix, Ay)-l-{H,Ax, y) = {Hx. A,y) -\- {HA,x, y), and - (Hx,y) = {PAx.y)-l-{x,PAy}iWx.y e D i ( ^ ) ) , wehave
-^{Qit)x.Ay}--^iQit)Ax.y)
= {H,Tit)x, AT(t)y) + {H,Tit)Ax. Tit)y)
= {HTit)x, ẠTiDy) + {HTit)AiX, Tit)y)
= -{PTit)x, A A , r ( t ) y ) - {Pr(t)AA, A , r ( t ) y ) - { P r ( t ) A , » , A r ( t ) y ) - ( P r ( t ) A A , x , T(t)y)
= —^{Rit)x, Ay) - —{Rit)AK. y) ix,ye DiiA)).
at, dl.
This implies
{Qit)x. Ay) + {Qit)Ax, y) = {Rit)x. Ay) •+• {Rit)Ax, y) ix, y e D2iA)).
By Lemma 2, we have ^ ( t ) ^ Rit) for all t € R*;., hence -^{Rit)x.y} = -{H,Tit)x,Tit)y)
= {PA,Tit)x,Tit)y} -^ {PTit)x, AiTit)y) {x.yeDiiA)), so dial, in particular - (H,x,y) = {PA,x,y) -F (Px, A,y)Vj;, y e D]iA). l e , P is a simultaneous solution of (15). The uniqueness of P follows from the uniqueness of the
solution of ÁP -\- PÂ-H U
3 Simultaneous Solutions of Sylvester Equations
It is a well-known fact in matrix theory, usually referred to as Sylvester theorem, that if A and B are matrices of dimensions n x /; and m x m. respectively, which do not have a common eigenvalue, then for every n x m matrix C, the matnx equation AX - XB = C has a umque solution. The same is true for bounded linear operators: if A and 6 are bounded linear operators on Banach spaces £ and T, respectively, such thatcr(A) ricr(fi) — 0, then for every C e LiT, £), there exists a unique solution X e LiT, £) of the operator equation AX — XB — C. Moreover, the solution X can be expressed as the following integral
27r/ Jj-ik-A)'^Cik-B)~*dk,
where P is a suitable contour surrounding oiA) and separating from ( T ( P ) (see [11, 27]). If CT(A) is in the open left half plane and criB) is in the open right half plane, then the solution X can be expressed in the following form [16]:
X= rếCe-'"dl.
Jo
Sylvester equations have many applications in vanous areas of analysis, differential equa- tions and control theory (see the survey article [7]). In particular, they can be used to obtain results on asymptotic behavior of solutions of linear differential equations in Banach spaces (see [35, 36]).
In [20,21 ], Ihe Sylvester-Krein-Rosenblum theorem was extended to commuting matri- ces and operators, respectivelỵ Namely, it is proved ihai if ^ = ( A | . . . , At) is a commuting /:-luple of bounded linear operators on £, B = iB]...., Bk) h a commuting S Springer
Extension of Lyaponov Theorem
t-tuple of bounded linear operators on J- such that their Taylor spectra are disjoint, then for every i-tuple (C,, . . . , C*) of bounded linear operators from J" to £, which satisfy the compatibility condition
AiC J -CjBi^AjC,-C,Bj (I <i,j <k). (16) there exists a unique bounded linear operator X: ^ —»• £, which is the simultaneous solution
of the system of Sylvester equations
A,X-XB, =C, (1 <i <k). (17) In this section, we consider simultaneous systems of Sylvester equations (17) for the case
A = iA],..., Ai) and —B = (—fii - B t ) generate commuting semigroups 7", and 5, on Banach spaces £ and J^, respectively, such that the fc-parameter semigroup T(t) = TiU])- • -TkUk) IS exponentially stable, and the i-parameter sermgroup 5(t) — Slit])- • -SkUk) IS bounded. We assume that the operators A, are bounded, and the operators 6, are generally unbounded. Note that in this case, both the notions of solution X of (17) and the compatibility condition (16) should be extended in the following natural way:
Xiscalledasolutionof (I7)ifforevery A- e /?](6) (= n*^jZ>(fi,)), wehave
A,Xx-XB,x=^C,x. (18) C = (C] Ck) is said to satisfy the compatibility condition if for every x e DiB,) n
F>iBj) (I < i, j < k) the following identities hold
AiCjX - CjB,x = AjC.x - C,BjX (1 < i.j < k). (19) Theorem 3 Suppose A — (A i , . . . , A*) are bounded generators ofan exponentially sta- ble k-parameter semigroup Tit), and—B = i—Bi,.... — Bk) are generators of a bounded k-parameter semigroups Sit) ii e M'\.). Then, for every k-luple id.. ..C/,) c LiT.£) which satisfies the compatibility condition (19), there exists a unique simultaneous solution X e HT.£) of the system of Sylvester i]7). Moreover, the solution X can be expressed in the following form
X ^ I TUDCSUDdt. (20) Jo
where C = E f = | C , .
