Frequency domain stability of nonlinear feedback systems with unbounded input operator Francesca Bucciy Abstract
We give a criterion for equiasymptotic stability in the whole of a class of holomorphic semigroup systems with nonlinear boundary feedback, under a frequency domain condition. The Lyapunov function for the closed loop is obtained via an arbitrary solution to the dissipation inequality arising in quadratic regulator problems with non-de nite cost functional.
Key words. absolute stability, analytic semigroup, boundary control, feedback system, frequency domain condition, Lyapunov function, quadratic regulator, unbounded operator 1991 Mathematics Subject Classi cation. 93C20, 93D10, 49J20
1 Introduction The absolute stability problem for abstract systems in Hilbert spaces has been investigated by Wexler in [17, 18]. In these papers, the following system is considered x0 = A x + b u (1) on a Hilbert space X , with nonlinear gain given in terms of the output by
u = '(); 0 = hc; xi ? u: (2) The coecients b and c are elements of X , is a given non-negative constant. As a minimal assumption, A : D(A) X ! X is the generator of a C0 -semigroup This research was supported by the Italian Ministero dell'Universit a e della Ricerca Scienti ca e Tecnologica within the program of GNAFA{CNR. y Universit a di Firenze, Dipartimento di Matematica Applicata, Via S. Marta 3, 50139 Firenze, Italy. (
[email protected])
1
on X . ' : IR ! IR belongs to the class of locally Lipschitz continuous functions such that r'(r) > 0 for r 6= 0; (3) which means that the graph of ' lies in an in nite sector contained in the rst and third quadrants. We shall refer to this class of sectorial non-linearities as to fSNLg. Note that (3) implies '(0) = 0, so that the system (1)-(2) admits the trivial solution. It is well known that a system of this form is termed absolutely stable if the zero solution is globally asymptotically stable, no matter how ' is choosed in fSNLg. The absolute stability problem of nonlinear feedback systems has been investigated by several authors, since the earlier paper by Lur'e and Postnikov [10]. Two main methods for solving this problem, both leading to results known as `frequency domain criteria', have beeen developed in the rst sixties. In the original paper of Popov [15] the desired stability properties are guaranteed if a certain function of complex variable { which does not depend on ', but only on the coecients of the system, { is positive real. We refer to the monographs [3, Ch. 3] and [16, Ch. 8] for a nice illustration of Popov's results and to [11] (and the reference contained therein) for an overview of the main contributions in a nite dimensional context, until the rst seventies. Stability of delay systems is treated in [6]. More recently, the `integral equation approach' has been successfully adapted in [17] to prove absolute stability of di erentiable semigroup systems of the form (1)-(2). Alternatively, another approach can be used, which relies on the application of the direct Lyapunov method, and takes advantage of Yakubovich's frequency theorems [19, 20, 21], or subsequent extensions to the Hilbert space setting ([8], [18], [9]). In this case a Lyapunov function for the system is constructed by means of an arbitrary solution to certain algebraic operator equations or inequations arising in quadratic regulator problems with non-de nite cost. It is worth recalling that the problem of stability of feedback systems motivated and pushed further the study of optimal control problems with general quadratic cost functional. In this paper we wish to examine a system of the form (1), but in the more challenging situation when the input operator is unbounded. Namely, we consider the following abstract model for holomorphic semigroup systems x0 = A(x ? du); x(0) = x0 2 X; (4) with nonlinear feedback u as in (2). Throughout the paper is a given positive constant, A and d satisfy the standard `analyticity assumption' (condition (H1) in x2). Morever, we restrict ourselves to the class of ' in fSNLg which satisfy a growth condition (see (26) below). Here, following the latter approach recalled before, we provide a criterion for equiasymptotic stability in the whole of the zero solution of the system (4)-(2), 2
which extends an analogous result for distributed systems by Wexler [18]. The paper is organized as follows. Existence, uniqueness and regularity of the solutions to the initial value problem derived from (4)-(2) are the subject of x2. In x3 we give an extension of a Yakubovich's frequency theorem to holomorphic semigroup systems. This result will play a crucial role in the achievement of stability properties of the system under consideration, but it is also of independent interest as it gives a criterion for solving quadratic regulator problems with general cost for boundary control systems of parabolic type. The main result of this paper, Theorem 4.1, is presented in x4. Preliminary results, detailing the steps of stability in the whole and of global attractivity of the trivial solution of the system (6), are provided in x4 as well, along with all related proofs. The frequency domain condition, namely assumption (27) of Theorem 4.1, is checked in a speci c case. A model of physical interest is presented below, in order to explain the signi cance of extending known frequency domain stability criteria to systems with unbounded input operator.
