STABILITY RADIUS AND INTERNAL VERSUS EXTERNAL STABILITY ...

STABILITY RADIUS AND INTERNAL VERSUS EXTERNAL STABILITY IN BANACH SPACES: AN EVOLUTION SEMIGROUP APPROACH S. CLARK, Y. LATUSHKIN, S. MONTGOMERY-SMITH, AND T. RANDOLPH Abstract. In this paper the theory of evolution semigroups is developed and used to provide a framework to study the stability of general linear control systems. These include autonomous and nonautonomous systems modeled with unbounded state-space operators acting on Banach spaces. This approach allows one to apply the classical theory of strongly continuous semigroups to timevarying systems. In particular, the complex stability radius may be expressed explicitly in terms of the generator of a (evolution) semigroup. Examples are given to show that classical formulas for the stability radius of an autonomous Hilbert-space system fail in more general settings. Upper and lower bounds on the stability radius are proven for Banach-space systems. In addition, it is shown that the theory of evolution semigroups allows for a straightforward operator-theoretic analysis of internal stability as determined by classical frequency-domain and input-output operators, even for nonautonomous Banach-space systems.

evolution semigroups, stability radius, exponential stability, external stability, spectral mapping theorem, transfer function 47D06, 34G10, 93C25, 93D09, 93D25 0. Introduction Presented here is a study of stability of in nite-dimensional linear control systems which is based on the relatively recent development of the theory of evolution semigroups. These semigroups have been used in the study of exponential dichotomy of time-varying dierential equations and more general hyperbolic dynamical systems; see [5, 20, 21, 24, 27, 32, 40] and the bibliographies therein. The intent of this paper is to show how the theory of evolution semigroups can be used to provide a clarifying perspective, and prove new results, on the uniform exponential stability for general linear control systems, x_ (t) = A(t)x(t) + B (t)u(t), y(t) = C (t)x(t), t  0. The operators A(t) are generally unbounded operators on a Banach space, X , while the operators B (t) and C (t) may act on Banach spaces, U and Y , respectively. In addressing the general settings, diculties arise both from the time-varying aspect and from a loss of Hilbert-space properties. This presentation, however, provides some relatively simple operator-theoretic arguments for properties that extend classical theorems of autonomous systems in nite dimensions. The topics covered here include characterizing internal stability of the nominal system in terms of appropriate input-state-output operators and, subsequently, using these properties to obtain new explicit formulas for bounds on the stability radius. Nonautonomous systems are generally considered, but some results apply only to autonomous ones, such as the upper bound for the stability radius (Section 3.3), the formula for the norm of the input-output operator in Banach spaces (Section 3.4) and a characterization of stability that is related to this formula (Section 4.2). Although practical considerations usually dictate that U and Y are Hilbert spaces (indeed, nite dimensional), the Banach-space setting addressed here may be motivated by the problem of determining optimal sensor (or actuator) location. For this, it may be natural to consider U = X and B = IX (or Y = X and C = IX ) [3]; if the natural state space X is a Banach space then, as will be shown in this paper, Hilbert-space characterizations of internal stability or its robustness The research of the second author was supported by the National Science Foundation, the Missouri Research Board and a Summer Research Fellowship. The research of the third author was supported by the National Science Foundation and the Missouri Research Board. The research of the fourth author was supported by the Missouri Research Board; a portion of the work by this author was carried out while visiting the Department of Mathematics, University of Missouri{Columbia.