Proof For the sake of simplicity of presentation, first we assume that the generators B, H < I < k) are also bounded. Since Till) is an exponentially stable one-parameter semigroup, with generator A = EE=i'^"'*"diS(;l) is a bounded one-parameter semigroup, with generator —6 — —E,= i ^ " '•^e integral (20} converges and represents a bounded solution of equation
AX-XB^C (21) WeshowUiatXisasimultaneoussoIutionof (17). From (16) wehave
it k
AiC - CB, = JliA,Cj - CjB,) = Y^iAjC, - C^fiy) ^ AC, - C,B. (22)
€ i Sprineer
428 Q -P, Vu Therefore
AiA,X - XB, - C) - (A,X - x e , - C,)B
= A,AX-AXB, -A,XB + XB,B-iAC, - C,B)
^ A,AX - AXB, - A,XB-\-XB,B -iA,C-CB,)
^ A,iAX -XB -C)~ (AX - XB - C)B, = 0, (23) sothat (A, X —XS, — C,) isasolutionof AX —Xfi = 0. From the uniqueness of the solution X of (21) it follows that (A,X - XS, - C,) = 0 (V( ^ 1 k), i.e., X is the simultaneous solution of (17). The uniqueness of the simultaneous solution of (17) follows from the uniqueness of the solution X of (21).
Now assume that S, are not necessarily bounded. As before, the integral (20) converges and represents a bounded solution of (21) (in the extended manner, i.e., A X J : - XBx = CJ: (VX G DiB), where ~B is the closure of S ^ B| H h 6*) To show that X satis- fies (18) for all i. we replace the operator equations in the formulas (22) and (23) by their corresponding vector equations. Namely, we have
A i C j : - C 6 , j : = Yl^^i'^j'^ -CjB,x)
= YliAjC,x - C.BjX) ^ AC,x - CjBx i^x e D](B)).
Therefore
AiA,X-XB, -C,)x -iA,X-XB, -C,)Bx
^ A,AXx - AXB,x - A,XB.\ -i-XB,Bx - iAC, -C,B)x
^ A , A X J : - AXB,x - A,XBx -h XB, Bx - iAjC - CB,)x
= A,iAX-XB~C)x-iAX-XB-C)B,x=^0 (Vx e D]iB)). (24) Let Y, ^ A,X - XBi - C, The identity in (24) imphes that AY,x -Y,Bx = 0(yx e D]iB). Since Si it)x e D]iB), wehave TUl)AY,Sitl)x - TUl)Y, BSUDx = 0 (V/ >
0, VJ: G DiiB)), which implies
—Titl)Y,Sitl)x = Titl)iAY, - Y,B)Sitl)x=0 (VJ: G D]iB)).
so that Till)Y,Sitl)x IS constant, which together with TiOl)Y,SiOl)x = Y,x imphes that Tiil)YiSitl)x = Y,x iVx e D]iB)) Since | | r ( ( I ) > ' , 5 ( ( l ) j : | | - \\TUl)iA,X - XB, - C,)Siil)x\\ < | | A ; r ( r l ) X 5 ( r l ) j || -|- \\Tin)XSitl)B,x\\ + | | r ( r I ) C , j : | | ^ 0 a s ; ^ oo, w e h a v e r , x ^ O ( V j ; e /)] (,B)). This implies that A,Xx - XB,x ^ Cix iVx G D]iB)).
I.e., X is a simultaneous (and unique) solution of (18). D Note that if we define
then K,,L, (1 < / < k) are unbounded linear operators with domains DiK,) = DiL,) = f ffi DiB,), and the compatibility condition (19) is equivalent to ^ ^ ( ^ l , . . , K^) being a commuting k-niple (i.e., K,Kjy = KjK,y Vy e DiK,Kj) n DiKjK,)). If X is a simul- taneous solution of (17), and the operator j2 on £• © ^ is defined by Q = ( V then Q is invertible and a straightforward calculation shows that for all v e DiK ) we have Q^ e DiL,) and QK,v = £„Qy. i.e.. ^ = iK]. ...Kk)aadC= (L,,.'..", L*) are
^ Springer
Extension of Lyaponov Theorem
simultaneously similar. Consequentiy, if iS(t) is a group (so that B = (Bj S() are gen- erators of 5(—t)), then the semigroup with generators AT — (A^i,. . . Aft) is similar to the semigroup T ( t ) ® *5(—t). Thus, from Theorem 3, we obtain the following corollary.