1.1 An example
Consider the following system with boundary feedback:
xt (t; ) = x (t; ) 0 < < 1; x(0; ) = x0 ( ); x(t; 0) = 0; x (t; 1) = '((t)); (5) R 1 d dt (t) = 0 x(t; ) d ? '( (t)); (0) = 0 : Here x(t; ) represents the temperature at time t and position along a rod of lenght 1. The problem consists of the heat equation in x with feedback acting
at one end of the rod through an actuator subject to saturation phenomena. To put (5) into the desired abstract framework, set X = L2 (0; 1) and introduce the operator A : D(A) X ! X de ned by
A f = f ;
D(A) = ff 2 H 2 (0; 1) : f (0) = 0; fx(1) = 0g:
Then, A is selfadjoint and negative, hence it is the in nitesimal generator of an analytic semigroup etA on X . Moreover A is boundedly invertible in X , which implies that etA is exponentially stable, and the free system is stable. The fractional powers (?A) are well de ned: in particular, by interpolation,
D((?A) ) = f f 2 H 2 (0; 1) : f (0) = 0 g for 1=4 < < 3=4: Let now d : (0; 1) ! IR be the solution to the boundary value problem x ( ) = 0 for 0 1, x(0) = 0; x (1) = 1; 3
and let c( ) 1. It is readily veri ed that the boundary control system (5) reduces to the abstract set-up (4)-(2) (with = 1). Finally, since d 2 D((?A)3=4? ), the basic assumptions (H1) ? (H2) of x2 hold true. The main theoretical result presented in this paper { Theorem 4.1 { will be applied to this model later in x4.
1.2 Notation
We set IR+ := fx 2 IR : x 0g and IR+ := fx 2 IR : x > 0g. The simbols h; i, j j and k k will denote inner products, norms and operator norms in abstract spaces, respectively. Unless speci ed, these spaces will be clear from the context. If A : D(A) X ! X is a linear closed operator and belongs to the resolvent set (A) of A, we will use R(; A) or ( ? A)?1 to denote the resolvent operator. Let X be a Banach space, A the generator of a C0 -semigroup on X , and let T > 0. The concepts of classical, strict, mild solutions of the initial value problem ( 0 x (t) = Ax(t) + f (t); t 2 [0; T ] x(0) = x0 1 for f 2 L (0; T ; X ) and x0 2 X are common in the literature and can be found, for instance, in [1].
2 Existence and regularity of solutions
Fix a locally Lipschitz continuous nonlinearity ' : IR ! IR and consider the initial value problem on X IR which is obtained plugging u = '() into (4) and taking into account (2): ( 0 x = A(x ? d'())
0 = hc; xi ? '()
x(0) = x0 2 X (0) = 0 2 IR:
(6)
In this section we study existence and regularity of solutions (x(t); (t)) to the system (6). The results of this section will be used in the study of stability properties of the zero solution of (6). Throughout the paper we assume that 8 (i) > > >
> > :
(H2)
A : D(A) X ! X is the generator of an analytic semigroup etA on X , which is exponentially stable, (ii) d 2 D((?A) ) for some 2 (0; 1);
c 2 X. 4
Recall that exponential stability of etA means that there exist M 1 and ! > 0 such that ketA k Me?!t for all t 0.