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S. CLARK, Y. LATUSHKIN, S. MONTGOMERY-SMITH, AND T. RANDOLPH

do not apply. We also show that even in the case of Hilbert spaces U and Y , known formulas for the stability radius involving the spaces L2 (R+ ; U ) and L2 (R+ ; Y ) do not apply if the L2 norm is replaced by, say, the L1 norm|see the examples in Subsection 3.5 below. In addition to the general setting of nonautonomous systems on Banach spaces, autonomous and Hilbert-space systems are considered. To motivate the methods, recall Lyapunov's stability theorem which says that if A is a bounded linear operator on X and if the spectrum of A is contained in the open left half of the complex plane, then the solution of the autonomous dierential equation x_ (t) = Ax(t) on X is uniformly exponentially stable; equivalently, the spectrum (etA ) is contained in the open unit disk fz 2 C : jz j < 1g, for t > 0. This is a consequence of the fact that when A is a bounded operator, then the spectral mapping theorem holds: (etA ) n f0g = et(A) ; t 6= 0: Diculties with Lyapunov's Theorem arise when the operators, A, are allowed to be unbounded. In particular, it is well known that there exist strongly continuous semigroups fetAgt0 that are not uniformly exponentially stable even though Re  ! < 0 for all  2 (A); see, e.g., [25, 27, 39]. For nonautonomous equations the situation is worse. Indeed, even for nite-dimensional X it is possible for the spectra of A(t) to be the same for all t > 0 and contained in the open left half-plane yet the corresponding solutions to x_ (t) = A(t)x(t) are not uniformly exponentially stable. In the development that follows we plan to show how these diculties can be overcome by the construction of an \evolution semigroup." This is a family of operators de ned on a superspace of functions from R into X , such as Lp (R; X ), 1  p < 1, or C0 (R; X ). Section 1 sets up the notation and provides background information. At the end of this section we give a brief synopsis of the main results of the paper. Section 2 presents the basic properties of the evolution semigroups. Included here is the property that the spectral mapping theorem always holds for these semigroups when they are de ned on X -valued functions on the half-line, such as Lp (R+ ; X ). A consequence of this is a characterization of exponential stability for nonautonomous systems in terms of the invertibility of the generator ? of the evolution semigroup. This operator, and its role in determining exponential stability, is the basis for many of the subsequent developments. In particular, the semigroup fetAgt0 is uniformly exponentially stable provided Re  < 0 for all  2 (?). Section 3 addresses the topic of the (complex) stability radius; that is, the size of the smallest disturbance,
(
), under which the perturbation, x_ (t) = (A(t) +
(t))x(t), of an exponentially stable system, x_ (t) = A(t)x(t), looses exponential stability. Results address structured and unstructured perturbations of autonomous and nonautonomous systems in both Banach and Hilbert space settings. Examples are given which highlight some important dierences between these settings. Also included in this section is a discussion about the transfer function for in nite-dimensional time-varying systems. This concept arises naturally in the context of evolution semigroups. In Section 4 the explicit relationship between internal and external stability is studied for general linear systems. This material expands on the ideas begun in [36]. A classical result for autonomous systems in Hilbert space is the fact that exponential stability of the nominal system (internal stability) is, under the hypotheses of stabilizability and detectability, equivalent to the boundedness of the transfer function in the right half-plane (external stability). Such a result does not apply to nonautonomous systems and a counterexample shows that this property fails to hold for Banach space systems. Properties from Section 2 provide a natural Banach-space extension of this result: the role of transfer function is replaced by the input-output operator. Moreover, for autonomous systems we provide an explicit formula relating the norm of this input-output operator to that of the transfer function. Finally, we prove two theorems|one for nonautonomous and one for autonomous systems|which characterize internal stability in terms of the various input-state-output operators. 1. Notation and Preliminaries Throughout the paper, L(X; Y ) will denote the set of bounded linear operators between complex Banach spaces X and Y . If A is a linear operator on X , (A) will denote the spectrum of A, (A) the resolvent set of A relative to L(X ) = L(X; X ), and kAk = kAk;X := inf fkAxk : x 2 Dom(A); kxk = 1g. In particular, if A is invertible in L(X ), kAk = 1=kA?1kL(X ) . Also, let C + = f 2 C : Re  > 0g.