Corollary 1 Suppose A = ( A | , . ., A^) are bounded generators ofan exponentially sta- ble k-parameter semigroup Til), and—B ~ i-B], ... —Bt) are generators of a bounded k-parameter semigroups S it) it e R \ . ) . Then for every k-tuple iCi Ci) C LiT.£) which satisfies the compatibility condition (19), the operators K, and L, defined b\ (25) are simultaneously similar In addition, if Sit) is a group, then the semigroup with generaior K. = iKt, Kt) defined by (25) is similar lo the semigroup Tit) ® 5(—t) Remark i If we define bounded hnear operators Qit) in an analogous manner as in the proof of Theorem 2, i e.,
Qit) = f T]is)C]Siis)ds-\- f~T]Ui)T2is)C2S2is)Siit])ds-l- • Jo Jo
•+• [' Tiit])--Tt.]in.])Tkis)Ci,Stis)S]it])--S,^-tiii.-])ds. (26) Jo
then, as can be seen by a direct (but somewhat unwieldy) symbol calculations with utiliza- tion of the compatibihty condition, that the definition of Q (t) in (26) is independenl of the order of variables t], lk (we get the same in (26) if we switch any two indices i andy").
Thus, if we denote by [0 —>• t] any continuous piecewise linear curve in K+from s — 0 tos = t, which consistsof ft edges /, of thefc-dimensional box [0. t], with /j being parallel to ihe edge f,- (in other words, if ( r | . . ., rj.) e /,. then, for j ^ i. Zj is either 0 or ij.
and T, changes from 0 to f,), and define a piecewise constant function C(t) on [0 -*• t] by C(t) = C, ift G /,,tben
Qit) - / Tit)Cit)Sit)dt.
JlO^t]
where the integral over [0 - * t} is understood as the sum of integrals over /,. Note that the above information about Qit) is not needed in the sequel; we give it here because it shows a stiong resemblance with a sinular integral, given in Section 4, which occurred in extending the well known variation-of-constants formula to the multi-time equations.
Note also that Qit) satisfies the following equations
= k). (27) To prove (27), e.g., for i — I, we take the partial derivative ^^|""' (j: G D\iB)),
use the compatibility condition to replace A|C, — C,B] by AjC] — C]B, (V2 < ( < k).
then use T,is)iA,C] - C I B , ) S | ( S ) J : = ^ r , ( j ) C ] S , ( s ) j : to evaluate the integrals using the Fundamental Theorem of Calculus. All terms will cancel each other, except the pre-lasl one. which is nUi,)- • -Fi Ui )C]Stilk)- • -S] U\)x = Tit)C]Sit)x. If X is the simultaneous solution of Sylvester (18). then from
- [r(t)X5(t) - X{x = Tit)C,Sit)x iVx e T)
© Sprmger
430 Q -P. Vu and 0(0) = r ( 0 ) X S ( 0 ) - X = O, we have
Qit)x = Tit)XSH)x - Xx iVx e J ) .
4 Asymptotic Behavior of Non-homogeneous Multi-time Equations
Consider the ahstract Cauchy problem
\V,i,it) = A,iiit) + fiit) (t=((i tl) e R^., 1 S i < e , , . . ^
\t.im=xe£, <^"
where V, are the partial derivatives with respect to the variables (,, A, are generators of commuting Co-semigroups Tiit) it > 0) on a Banach space £ and / , ( t ) are continuous functions on R^ with values in £.
Recall that for one-time linear differential equation «'(() — AuU) -\- fU), H(0) = J:, where A is the generator of a Cn-semigroup TU) (r>0), if a function w(l) is a classical solution, then it satisfies the following variation-of-constants formula
uit) = Tit)x-\- [ TU-.'!)fis)ds U>0) (29) JO
In general, a function u given by (29) is called a mild solution Below, we extend the vanation-of-conslants fonnula and the notion of mild solution to the mulli-time system (28).
A function M(t) (t G R ^ ) is called a classical solution of (28) if uit) G nf^, DiA,), has partial derivatives and (28) holds for every t G E ^ . It is clear that the sys- tem (28) is over determined, hence for the existence of solutions the functions /,(t) and the operators A, must satisfy appropriate compatibility condilions. For instance, if / , ( t ) (1 < ' < k) have continuous partial derivatives, fit) e n*^,D(A,) and a classical solution uit) exists, then /,(!) must satisfy the following compatibility conditions
iV, - A,)fj = iVj - Aj)f (VI < ;, J < k), (30) where A, is defined by DiA,) = {/ : R*. - * £. fit) e £»(A,){Vt)l, ( A , / ) ( t ) :=
A , / ( t ) ( V / G DiA,)). However, conditions (30) are too restnctive and do not allow to use functional-analytic methods. The appropnate compatibility conditions must be suffi- ciently general in order to allow non-differenliable functions / , and functions with ranges not necessanly in n*^| DiA,), and must be equivalent to (30) if / , are differentiable and fiii) e nj^^j DiA,). Such conditions will be given below, at die same time as we gener- alize the van ation-of-con slants formula and the notion of mild solutions lo the multi-time situation.