By formal integration of the initial value problem (6) on a certain interval (0; t), one easily gets the following system of integral equations: Zt x(t) = etA x0 ? A e(t?s)A d'((s)) ds; 0
(7)
(t) = 0 + hc; etA A?1 x0 ? A?1 x0 i +
Zt 0
[hc; (I ? e(t?s)A )di ? ]'((s)) ds:
(8)
It is natural to introduce the following De nition 2.1 A continuous solution (x; ) of the system (7)-(8) will be called a mild solution to the initial value problem (6). We concentrate rst on (8), which is a nonlinear evolution equation in . Since it is assumed that ' : IR ! IR satis es a local Lipschitz condition, we will use that there exists a strictly increasing, continuous function l : IR+ ! IR+ , such that j'(1 ) ? '(2 )j l(r)j1 ? 2 j (9) holds for all 1 , 2 with j1 j r, j2 j r. In order to show existence and regularity results of the solutions to system (6), the sector condition (3) is unnecessary. Our rst result, providing existence and uniqueness of maximal solutions to the equation (8), can be easily obtained arguing as in [14, Ch. 6, Theorem 1.4]. We omit the proof. Theorem 2.1 (cf. [14]) Assume that (H1) ? (H2) are satis ed and let ' : IR ! IR be a locally Lipschitz continuous function. Then for every r > 0 there exists Tmax +1 such that the integral equation (8) has a unique continuous solution on [0; Tmax[, for any initial data (x0 ; 0 ) 2 X IR with j(x0 ; 0 )j r. Moreover if Tmax < +1 then lim j(t)j = 1:
t"Tmax
To proceed, we need the following Lemma 2.2 Assume that (H1) ? (H2) are satis ed and let ' : IR ! IR be a locally Lipschitz continuous function. Fix r > 0 and let 2 C ([0; Tmax[; IR) be 5
the solution to (8) with initial data (x0 ; 0 ) such that j(x0 ; 0 )j r. Denote by v the convolution
v(t) := (eA d' )(t) =
Zt 0
e(t?s)A d'((s)) ds:
(10)
Then v 2 C ([0; Tmax[; D(A)) \ C 1 ([0; Tmax[; X ) is the strict solution of (
v0 (t) = Av(t) + d'((t)); t > 0 (11) v(0) = 0 Proof. It is sucient to show that v(t) 2 D(A) for t 2 [0; Tmax[ and that Av(t) is continuous for t 2 [0; Tmax[: this implies v 2 C 1 ([0; Tmax[; X ), and that (11)
holds. Fix T 2]0; Tmax[. Set f (t) = d'((t)) and note that j(?A) f ()j is bounded on [0; T ]. Hence we can argue as in Theorem 3.6 in [14, Ch. 4] to show the rst issue and obtain the estimate Zt
jAv(t)j k (t ? s) ?1 ds: 0
Therefore Av 2 L1 (0; T ; X ) which implies Av 2 C ([0; T ]; X ) by a standard density argument (see, for instance, [2, p. 196]). Since T is arbitrary, the proof is complete. As a consequence of the previous results we can prove Theorem 2.3 Let assumptions (H1) ? (H2) be satis ed and let ' : IR ! IR be a locally Lipschitz continuous function. Then for every r > 0 there exist Tmax +1 and a unique mild solution (x(); ()) to the system (6) on [0; Tmax[, for any initial data (x0 ; 0 ) 2 X IR, with j(x0 ; 0 )j r. Proof. Fix r > 0, take (x0 ; 0 ) 2 X IR with j(x0 ; 0 )j r and let Tmax and 2 C ([0; Tmax[; IR) as from Theorem 2.1. Existence of x(t) as de ned by the variation of constants formula (7) is guaranteed by assumption (H1). In order to show that in fact x 2 C ([0; Tmax[; X ), it is sucient to observe that x(t) = etAx0 ? Av(t) on [0; Tmax[, with v de ned by (10). Conclusion follows from Lemma 2.2, by using Av 2 C ([0; Tmax[; X ). Further investigation of the regularity properties of solutions (x(); ()) to the system (6) is motivated later in x4. First of all, it is readily veri ed that any continuous solution to the integral equation (8) on [0; Tmax[ is in fact continuously di erentiable on [0; Tmax[: Theorem 2.4 Assume that (H1) ? (H2) are satis ed and let ' : IR ! IR be a locally Lipschitz continuous function. Fix r > 0 and let 2 C ([0; Tmax[; IR) be the solution to (8) produced in Theorem 2.1. Then 2 C 1 ([0; Tmax[; IR) and 0 (t) = hc; x(t)i ? '((t)); 0 t < Tmax: (12) 6
Proof. Note that (t) = u(t) ? hc; v(t)i, where u is given by
u(t) := 0 + hc; etA A?1 x0 ? A?1 x0 i + (hc; di ? )
Zt
'((s)) ds; (13) which is apparently a continuously di erentiable function for any t 0, and v 0
is the convolution (10), respectively. Apply Lemma 2.2 and compute 0 (t) = u0 (t) ? hc; v0 (t)i = hc; etA x0 i + (hc; di ? )'((t)) ? hc; Av(t)i ? hc; di'((t)) = hc; etA x0 ? Av(t)i ? '((t)); which is nothing but (12).