STABILITY IN BANACH SPACES

3

If A generates a strongly continuous (or C0 ) semigroup fetAgt0 on a Banach space X the following notation will be used: s(A) = supfRe :  2 (A)g denotes the spectral bound; s0 (A) = inf f! 2 R : f : Re  > !g  (A) and supRe >! k(A ? )?1 k < 1g is the abscissa of uniform boundedness of the resolvent; and !0 (etA ) = inf f! 2 R : ketAk  Met! for some M  0 and all t  0g denotes the growth bound of the semigroup. In general, s(A)  s0 (A)  !0 (etA ) (see, e.g., [27]) with strict inequalities possible; see [25, 27, 39] for examples. However, when X is a Hilbert space, the following spectral mapping theorem of L. Gearhart holds (see, e.g., [25, p. 95] or [27, 31]): Theorem 1.1. If A generates a strongly continuous semigroup fetAgt0 on a Hilbert space, then s0 (A) = !0 (etA ). Moreover, 1 2 (e2A ) if and only if iZ  (A) and supk2Zk(A ? ik)?1 k < 1. In particular, this result shows that on a Hilbert space X the semigroup fetA gt0 is uniformly exponentially stable if and only if sup2C + k(A ? )?1 k < 1 [16]. Now consider operators A(t), t  0, with domain Dom(A(t)) in a Banach space X . If the abstract Cauchy problem (1) x_ (t) = A(t)x(t); x( ) 2 Dom(A( )); t    0; is well-posed in the sense that there exists an evolution (solving) family of operators U = fU (t;  )gt on X which gives a dierentiable solution, then x(
) : t 7! U (t;  )x( ), t   in R, is dierentiable, x(t) is in Dom(A(t)) for t  0, and (1) holds. The precise meaning of the term evolution family used here is as follows. De nition 1.2. A family of bounded operators fU (t;  )gt on X is called an evolution family if (i) U (t;  ) = U (t; s)U (s;  ) and U (t; t) = I for all t  s   ; (ii) for each x 2 X the function (t;  ) 7! U (t;  )x is continuous for t   . An evolution family fU (t;  )gt is called exponentially bounded if, in addition, (iii) there exist constants M  1, ! 2 R such that kU (t;  )k  Me!(t? ); t  : Remarks 1.3. (a) An evolution family fU (t;  )gt is called uniformly exponentially stable if in part (iii), ! can be taken to be strictly less than zero. (b) Evolution families appear as solutions for abstract Cauchy problems (1). Since the de nition requires that (t;  ) 7! U (t;  ) is merely strongly continuous the operators A(t) in (1) can be unbounded. (c) In the autonomous case where A(t)  A is the in nitesimal generator of a strongly continuous semigroup fetAgt0 on X then U (t;  ) = e(t? )A, for t   , is a strongly continuous exponentially bounded evolution family. (d) The existence of a dierentiable solution to (1) plays little role in this paper, so the starting point will usually not be the equation (1), but rather the existence of an exponentially bounded evolution family. In the next section we will de ne the evolution semigroup relevant to our interests for the nonautonomous Cauchy problem (1) on the half-line, R+ = [0; 1). For now, we begin by considering the autonomous equation x_ (t) = Ax(t), t 2 R, where A is the generator of a strongly continuous semigroup fetA gt0 on X . If FR is a space of X -valued functions, f : R ! X , de ne (2) (ERt f )( ) = etA f ( ? t); for f 2 FR: If FR = Lp (R; X ), 1  p < 1, or FR = C0 (R; X ), the space of continuous functions vanishing at in nities (or another Banach function space as in [32]) this de nes a strongly continuous semigroup of operators fERt gt0 whose generator will be denoted by ?R. In the case FR = Lp (R; X ), ?R is the closure (in Lp(R; X )) of the operator ?d=dt + A where (Af )(t) = Af (t) and Dom(?d=dt + A) = Dom(?d=dt) \ Dom(A) = fv 2 Lp (R; X ) : v 2 AC (R; X ); v0 2 Lp (R; X ) ; v(s) 2 Dom(A) for almost every s, and ?v0 + Av 2 Lp (R; X ) g:

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S. CLARK, Y. LATUSHKIN, S. MONTGOMERY-SMITH, AND T. RANDOLPH

The important properties of this \evolution semigroup" are summarized in the following theorem from [20]; see also further developments in [27, 32, 40]. The unit circle in C is denoted here by T = fz 2 C : jz j = 1g. Theorem 1.4. The spectrum (ERt ), for t > 0, is invariant with respect to rotations centered at the origin, and (?R) is invariant with respect to translations along iR. Moreover, the following are equivalent. (i) (etA ) \ T = ; on X ; (ii) (ERt ) \ T = ; on FR; (iii) 0 2 (?R) on FR. As a consequence, (3) (ERt ) n f0g = et(?R) ; t > 0: Note that fERt gt0 has the spectral mapping property (3) on FR even if the underlying semigroup fetAgt0 does not have the spectral mapping property on X . In the latter case, it may be that the exponential stability of the solutions to x_ = Ax on R are not determined by the spectrum of A. However, such stability is determined by the spectrum of ?R. This is made explicit by the following corollary of Theorem 1.4: The spectral bound s(?R) and the growth bound !0 (ERt ) for the evolution semigroup coincide and are equal to the growth bound of fetAgt0 : s(?R) = !0 (ERt ) = !0 (etA ): One of the diculties related to nonautonomous problems is that their associated evolution families are two-parameter families of operators. From this point of view, it would be of interest to de ne a one-parameter semigroup that is associated to the solutions of the nonautonomous Cauchy problem (1). For such a semigroup to be useful, its properties should be closely connected to the asymptotic behavior of the original nonautonomous problem. Ideally, this semigroup would have a generator that plays the same signi cant role in determining the stability of the solutions as the operator A played in Lyapunov's classical stability theorem for nite-dimensional autonomous systems, x_ = Ax. This can, in fact, be done and the operator of interest is the generator of the following evolution semigroup that is induced by the two-parameter evolution family: if U = fU (t;  )gt is an evolution family, de ne operators ERt , t  0, on FR = Lp (R; X ) or FR = C0 (R; X ) by (4) (ERt f )( ) = U (;  ? t)f ( ? t);  2 R; t  0: When U exponentially bounded, this de nes a strongly continuous evolution semigroup on FR whose generator will be denoted by ?R. As shown in [20] and [32] the spectral mapping theorem (3) holds for this semigroup. Moreover, the existence of an exponential dichotomy for solutions to x_ (t) = A(t)x(t); t 2 R, is characterized by the condition that ?R is invertible on FR. Note that in the autonomous case where U (t;  ) = e(t? )A, this is the evolution semigroup de ned in (2). In the nonautonomous case, the construction of an evolution semigroup is a way to \autonomize" a time-varying Cauchy problem by replacing the time-dependent dierential equation x_ = A(t)x on X by an autonomous dierential equation f_ = ?f on a superspace of X -valued functions. This section concludes with a brief synopsis of the main results. The characterization of uniform exponential stability in terms of an evolution semigroup and its generator is given in Theorem 2.2, Theorem 2.5, and Corollary 2.6. Although these results are essentially known, the proofs are approached in a new way. In particular, Theorem 2.5 identi es the operator G = ???1 used to determine stability throughout the paper. Theorem 3.2 records the main observation that the input-output operator, L = C G B, for a general nonautonomous system is related to the inverse of the generator of the evolution semigroup. A very short proof of the known fact that the stability radius for such a system is bounded from below by kLk?1 is also provided here. The upper bound for the stability radius, being given in terms of the transfer function, applies only to autonomous systems and is proven in Subsection 3.3. The upper bound, as identi ed here for Banach spaces, seems to be new although our proof is based on the idea of the Hilbert-space result of [15, Thm. 3.5]. In Subsection 3.3 we also introduce the pointwise stability radius and dichotomy radius. Estimates for the former are provided by Theorems 3.3 and 3.4 while the latter is addressed in Lemma 3.5. Examples 3.13 and 3.15 show that, for autonomous Banach space systems, both inequalities for