Applying (29) to (28) for t = I, we see that the solution uit) satisfies
"(ri, 0 0) = r,it])M(0) -\- \ T]it]-s)fiis,0 0)ds.
Jo
Q Springer
Extension of Lyaponov Theorem
Applying the variation-of-constants formula (29) to (28) for/ = 2 we get r!i
uU\,t2,0.. ..0) = T2it2)uU].0 0)-|- / T2it2-s)f2Ui.s,Q,...,0)ds Jo
- T2U2)\TiU])ui(})-\- j " T]Ui-s)f]is,0....,0)ds]
+ f'T2it2-s)f2il],s.O,....0)ds Jo
- 72(f2)7-|((,)«{0)-|- r T2itl)Tiil] -s)f]is,0,...,0)ds Jo
+ r T2it2-s)f2U].s.0 0)ds.
Jo Continuing this process, we obtain
uit)^Tit)uiO)-\-Yl [' n TjUj)T,it,-s)f,U[ f,-i.s.O. . ,0)ds. (31) Note that formula (31) presents the evolution from the initial state w(0) to the state K(t) following the evolution line from (0, 0 0) to Ui.0 0) along the first-time coor- dinate, then from (f,, 0 0) to ((|, ti, 0, . ,0) along die second-time coordinate, etc , until t = (fl Ik) is reached. Since the system is deterministic, die state H(t) should be independent from the choice of the directions from 0 to t. Hence, if we denote by 5 a per- mutation of { 1 , . . .k], then the following expressions should give the same values for all s-
k y's, *
uit) = Tit)uiO)-\-Yl / n ^^('^j)^i,(''5, -^)fs,{ii'& is,_i.s))ds, (32) ,=i -^^ j=,+i
where we use the convention that /a, itUs ts,_^, s)) is the value of the function fs, at the time variable t — tilg, ts,_^.s) = ( r i , . . . n ) such that r^ — tg for j = 1,...,/—l,rs, = 5 (the floating variables ofthe integral) and r, — Ofoii ^ [S] S,].
It is convenient to represent the formula (32) in the following form
u (t) = Tit)u (0) + ( nt-s)F is)ds. (33) J[0^tl
where [0 - * t] is understood as an increasing (in the order in R^) continuous piecewise hnear curve from 0 tot, which consists o f t segments /, — ({ri T , _ | . 5 . T , + ] , . . , r^) ; 0 < J < /,} (zj, j ^ i, are the fixed time coordinates of the points on /,, Zj = 0 or ij^J ^ ' ) , and Fis) is a function defined on [0 - * t] by F(t) = /,(t) if t e /,. The line integral /||,_,,| Tit — s)Fis)ds is defined as sum of the integrals on the composing segments (cf Remark 3).
Note that although (32) contains ^! formulas, many terms in them are identical and hence cancel each other, so that f(t) - if\it). ..fkit)) e BCC^'dR^,. 5) is compatible if and onlyif/,(] < ( < A) satisfy the following (t^ — fe)/2 relations
T,iti) f'TjUj-s)fjitit,..-i))cls- f'TjUj-s)fjitUj.s))ds Jo Jo
= TjUj) [' TAt,-s)fitUj.s))ds- f T,U,-s)fitU,,s))ds. (34) Jo Jo
© Springer
432 Q -?• Vu where, for simplicity of notations, we use the convention that/j(t(f,, J ) ) is the value of tiie function/j-at the time variable t = tUi.s) — (TI T^-) such that r, = O.Zj = j (the floating variable s of die integral) and rp — tp if p ^ i, j , and fjitUj.s)) ii ^ j) is the value of the function/, al the time variable t = tUj.s) = (TI rt) such that Zj = s and Xp = ipif p / j -
It is obvious that if the system (28) has a classical solution w(t), then f(t) — if] (t) fkit)) is compatible and (32) holds. In the general case, for every com- patible it-tuple of functions f(t) = (/i(t) , fkit)), a function w(t) given by (32) for any (hence all) permutation S is called a mild solution of (30), and (32)-(33) is called the varialion-of-conslants formula for the multi-time systems. From the determinism and time invanant properties it follows that the function M(t) given by (32)-(33) also satisfies this equation in which the initial time 0 is replaced by any s. i.e.,
H(t) ^ Tii-s)uis)-\-Yl / * ' n ^^(fs,)T5,(ri, -s)fs,iti%,...,iB,^^.s))ds
- TU - S)H(S) + f TU- T)Fix)dr (0 < s ^ t). (35) J\s^l\
In addition, m the integral over (s -» t] one can define [s -^ t] as any continuous increasing (in the orderofR+) piecewise linear curve from 0 to t, which consists ofan arbitiary number of segments parallel to the time coordinate axes.