Remark 2.5 In contrast, x() is not a C 1 map in general, when ' is merely locally Lipschitz continuous, even if x0 2 D(A). However, it will suce to show the following
Theorem 2.6 Assume that (H1) ? (H2) are satis ed and let ' : IR ! IR be a locally Lipschitz continuous function. Fix r > 0 and let (x; ) be the solution to the system (6) with (x0 ; 0 ) such that j(x0 ; 0 )j r, de ned on [0; Tmax[. Then x(t) = (t) + d'((t)); t 2 [0; Tmax[; (14)
where is a classical solution in L1 loc (0; Tmax ; X ) of the problem 0 (t) = A (t) ? d'0 ((t))0 (t) a.e. in ]0; Tmax[ (15) (0) = x0 ? d'(0 ); (with '0 2 L1 loc (IR)). Moreover x is di erentiable almost everywhere on ]0; Tmax [, with x0 (t) = A (t) a.e. on ]0; Tmax[. Proof. Fix r > 0, take (x0 ; 0 ) with j(x0 ; 0 )j r and let (x; ) be the solution to the system (7)-(8) on [0; Tmax[. Since ' is a locally Lipschitz continuous function, '() is di erentiable almost everywhere on IR, with '0 2 L1 loc (IR). Note that ' is di erentiable almost everywhere as well, due to Theorem 2.4. If d 2 D(A), then, by using integration by parts, we have
x(t) = etAx0 ? = etAx0 +
Zt
0 Zt 0
Ae(t?s)A d'((s)) ds d (t?s)Adg'((s)) ds ds fe
= etAx0 + d'((t)) ? etA d'(0 ) Zt ? e(t?s)A d'0 ((s))0 (s) ds: 0
7
De ne
(t) = etA (x0 ? d'(0 )) ?
Zt 0
e(t?s)A d'0 ((s))0 (s) ds;
(16)
and observe that x(t) = (t) + d'((t)). Consequently (14) holds true { for d 2 D(A), hence for d 2 D((?A) ) by a density argument { with given by (16), which implies the rst statement of the Theorem. To conclude the proof, note that x(t) turns out to be the sum of a continuously di erentiable function on ]0; Tmax[ and a locally Lipschitz function on ]0; Tmax[, and compute the derivatives.
3 A frequency theorem In this section we state a version of the sucient part of the Kalman-Yakubovich lemma. Speci cally, we prove existence of solutions to the dissipation inequality associated with general quadratic regulator problems for holomorphic semigroup systems, under a frequency domain condition in strong form. We refer to [12, 13] for an historical survey on the Kalman-Yakubovich lemma and the latest extensions to boundary control systems of hyperbolic or parabolic type. Denote by X and U two complex Hilbert spaces and consider a continuous quadratic form on X U
F (x; u) = hQx; xi + 2RehSu; xi + hRu; ui; where h; i denotes all inner products, Q, S , R are linear bounded operators in the proper spaces and Q and R are selfadjoint operators. No de nitess properties are required. Consider the Cauchy problem x0 = A(x ? Du); x(0) = x0 2 X;
(17)
under the `analytic assumption' ([7]) 8 (i) > > < > > :
A : D(A) X ! X is the generator of an analytic semigroup etA on X ; (ii) D 2 L(U; D((?A) )) for some 2 (0; 1).