STABILITY IN BANACH SPACES

5

the upper and lower bounds on the stability radius (see Theorem 3.1) can be strict. In view of the possibility of the strict inequality kLk > sups2R kC (A ? is)?1 B k, Theorem 3.11 provides a new Banach space formula for kLk in terms of A, B , and C . In Section 4 this expression for kLk is used to relate state-space versus frequency-domain stability|concepts which are not equivalent for Banach-space systems. A special case of this expression gives a new formula for the growth bound of a semigroup on a Banach space; see Theorem 4.4 and the subsequent paragraph. Finally, Theorem 4.3 extends a classical characterization of stability for stabilizable and detectable control systems as it applies to nonautonomous Banach-space settings. 2. Evolution Semigroups and Cauchy Problems In order to tackle the problem of characterizing the exponential stability of solutions to the nonautonomous Cauchy problem (1) on the half-line, R+ , the following variant of the above evolution semigroup is needed. As before, let fU (t;  )gt be an exponentially bounded evolution family, and de ne operators E t , t  0, on functions f : R+ ! X by ( t (E f )( ) = U (;  ? t)f ( ? t); 0  t   (5) 0; 0   < t: This de nes a strongly continuous semigroup of operators on the space of functions F = Lp (R+ ; X ), and the generator of this evolution semigroup will be denoted by ?. This also de nes a strongly continuous semigroup on C00 (R+ ; X ) = ff 2 C0 (R+ ; X ) : f (0) = 0g. For more information on evolution semigroups on the half-line see also [24, 26, 27, 40]. 2.1. Stability. The primary goal of this subsection is to identify the useful properties of the semigroup of operators de ned in (5) which will be used in the subsequent sections. In particular, the following spectral mapping theorem will allow this semigroup to be used in characterizing the exponential stability of solutions to (1) on R+ . See also [32, 40] for dierent proofs. The spectral symmetry portion of this theorem is due to R. Rau [34]. Theorem 2.1. Let F denote C00(R+ ; X ) or Lp(R+ ; X ). The spectrum (?) is a half plane, the spectrum (E t ) is a disk centered at the origin, and (6) et(?) = (E t ) n f0g; t > 0: Proof. The arguments for the two cases F = C00 (R+ ; X ) and F = Lp (R+ ; X ) are similar, so only the rst one is considered here. We rst note that (?) is invariant under translations along iR and (E t ) is invariant under rotations about zero. This spectral symmetry is a consequence of the fact that for  2 R, (7) E t ei
f = ei
e?it E t f; and ?ei
= ei
(? ? i ): The inclusion et(?)  (E t ) n f0g follows from the standard spectral inclusion for strongly continuous semigroups [25]. In view of the spectral symmetry, it suces to show that (E t ) \ T = ; whenever 0 2 (?). To this end, we replace the Banach space X in Theorem 1.4 by C00 (R+ ; X ) and consider two semigroups fE~ t gt0 and fE t gt0 with generators ?~ and G , respectively, acting on the space C0 (R; C00 (R+ ; X )). These semigroups are de ned by ( t (E~ h)(; ) = U (;  ? t)h( ? t;  ? t) for   t; 0 for 0   < t; ( (E t h)(; ) = U (;  ? t)h(;  ? t) for   t; 0; for 0    t; where  2 R and h(;
) 2 C00 (R+ ; X ). Note that if H 2 C0 (R; C00 (R+ ; X )), then h(;
) := H ( ) 2 C00 (R+ ; X ) and we recognize fE~ t gt0 as the evolution semigroup induced by fE t gt0 , as in (2): (E~ t H )( ) = E t H ( ? t): Also, the semigroup fE t gt0 is the family of multiplication operators given by (E t H )( ) = E t H ( ):