Below, let Z = 6 t / C ( R ^ , £) be the Banach space of uniformly continuous bounded functions / : R+ -* £ and
BU C''iR'l, £) = Z^ = Zx ••• X Z ,
Deflnition 4 (i) A i-tuple of functions f(t) - ( / i ( t ) fkit)) e S [ / C * ( R ^ , £ ) is called compatible (with respect to the ^-parameter semigroup Tit)) ifthe expression (32) is independent of 6. In this case the line integral in (33) is path-independent and the function w(t) given by (33) is called a mild solution ofthe (28).
(ii) A it-tiiple of functions f(t) = if] (t) fkit)) e BUC''iR''.£) is called compati- ble on R* (with respect to the ^-parameter semigroup Tit)) if the expression in (35) is independent of S. In this case, the line integral in (35) is path-independent and a func- tion M(t) (t e R*^) is called a mild solution on R* of (28)ifH(t) satisfies (35) for all s , t ( s -< t).
Let A^^ iM'") be the subspace of all compatible t-tuple of functions f(t) in BUCHR-+.£) (resp. Bf/C*(R*.f)), which satisfy the compatibihty condition on R^
(resp. R*). It IS easy to see that M+, A^* are non-tnvial closed subspaces of B (7 C* (R^. £) andBf/C*(K*,£:),respectively. l n d e e d , l e i A ^ (Xi kt) G C * , j : e n f ^|D(A,Aj), Xi = (A, - k,)x and define / , (t) = exp(t • kjx,. Since ik, — A,)xj = {kj ' - Aj)jc,, the it-tuple f(t) = ( / i ( t ) , . . , / t ( t ) ) is compatible.
Below, let Sis) (s G R ^ ) be the semigroup of translations in 6 ( 7 C ( R ^ , £ ) defined by iSis)f)it) = fit -1- s), and let A, be defined by DiA,) = {f e B ( / C ( l * . , £) : fit) e D(A,)(Vt),A,/(t) e B f / C ( R * , 0 ) , ( A , / ) ( t ) : = A.fiDiWf G DiA,)).
Extension of Lyaponov Theorem
Lemma 3 A^^(At*) is a closed subspace of BUC''iR''+.£) (resp. Bt/C*(R*. £)). which has the following properly: ij fit) = (/i(t) fkU)) e J^\_iA4''). and C is a bounded linear operator on fiCC(R*.,f) (resp. BUCiR''.£)) which commutes tuth Sis) and A,, then [C"-''r](t) : - (C/i(t) Cfiit)) e M% iresp e M'-).
Define operators V, • M'' - * A^* and A, : M'' ^ i^ by
DiV,) = {t=ifi,...,fOeA4'': fj&DiV,) (V; ^ l , . . . ; t ) | . V,f= iV.fi V,fk) Vie DiV,).
A , f - / , ( 0 ) iV(=if],...,fk)eM'').
It is straightforward to verity (hat the compatibility condition is satisfied for Ihe following system of simultaneous Sylvester equations
A,X-XV, - - A , - ((•= 1 it).
Note that -X>, (I < i < A) are generators of the bounded translation semigroups (in fact, groups)5,(-f)onAf*", defined by 5,(-r)f(s) ^ f(.^| 5,-|..T, -t,s,^], . , Sk)is = is],...,sk) G R*, ( e R); indeed, 5,(r) ^ 5(fi)(r > 0), where i ^ (5,i 5/*) iSjj is the Kronecker symbol). Therefore, from Theorem 3 and Corollary 1, we obtain the following result.
Theorem 4 Suppose A = ( A | , . , . , A^) are bounded generators of an exponentially stable multi-lime system (4) on a Banach space £, andf = if] fi) G M'' is such that fit) are almost periodic fiinctions on R* (in the sense of Bohr). Then every (mild) solution uit) of (28) IS asymptotically almost periodic.