(18)
It is well known that in this case (17) admits a unique mild solution x given by
x(t) = etA x0 ? A
Zt 0
e(t?s)A Du(s) ds;
for any u 2 L2 (0; 1; U ). In view of our applications, it is not restrictive to assume that the semigroup etA is exponentially stable on X . 8
(19)
(20)
In this case x(t) = x(t; x0 ; u) belongs to L2(0; 1; X ) (see, for instance, [7]), hence the quadratic functional
J (x0 ; u) =
Z1 0
F (x(s); u(s)) ds
is well de ned for any u 2 L2(0; 1; U ). Theorem 3.1 Let A and D satisfy (18) and (20). Assume moreover that there exists > 0 such that F (?A (i! ? A)?1 Du; u) juj2 for all ! 2 IR, u 2 U: (21) Then there exists a linear bounded operator P = P on X which solves the linear operator inequality 2RehP (x + Du); Axi + F (x + Du; u) 0; (x; u) 2 D(A) U: (22) In this case, the maximal solution P^ to (22) is such that ^ 0i = hx0 ; Px inf J (x0 ; u); x0 2 X: 2 u2L (0;1;U )
Proof. Let x0 2 X , u 2 L2 (0; 1; U ) and consider the solution x to (17) in mild form as given by (19). The exponential stability of the semigroup etA implies that 0 belongs to the resolvent set (A), hence A?1 is a bounded operator and we can write Zt ? 1 ? 1 tA A x(t) = A e x0 ? e(t?s)A Du(s) ds: (23) 0
Since x 2 L2(0; 1; X ), we can apply the Fourier transform to both sides of (23), obtaining A?1 x^(i!) = A?1 R(i!; A)x0 ? R(i!; A)Du^(i!); hence x^(i!) = R(i!; A)x0 ? AR(i!; A)Du^(i!): (24) From Parceval inequality and (24) it follows 2J (x0 ; u) =
Z +1
?1
F (R(i!; A)x0 ? AR(i!; A)Du^(i!); u^(i!)) d!;
which can be rewritten, if we set K (!) = ?AR(i!; A)D, 2J (x0 ; u) =
Z +1
?1
hQR(i!; A)x0 ; R(i!; A)x0 ; i d!
+ 2 Re +
Z +1
Z +1
?1
?1
h(QK (!) + S )^u(i!); R(i!; A)x0 i d!
F (K (!)^u(i!); u^(i!))d!
= I1 + I2 + I3 : 9
Thus the frequency domain condition (21) implies that
F (K (!)^u(i!); u^(i!)) ju^(i!)j2 ;
! 2 IR
which in turn yields { again via the Parceval inequality
I3
Z +1
?1
ju^(i!)j2 d! = 2
Z1 0
ju(s)j2 ds = 2 juj2L2 :
On the other hand, it is readily veri ed that there exist positive constants c1 and c2 , independent of u and x0 , such that jI1 j c1 jx0 j2 , jI2 j c2 jx0 j jujL2 . Consequently we have
J (x0 ; u) juj2L2 ? 2c2 jx0 jjujL2 ? 2c1 jx0 j2 ;
hence there exist a constant c, independent of x0 , such that
J (x0 ; u) cjx0 j2
for any u 2 L2 (0; 1; U ):
Conclusion follows by applying a recent result by Pandol [13], which we recall explicitly for the sake of completeness: Theorem 3.2 ([13]) There exist a number 2 IR such that for each x0 2 X we have
inf J (x0 ; u); u 2 L2 (0; 1; U ) such that x(; x0 ; u) 2 L2 (0; 1; X ) jx0 j2 if and only if there exists a linear bounded operator P = P on X such that the following holds true for each x 2 D(A), u 2 U : 2RehP (x + Du); Axi + F (x + Du; ui 0: In this case, the maximal solution P^ to (25) is such that
(25)
^ 0i = hx0 ; Px inf J (x ; u); x0 2 X: u2L2 (0;1;U ) 0
4 Absolute stability We start with recalling the concepts of stability that we want to estabilish for the system under consideration, cf. [5]. In the rst two de nitions it is assumed that a sectorial nonlinearity ' is given. De nition 4.1 The zero solution to (6) is stable in the whole if for any (x0 ; 0 ) 2 X IR there exists a unique solution (x; ) to (6) de ned for every t 0; 10
there exists a strictly increasing, continuous function : IR+ ! IR+ , with (0) = 0, such that for any solution (x; ) to (6) with initial data (x0 ; 0 ) and r > 0 j(x0 ; 0 )j r implies j(x(t); (t))j (r); 8t 0: The function is usually called comparison function. De nition 4.2 The zero solution to (6) is equiasymptotically stable in the whole if it is stable in the whole and if, for any bounded set B 2 X IR, each solution to (6) with initial data in B tends to zero as t ! +1, uniformly with respect to (x0 ; 0 ) 2 B. As pointed out in x1, the goal of the present work is to prove stability properties of the zero solution of system (6) no matter how we choose ' in a speci ed class of sectorial nonlinearities. De nition 4.3 We say that the system (6) is absolutely stable if for any ' 2 fSNLg such that Z lim '(s) ds = +1; (26) jj!+1 0
the zero solution is stable in the whole and globally attractive. We shall refer to the class of ' 2 fSNLg which satisfy (26) as to fSSNLg (special sectorial non-linearities).