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S. CLARK, Y. LATUSHKIN, S. MONTGOMERY-SMITH, AND T. RANDOLPH

The generator G of this semigroup is the operator of multiplication by ?: (G H )( ) = ?(H ( )), where H ( ) 2 Dom(?) for  2 R. In particular, if 0 2 (?) on F , then (G ?1 H )( ) = ??1 (H ( )), and so 0 2 (G ). Let J denote the isometry on C0 (R; C00 (R+ ; X )) given by (Jh)(; ) = h( + ; ) for  2 R,  2 R+ . Then J satis es the identity: (E t Jh)(; ) = (J E~ t h)(; );  2 R;  2 R+ : It follows that G JH = J ?~ H for H 2 Dom(?~ ), and J ?1 G H = ?~ J ?1 H for H 2 Dom(G ). Consequently (G ) = (?~ ) on C0 (R; C00 (R+ ; X )). In particular, 0 2 (?~ ). Therefore, (E t ) \ T = ; follows from Theorem 1.4 applied to the semigroup fE t gt0 on F in place of fetAgt0 on X . The fact that (?) is a half plane (E t ) is a disk follow from the spectral mapping property (6) and [34, Proposition 2]. An important consequence of this theorem is the property that the growth bound !0 (E t ) equals the spectral bound s(?). This leads to the following simple result on stability. Theorem 2.2. Let F denote C00 (R+ ; X ) or Lp(R+ ; X ). An exponentially bounded evolution family fU (t;  )gt is exponentially stable if and only if the growth bound !0 (E t ) of the induced evolution semigroup on F is negative. Proof. Let F = C00 (R+ ; X ). If fU (t;  )gt is exponentially stable, then there exist M > 1, > 0 such that kU (t;  )kL(X )  Me? (t? ), t   . For   0 and f 2 C00 (R+ ; X ), kE  f kC00(R+;X ) = sup kE  f (t)kX = sup kU (t; t ?  )f (t ?  )kX t>0

t>

 sup kU (t; t ?  )kL(X ) kf (t ?  )kX t>  Me?  kf kC00(R+;X ) : Conversely, assume there exist M > 1,  > 0 such that kE t k  Me?t, t  0. Let x 2 X , kxk = 1. For xed t >  > 0, choose f 2 C00 (R+ ; X ) such that kf kC00(R+;X ) = 1 and f ( ) = x. Then,

kU (t;  )xkX = kU (t;  )f ( )kX = kE (t? )f (t)kX  sup kE (t? )f ()kX >0 = kE (t? )f kC00(R+;X )  Me?(t? ): A similar argument works for F = Lp(R+ ; X ).

The remainder of this subsection focuses on the operator used for determining exponential stability. In fact, stability is characterized by the boundedness of this operator which, as seen below, is equivalent to the invertibility of ?, the generator of the evolution semigroup. We begin with the autonomous case. R. Datko and J. van Neerven have characterized the exponential stability of solutions for autonomous equations x_ = Ax, t  0, in terms of a convolution operator, G , induced by fetA gt0. In this autonomous setting, ( tA t (E f )( ) = e f ( ? t); 0  t   (8) 0; 0   < t; and the convolution operator takes the following form: for f 2 L1loc (R+ ; X ), (9)

(G f )(t) :=

Z

0

t

eAf (t ?  ) d =

Z

0

1

(E  f )(t) d; t  0:

Theorem 1.3 of [26] (see also [11]) is as follows. Theorem 2.3. If fetAgt0 is a strongly continuous semigroup on X , and 1  p < 1, then the following are equivalent:

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(i) !0 (etA ) < 0; (ii) G f 2 Lp (R+ ; X ) for all f 2 Lp (R+ ; X ); (iii) G f 2 C0 (R+ ; X ) for all f 2 C0 (R+ ; X ). Remarks 2.4. (a) Note that condition (ii) is equivalent to the boundedness of G on Lp(R+ ; X ). To see this, it suces to show that the map f 7! G f is a closed operator on Lp (R+ ; X ), and then apply the closed graph theorem. For this, let fn ! f and G fn ! g in Lp (R+ ; X ). Then (G fn )(t) ! (G f )(t) for each t 2 R. Also, every norm-convergent sequence in Lp (R+ ; X ) contains a subsequence that converges pointwise almost everywhere. Thus, (G fnk )(t) ! g(t) for almost all t. This implies that G f = g, as claimed. (b) Also, condition (iii) is equivalent to the boundedness of G on C0 (R+R ; X ). This follows from the uniform boundedness principle applied to the operators G t : f 7! 0t eAf (t ?  ) d . We now extend this result so that it may be used to describe exponential stability for a nonautonomous equation. For this de ne an operator G in an analogous way: let fU (t;  )gt be an evolution family and fE tgt0 the evolution semigroup in (5). Then de ne G for f 2 L1loc(R+ ; X ) as (10)

(G f )(t) := =

1

Z

0

Z

0

t

(E  f )(t) d =

t

Z

U (t;  )f ( ) d;

0

U (t; t ?  )f (t ?  ) d t  0:

For G acting on F = C00 (R+ ; X ) or Lp (R+ ; X ), standard semigroup properties show that G equals ???1 provided the semigroup fE t gt0 or the evolution family is uniformly exponentially stable. Parts (i) , (ii) of Theorem 2.3 and the nonautonomous version below are the classical results by R. Datko [11]. Our proof uses the evolution semigroup and creates a formally autonomous problem so that Theorem 2.3 can be applied. Theorem 2.5. The following are equivalent for the evolution family of operators fU (t;  )gt on X. (i) fU (t;  )gt is exponentially stable; (ii) G is a bounded operator on Lp (R+ ; X ); (iii) G is a bounded operator on C0 (R+ ; X ). Before proceeding with the proof, note that statement (ii) is equivalent to the statement: G f 2 Lp (R+ ; X ) for each f 2 Lp (R+ ; X ). This is seen as in Remark 2.4, above. See also [4] for similar facts. Proof. By Theorem 2.2, (i) implies that fE t gt0 is exponentially stable, and formula (10) implies (ii) and (iii). The implication (ii))(i) will be proved here; the argument for (iii))(i) is similar. The main idea is again to use the \change-of-variables" technique, as in the proof of Theorem 2.1. Consider the operator G~ on Lp (R; Lp (R+ ; X )) = Lp (R  R+ ; X ) de ned as multiplication by G . More precisely, for h 2 Lp (R  R+ ; X ) with h() := h(;
) 2 Lp(R+ ; X ), de ne

(G~ h)(; t) = G (h())(t) =

Z

t

0

U (t; t ?  )h(; t ?  ) d; t 2 R+ ;  2 R:

In view of statement (ii), this operator is bounded. For the isometry J de ned on the space

Lp (R; Lp (R+ ; X )) by (Jh)(; t) = h( + t; t), we have (11)

(J ?1 G~ Jh)(; t) =

Z

0

t

U (t; t ?  )h( ? ; t ?  ) d:

Next, let fE t gt0 be the evolution semigroup (5) induced by fU (t;  )gt , and de ne G  to be the operator of convolution with this semigroup as in (9); that is, (12)

(G  h)() =

Z

0

1

E  h( ?  ) d; h 2 Lp (R; Lp (R+ ; X )):

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S. CLARK, Y. LATUSHKIN, S. MONTGOMERY-SMITH, AND T. RANDOLPH

If h(;
) = h() 2 Lp (R+ ; X ), then by de nition (5), evaluating (12) at t gives (13)

[(G  h)()] (t) = (G  h)(; t) =

Z

0

t

U (t; t ?  )h( ? ; t ?  ) d; t 2 R+ ;  2 R:

From (11) it follows that G  = J ?1 G~ J is a bounded operator on Lp (R; Lp (R+ ; X )). Now, each function h+ 2 Lp (R+ ; Lp(R+ ; X )) is an Lp(R+ ; X )-valued function on the half line R+ . We extend each such h+ to a function h 2 Lp (R; Lp (R+ ; X )) by setting h() = h+ () for   0 and h() = 0 for  < 0. Note that G  h 2 Lp (R; Lp (R+ ; X )) because G  is bounded on Lp (R; Lp (R+ ; X )). Consider the function f+ : R+ ! Lp (R+ ; X ) de ned by

f+(t) =

Z

t

0

E  h+ (t ?  ) d =

1

Z

0

E  h(t ?  ) d; t 2 R+ :

To complete the proof of the theorem, it suces to prove the following claim:

f+ 2 Lp(R+ ; Lp(R+ ; X )): Indeed, the operator h+ 7! f+ is the convolution operator as in (9) de ned by the semigroup operators E t instead of etA . An application of Theorem 2.3 to E t on Lp (R+ ; X ) (in place of etA on X ) shows that the semigroup fE t gt0 is exponentially stable on Lp (R+ ; X ) provided:

f+ 2 Lp(R+ ; Lp(R+ ; X )) for each h+ 2 Lp(R+ ; Lp(R+ ; X )): But if fE t gt0 is exponentially stable, the evolution family fU (t;  )gt is exponentially stable by Theorem 2.2. To prove the claim, apply formula (13) for h(; t) = h+ (; t);   0 and h(; t) = 0,  < 0, t 2 R+ , where h+ () = h+ (;
). This gives (G  h)(; t) =