Remark 4 Before giving a proof of Theorem 4, let us give the definition of the notion of asymptotically almost penodic function. A function fit) e BiyC(R+,]E) is called asymptotically almost periodic if it can be written in the form
fit) ^ gii) ^-hit).
where git) can be extended to an almost penodic function on E*^ and Vim^k hit) = 0. This IS a natural generalization, in the framework of stability of multi-time systems, of the well known notion of asymptotical iy almost periodic functions on R+ (which was firsl intro- duced by Frechet [14] and plays an imponant role in the qualitative theory of differential equations).
A bounded ft-parameier semigroup T(t) (or deterministic multi-time system (4)) is called asymptotically almost periodic if every trajectory (i e., solution of (4)) uil) = 7'(t)j:,j: G £, is an asymptotically almost periodic function. Equivalently, a bounded semigroup TU) is asymptotically almost periodic if and only if there is a decomposi- tion £• — £'o © £], where £o = {x e £ - lim^k l[T(t).v|[ — 0} and £] — spaii{j: e
£ : 3k ^ ik] Xt) e ( i R ) ^ A,x ^ k.xiVi)]. The subspace £o and ^^i are called the asymptotically stable pari and (he peripheral part of Tit), respectively. Note thai Tit) is called almost penodic semigroup (in the sense of deLeeuw-Ghcksberg [12]) if each orbit £2(J:) = !T(t)j: t G S^) is relatively compact in £. It is well known that if Tit) is an almosi periodic semigroup, then the above decomposition holds (see.
e.g., [18, 22]), so that T(t) is asymptotically almost periodic. However, the following example shows tiiat asymptotic almost penodicity does not imply almost periodicity, in general,
^ Springer
434 Q--P. Vu Example 1 Let Uil) : / / - » • / / be a unitary group with continuous spectrum and V(;) H ->- Hbea nilpotent semigroup on a Hilbert space H. Then, tiie two-parameter semigroup T(t) — Uiti)Vit2) (t = ((1,(2) G K^) IS asymptotically almost periodic, but is not almost periodic.
Spectral conditions for asymptotic almost penodicity of representations of Abehan semigroups (which include the case of multi-parameter semigroups) were obtained by Lyubich-Vu [32, 33] and Batty-Vu [6]. Since the system (4) is asymptotically stable if and only if it is asymptotically almost periodic and has zero peripheral part, the spectral cntena of asymptotic almost periodicity also give Ihe corresponding spectral criteria of asymptotic stability (cf [2, 24. 29, 34]).
Proof of Theorem 4 Consider Alf, a closed subspace of TM* which is spanned by the trans- lations of f, and the restrictions of 5 ( t ) and D, on Alf, for which we use the same notations.
Applying Theorem 3 and Corollary I we see ihat the semigroup 7?.(t) defined on f ffi Mt, with generators
in:) <'-^-«-
is similar to the semigroup T(t) ffi Sit). Since T ( t ) is asymptotically (even exponentially) stable and Sit) (restricted on Mf) is almost penodic, it follows thai T ( t ) ffi Sit), and hence 7^(t), is asymptotically almost periodic It remains to note that ii(t) = 7^(t)ii(0), where
so thatu(t), and hence uit), is asymptotically almosi periodic. D In conclusion, we note that, to our knowledge. Theorem 4 gives a first kind of results
on asymptotic behavior of linear differential equations with multi-time, even for the case of finite dimensional phase space £. Moreover, it presents only an example of applications of our approach of using simultaneous Sylvester equations for the study of asymptotic behavior of multi-time differential equations Further applications of this approach will be given in other publications
Acknowledgments The author wouW like to thank an anonymous referee for the many useful and e remarks
Appendix
In this Appendix, we present aproof of Proposition 1.
Proposition 1 Let T,U) (1 < ' < k) be commuting Co-semigroups, with generators A,.
Then the operators A, are commuting, i.e., A,AjX = AjA,j:(Vj; e DiA, A,) n DiAjA,)), the operator A = A] -1- • • • -i- Ak is closable, densely defined, and its closure 'A is the generaior of tlie semigroup TU) = T]it)-•-TkU). Moreover, the linear manifold D2{A):^ (^'1 j^] DiA, AJ) also is dense.
^ Springer
Extension of Lyaponov Theorem
First, recall die following well-known facts. If TU) (f > 0) is a one-parameter Co semi- group in a Banach space £, then its generator A is a closed, densely defined hnear operator.
Moreover, the following hold:
(i) If X e DiA).thenTit)x e DiA) and ^TU)x = ATU)x = TiDAx- (ti) For allx e £ we have f^Tis)xds e DiA) and Af^Tis)xds ^ r ( r ) j : - x; if, in
addition, x e DiA), then fQTis)Axds = TU)x - x;
(iii) \im,-.0 Jf(',Tis)xds = x iVx e E).
Lemma4 Ifx e DiA,), then Tit)x e DiA,) (Vt e R^.).