Our main result is the following Theorem 4.1 Assume that (H1) ? (H2) are satis ed. If, in addition,
9 > 0 such that + Rehc; AR(i!; A)di 8! 2 IR
(27)
holds, then the zero solution of the system (6) is equiasymptotically stable in the whole for every ' 2 fSSNLg. To illustrate the applicability of Theorem 4.1, we return to the PDE model (5) described in Section 1.1. In order to verify condition (27), it is necessary to compute the transfer function
T (z ) = hc; ?A(z ? A)?1 di on the imaginary axis. After solving a boundary value problem for a second order ODE, one easily gets
? 1; T (i!) = cosh 2 cosh
( i p e4 ! = ?i p e 4 ?!
11
! > 0; ! < 0;
which in turn yields, by elementary computations, the explicit representation p
p
j) ; k = p2=2; ! 6= 0: sin(k pj!j) sinh(k j!p Re T (i!) = 2 j!j(cos2 (k j!j) + sinh (k j!j)) Numerical computations show that ?10?3 < Re T (i!) 21 for all ! in IR. Consequently, since = 1 > 12 , then ? Re T (i!) for some > 0. Therefore (27) is satis ed and the system (5) is equiasymptotically stable in the whole (for all sectorial nonlinearities in the class fSSNLg). In the proof of Theorem 4.1, we will need the following lemmas, where the Lyapunov function candidate for the system (6) is introduced, and existence of a comparison function for the corresponding solutions (x; ) is provided. Lemma 4.2 Let (H1) ? (H2) be satis ed. If (27) holds, then there exists p > 0 such that for every ' 2 fSNLg there exists a continuous, strictly increasing function l : IR+ ! IR+ , such that, given r > 0, Z (t) 0
'(s) ds r2 (p + l(r)=2); t 2 [0; Tmax[;
(28)
where () is the solution to the equation (8) corresponding to initial data (x0 ; 0 ), with j(x0 ; 0 )j r. The constant p depends only on the coecients of the system (6) and on . Proof. By assumption (27) there exists a positive constant 0 such that
( ? 0 ) + Rehc; AR(i!; A)di 2
(29)
F (; u) := ?Rehc; iu + ( ? 0 )juj2
(30)
(take, e.g., 0 = =2). Next, introduce the quadratic form on X IR de ned by and compute
F (?AR(i!; A) du; u) = Rehc; AR(i!; A)diu2 + ( ? 0 )u2 : From (29) it follows that
F (?AR(i!; A) du; u) 2 juj2 ;
hence Theorem 3.1 applies, yielding the existence of a linear bounded operator P = P on X such that 2RehP (x + du); Axi + F (x + du; u) 0; (x; u) 2 D(A) IR; 12
(31)
holds true, with F as in (30). It is readily veri ed that P is a negative de nite operator. Indeed, rewrite (31) with u = 0, namely 2RehPx; Axi 0; x 2 D(A); and use exponential stability of eAt , to get P 0 (cf. [4]). De ne on X IR the Lyapunov function candidate in Lur'e-Postnikov form
W (x; ) := ?hPx; xi +