Pmof Let.v e /)(A,). Wehave
^lim - [rK/i)r,(f)j: - TjiOx] = TjU) hm^ ^ [T,ih)x -x] = Tjil)A,x.
which implies that r,(r)A- e DiA,) and A, TjU)x = Tjit)A,x.Rence,Tit)x e DiA,)
a n d A , r ( t ) j : =- r ( t } A , j : (Vt G R*J.) D Proof of Proposition 1 By Fubini's Theorem, we have
/ Tit)xdt ^ l'TkiSk)f''Tk-]ist-i)-- f'T]is])xdsvdsk iVxe£).
J\O.T] Jo Jo Jo Sincej-] := /y' T]is])xds] e / ) ( A | ) , we have, by (ii) and Lemma4
X2 : - / T2{S2) / T]is])xds]ds2 = I ' T2is2)x]ds2 e DiA]) D D(A2).
JO Jo Jo Continuing this process, we eventually obtain that
T(t).v^t enf^, D(A,-).
/
J[01
J[liT]
Since, for every .v e E. we have
Tit)xdt ^ A-, (36) '|(i./i|^
where by [0, ft]* we denote the generalized cube [0, h] y. • - • x. [0. /i], we conclude that D]iA) = n*^i 0 ( A , ) is dense in f. Hence, the operator A — A]-\--- -|-A^-is densely defined. Consider now the product semigroup T ( ' l ) = Pi(') • •7i(l). Since 7'(rl)_is a strongly continuous one-parameter semigroup, its generator, denoted (temporanly) by A, is closed and densely defined. Moreover, for every x e D]iA), Titl)x is differentiable and
— r ( r l ) j : = ^T]it)-• Tkit)x = TiU) ..Ttit)iAi-\---- + Ak)x dt dt
- T] (()• • TkiDAx = Tiit)- • -TkiDAx
This implies that A.v — A-V{VJ: e £>i(>^)). hence the operator A is closable and Z = A Next we show tiiatn*j^|£>(A,A^) is dense. Let J: e D]iA). We have, as shown above
v : ^ / Tit)xdte D]iA).
J[OT]
v ^ A , / Tit)xdt= I Tit)A,x
J[O.T] J\0,T]
T h i s i m p h e s that y G n f ^ ^ , / ) ( A ; A , ) . N o w t h e i d e n t i t y ( 3 6 ) i m p l i e s tiiat n f ^ _ , D ( A j A , )
IS d e n s e D]iA), h e n c e a l s o d e n s e in £. D
1. Anastassiou, G A . Goldstein, G.R.. Goldstein, J.A Uniqueness for evoluuon in muliidimensiona! lime Nonlinear Anal 6 4 , 3 3 ^ 1 (2006)
z. Arendt, W.. Batty. C J.K , Tauberian theorems and stability of one-parameter semigroups. Trans. Am, Math-Soc 306,837-852(1988)
3. Arendt, W , Batty, C J.K , Hieber, M , Neubrander, F Vector-Valued Laplace Transforms and Cauchy Problems Monographs m Maihemaiies, 2nd, vol 96 Birkhauser, Basel (2011)
4. Baillon, J . E , Clement, P.ti., Examples of unbounded imjginjiy powers of operators J Funct. Anal 10(1.
4 1 9 ^ 3 4 ( 1 9 9 1 )
5. Barles, G .Tourin, A Commuialion piopciliesof semigroups for first-order Hamilton-Jacobi equaUons and application to mulii-iimeequ.iiions. Indiana Univ Math J 50, 1523-1544(2001) 6. Batty, C J K , Vu, Q P Stability of strongly continuous representations ot Abehan semigroups Math. Z
209.75-88(1992)
7 Bhatia, R , Rosenthal, P., How and why to solve the operator equation AX — X S = )', Bull. Lond, Math Soc 29. 1-21 (1997)
8 Cardm, F , Viterbo, C Commuting Hamiltonians and Hamilton-Jacobi muiti-iime equations, Duke Math. J 144, 235-284 (2008)
9 Choi, M D , Davis. C : The spectral mapping theorem for joint approximate point spectrum Bull Am Math Soc 80,317-321(1974)
10 Datko, V. Extending a theorem of A,M. Lyapunov to Hilben spaces. J Math. Anal. Appl. 32, 61&-616 (1970)
11. Daleckii, Ju.L., Krein. M G Stabihty Theory of Solutions of Differential Equauons in Banach Space Transactions on Mathematical Monographs, vol. 43. American Mathematical Society, Providence, R.I.