Z 0
'(s) ds;
(32)
where P is a solution to (31) and ' 2 fSNLg is given. Fix r > 0, and let (x; ) the solution to the system (6) with initial data (x0 ; 0 ) which satisfy j(x0 ; 0 )j r, de ned on [0; Tmax[. We will show that W , computed along solutions of the system (6), is a monotone decreasing function. Note preliminarly that t 7! W (x(t); (t)) is a continuous function on [0; Tmax[. In fact from Theorem 2.4 and Theorem 2.6 it follows that this map is di erentiable almost everywhere on ]0; Tmax[, with
d W (x(t); (t)) = ?2 RehP ( (t) + d'((t)); A (t)i dt + '((t))[hc; x(t)i ? '((t))] = ?2 RehP ( (t) + d'((t)); A (t)i ? F ( (t) + d'((t)); '((t))) ? 0 j'((t))j2 for almost every t 2]0; Tmax[. In the above computation has the meaning of
formula (14); the last equality is obtained, as usual, adding and subtracting equal terms. By using (31) we nally obtain
d 0 2 dt W (x(t); (t)) ? j'((t))j 0;
(33)
and W is a monotone non-increasing function along trajectories of (6). Therefore W (x(t); (t)) W (x0 ; 0 ) on [0; Tmax[ and this implies, estimating both the left and right hand sides, Z (t) 0
'(s) ds ?hPx(t); x(t)i +
?hPx0 ; x0 i + kP kr2 + l(r)
Z (t)
Z 0 0
0
Z j0 j 0
'(s) ds
'(s) ds sds
r2 (kP k + l(r)=2); with l(r) as de ned by (9). Hence (28) holds with p = kP k. 13
(34)
Lemma 4.3 Let (H1) ? (H2) be satis ed. If (27) holds, then for every ' 2 fSSNLg there exist two continuous, strictly increasing functions , : IR+ ! IR+ , (0) = (0) = 0, such that, given r > 0, the solution (x(t); (t)) to system (6) corresponding to ' satisfy, on the maximal existence interval [0; Tmax[,
j(x0 ; 0 )j r implies jx(t)j (r); j(t)j (r): (35) The functions and do not depend on Tmax. Proof. Let ' 2 fSSNLg be given. Fix r > 0, take j(x0 ; 0 )j r, and let (x(t); (t)) be the maximal solution to the system (6), de ned on [0; Tmax[. We
argue as in [18, Theorem R 2, step II]. From continuity and Rstrict monotonicity of the functions 7! 0 '(s) ds from IR? to IR+ and 7! 0 '(s) ds from IR+ to IR+ it follows that they are invertible, with inverse functions ? : IR+ ! IR? and + : IR+ ! IR+ , respectively. Set = maxf? ?; + g, and observe that : IR+ ! IR+ is continuous, strictly increasing, (0) = 0 and by (26) () ! +1 as ! +1. Hence (28) implies
j(t)j (r2 (p + l(r)=2)) =: (r); t 2 [0; Tmax[: (36) An analogous estimate for x follows from (36), using again analiticity and uniform stability of the semigroup etA. Indeed, jx(t)j Me?!tjx0 j +
Zt 0
k(?A)1? esA kj(?A) djj'((t ? s))j ds
Me?!tjx0 j + j(?A) djl( (r)) (r)
Mr + M j(?A) djl( (r)) (r)
Z t 0
k(?A)1? esA k ds
Z t ?!s e ds) s 0 1?
= M (r + k l( (r)) (r)) =: (r):
We prove separately the core of Theorem 4.1. Theorem 4.4 Let (H1) ? (H2) be satis ed. If (27) holds, then the system (6) is absolutely stable. Proof. Fix ' 2 fSSNLg. Let r > 0, j(x0 ; 0 )j r, and let (x(t); (t)) be the solution to (6), de ned on the maximal interval [0; Tmax[. We invoke Lemma 4.3 and set p (r) = ((r))2 + ( (r))2 : Then j(x(t); (t))j (r); for t 2 [0; Tmax[, (37) 14
as long as initial data satisfy j(x0 ; 0 )j r. Note that : IR+ ! IR+ is a continous, strictly increasing function, with (0) = 0. Thus we immediately obtain Tmax = +1, since otherwise from Theorem 2.1 it follows lim j(t)j = 1; t"T max
which contradicts (37). In conclusion, we have obtained global existence of solutions of the system (6). Therefore (37) holds true for all t 0 and system (6) is stable in the whole for any ' 2 fSSNLg. It remains to be shown that all solutions (x(t); (t)) tend to zero as t ! +1. By integrating (33) between 0 and t one obtains
W (x(t); (t)) W (x0 ; 0 ) ? 0
Zt 0
j'((s))j2 ds; t > 0:
(38)
Thus, taking into account that W (x(t); (t)) is non-negative and estimating jW (x0 ; 0 )j as in (34), we have
0 In particular
Zt 0
j'((s))j2 ds r2 (p + l(r)=2); t > 0:
(39)
Z1
j'((s))j2 ds < +1: Note now that g(t) = j'((t))j2 is a uniformly continuous function on IR+ . 0
Indeed, it is readily veri ed that jg(t) ? g(s)j j'((t)) ? '((s))j (j'((t))j + j'((s))j) 2[l((r))]2 (r)j(t) ? (s)j 2[l((r))]2 [(r)]2 (jcj + l((r)))jt ? sj; where we have combined stability in the whole, the local Lipschitz property of ', and boundedness of j0 ()j. Thus we can invoke Barbalat lemma to deduce that '((t)) = 0; t!lim +1
hence (t) tends to zero as t ! +1 as well, due to the sector property (for a proof of Barbalat lemma see, for instance, [3, p. 89]). To conclude the proof it remains to show that x(t) tends to zero, as t ! +1, as well. Given > 0, there exists T 0 such that j(t)j for t T . We use again analiticity and exponential stability of the semigroup and nally get jx(t)j Me?!t jx0 j + k1 l((r))(r)e?!(t?T ) + k2 l() for all t T , where k1 , k2 are constants which depend only on the coecients of the system. The proof is complete. 15
We conclude the proof of our major result. Proof of Theorem 4.1: Global asymptotic stability of the zero solution of the system (6) has been estabilished in Theorem 4.4. In order to show that attractivity of the trivial solution of the system (6) is in fact uniform with respect to initial data, we follow the approach used by Wexler in [17, Theorem 1, step II] to deduce the same property for a class of distributed systems. Let ' 2 fSSNLg be given. Assume, by contradiction, that there exists a bounded set B contained in X IR, a positive constant , a sequence Tn ! +1 and a sequence f(x0n ; 0n )g 2 B such that the solution (xn ; n ) to (6) corresponding to initial data (x0n ; 0n ) satis es
j(xn (Tn ); n (Tn ))j > for all n 2 IN.
(40)
Since f(x0n ; 0n )g belongs to a bounded set, from stability in the whole it follows that the sequences fxn g and fn g are uniformly bounded. Moreover, since fn0 g is also uniformly bounded, we can apply the Ascoli-Arzela theorem to obtain the existence of a subsequence fnk g which converges to a continuous function ^ uniformly on compact intervals. We denote the subsequence nk still by n . Since the sequence f(x0n ; 0n )g is uniformly bounded in X IR, then it converges weakly to a suitable pair (^x0 ; ^0 ), as n ! +1, hence x0n * x^0 in X , while 0n ! ^0 in IR. Thus it is readily veri ed that ^ is the solution to the equation (8) corresponding to the initial data (^x0 ; ^0 ). De ne Zt x^(t) := etA x^0 ? A e(t?s)A '(^ (s)) ds 0
and recall that from Theorem 4.4 it follows that all solutions tends to zero, as t ! +1. Therefore there exists T 0 0 such that j(^x(t); ^ (t))j 21 ?1 () for all t T 0 . Uniform convergence of (xn ; n ) to (^x; ^ ) on compact intervals yields the existence of T T 0 and an integer N such that
j(xn (T ); n (T ))j ?1 () for all n N .
Use again stability in the whole to get
j(xn (t); n (t))j for all n N and all t T , which contradicts (40).
Acknowledgements I wish to acknowledge several stimulating conversations with Luciano Pandol from Politecnico di Torino. 16
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