(1974)
12. DeLeeuw, K., Glicksberg. 1.. Applications of almost periodic compaciificaiions. Acta Maih 105,63-97 (1961)
13 Engel, K.-J, Nagel, R : One-Parameter Semigroups for Linear Evolution Equations Springer, New York-Beriin-Heidelberg (2000)
14 Fr6chet,M. Les fonctions asymptotiquemenl presque-penodiques Rev. Sci 79,341-354(1941) 15 Esterle, J., Strouse, E., Zouakia, F Siabilite asympiotiques des cenains semigroupes d'operateurs et
i d e a u x p n m a i r e s d e L ' ( M 4 . ) J Oper. Theory 28, 203-227 (1992)
16 Heinz,E Beitrager zur Storunglheone dei Spektralzerlegung Math. Ann 1 2 3 , 4 1 5 ^ 3 8 ( 1 9 5 1 ) 17 Krein, S.G Linear Differential Equations in Banach Space Transactions Mathematical Monographs
American Mathematical Society, Providence, R I (1972) 18, Krengel, U • Ergodic Theorems, Walter de Gruyier, Berlin (1985)
19, Lions, P.-L., Rochet, J - C • Hopf forumula and multitime Hamilton-Jacobi equations P r o c A m Math.
Soc 9 6 , 7 9 - 8 4 ( 1 9 8 6 )
20, Lee, S - G , V u , Q , - P , : Simultaneous solutions of Sylvester equations and idempoient matrices .separating the joint spectrum Linear Algebra Appl. 435, 2097-2109 (2011)
21, Lee, S.-G., Vu, Q.-P.. Simultaneous solutions ot operator Sylvester equations. Stud, Math. 222, 87-96 (2014). Vietnam Institute for Advanced Study in Mathematics, prepnnt VIASM 1334, July 11. 2013 ftp-//fiie-viasmorgAVebmenAnPham-13/Preprim.l334,pdf
22, Lyubich, YI. Introduction to the Theory of Banach Representations of Groups Birkhauser, Berlin (1988)
^_. Lyubich, M.Y. Lyubich, Yl.; Splitting-oft the boundary spectrum for almost periodic operators and representauonsof semigroups. Teor. Funklsii, Funktsional'nyi Anal Ikh Pnlozheniya 45, 69-84(1986) (in Russian)
24 Lyubich. YI.. Vu. Q.P Asymptotic stability of linear differential equations in Banach spaces Slud.
Math 8 8 . 3 7 - 4 2 ( 1 9 8 8 )
© Springer
Extension of Lyaponov Theorem
25 Pazy, A Semigroups of Linear Operators and Apphcations to Partial Differential Equations Spnnger- Veriag, New York (1983)
26 Rochet, J.-C.: The taxation principle and muhi-lime Hamilton-Jacobi equations J. Maih Econ 1 4 , 1 1 3 - 128(1985)
27. Rosenblum, M ; On the operator equation BX - XA = Q- Duke Math. J 23, 263-269 (1956) 28. Ruess, W., Vu, Q.P, Asymplolic ally almosi periodic solutions of evolution equations in Banach spaces
J Differ. Equ. 122, 282-301 (1995)
29. Sklyar, G.M., Shirman, V.A.. On the asymplolic stability of a linear differential equations in a Banach space. Teor. Funktsii, Funktsional. Anal, Ikh Pnlo.;hcnia 37, 127-132 (1982). la Russian 30. Slodkowski,Z.,Zelasko,W.Onjoinl spectra of commuting families of operators.Stud Madi. 50, 127-
148(1974)
31 Udnjte, C : Multitime controllability, observabihty and the bang-bang principle J. Optim. Theory Appl 139,141-157(2008)
32 Vu, Q.P. Lyubich, YL: A spectral cnierion for asymptotic almost penodicity of uniformly conlinu- ous representation of Abelian semigroups Teor. Funklsii. Funkisional. Anal Ikh Pnlozhenia 50, 3 8 ^ 3 (1988) a n Russian). English trans, J. Soviei Math. 5 1 . 1263-1266(1990)
33. Vu, Q P., Lyubich, Y L : A spectral critenon for almost periodicity of one-parameter semigroups. Teor Funktsi', Funktsional. Anal, i Pnlozhenia 47, 36-41 (1987) (In Russian}. English trans in J. Soviet Math 48.644-647(1990)
34 Vu. Q.P Theorems of Katznelson-Tzafriri type for semigroups of operators J Funct Anal 103,74-84 (1992)
35, Vu, Q.P' The Operator equation AX — XB = C with unbounded operators A a n d S and related abstract Cauchy problems. Math, Z. 208, 567-588 (1991)
36. Vu. Q.P. Schnler, E ' The operator equ.ilion AX — XB = C, admissibility, and asymptotic behaviour of differential equations J Differ Equ 145,394-419(1998)
® Sprineer