(R minf;tg

U (t; t ?  )h+ ( ? ; t ?  ) d 0 (G  h)(; t) = 0;

for   0; t 2 R+ for  < 0; t 2 R+

Thus, the function

 7! (G  h)(;
) = (G  h)() 2 Lp(R+ ; X ) is in the space Lp (R+ ; Lp (R+ ; X )). On the other hand, denoting f+ (;
) := f+ () 2 Lp(R+ ; X ), we have that

f+ (; t) =

Z

0

minf;tg

U (t; t ?  )h+ ( ? ; t ?  ) d; ; t 2 R+ :

Thus,  7! f+ (;
) = (G  h)(;
) is a function in Lp (R+ ; Lp(R+ ; X )), and the claim is proved. This theorem makes explicit, in the case of the half line R+ , the relationship between the stability of an evolution family fU (t;  )gt and the generator, ?, of the corresponding evolution semigroup (5). Indeed, as shown above, stability is equivalent to the boundedness of G , in which case G = ???1 . Combining Theorems 2.1, 2.2 and 2.5 yields the following corollary.

Corollary 2.6. Let fU (t;  )gt be an exponentially bounded evolution family and let ? denote the generator of the induced evolution semigroup on Lp(R+ ; X ), 1  p < 1, or C00 (R+ ; X ). The following are equivalent: (i) fU (t;  )gt is exponentially stable; (ii) ? is invertible with ??1 = ?G ; (iii) s(?) < 0.

For more information on stability and dichotomy of evolution families on the semiaxis see [24].

STABILITY IN BANACH SPACES

9

2.2. Perturbations and robust stability. This subsection brie y considers perturbations of (1) of the form (14) x_ (t) = (A(t) + D(t))x(t); t  0: It will not, however, be assumed that (14) has a dierentiable solution. For example, let fetA0 gt0 be a strongly continuous semigroup generated by A0 , let A1 (t) 2 L(X ) for t  0, and de ne A(t) = A0 + A1 (t). Then even if t 7! A1 (t) is continuous, the Cauchy problem (1) may not have a dierentiable solution for all initial conditions x(0) = x 2 Dom(A) = Dom(A0 ) (see, e.g.,[29]). Therefore we will want our development to allow for equations with solutions that exist only in the following mild sense. Let fU (t;  )gt be an evolution family of operators corresponding to a solution of (1), and consider the nonautonomous inhomogeneous equation (15) x_ (t) = A(t)x(t) + f (t); t  0; where f is a locally integrable X -valued function on R+ . A function x(
) is a mild solution of (15) with initial value x() = x 2 Dom(A()) if

x(t) = U (t; )x +

Z

t

0

U (t;  )f ( ) d; t  :

Given operators D(t), the existence of mild solutions to an additively perturbed equation (14) corresponds to the existence of an evolution family fU1(t;  )gt satisfying (16)

U1(t; )x = U (t; )x +

Z

s

t

U (t;  )D( )U1 (; )x d:

for all x 2 X . It will be assumed that the perturbation operators, D(t), are strongly measurable and essentially bounded functions of t. In view of this, we use the notation Ls (X ) to denote the set L(X ) endowed with the strong operator topology and use L1 (R+ ; Ls (X )) to denote the set of bounded, strongly measurable L(X )-valued functions on R+ . A function D(
) 2 L1 (R+ ; Ls (X )) induces a multiplication operator D de ned by Dx(t) = D(t)x(t), for x(
) 2 Lp(R+ ; X ). In fact, D is a bounded operator on Lp (R+ ; X ) with kDk  kD(
)k1 := ess supt2R+ kD(t)k. Evolution semigroups induced by an evolution family as in (5) have been studied by several authors who have characterized such semigroups in terms of their generators on general Banach function spaces of X -valued functions (see [35, 40] and the bibliography therein). The sets F = Lp (R+ ; X ) or F = C00 (R+ ; X ) considered here are examples of more general \Banach function spaces." In the development that follows we use a theorem of R. Schnaubelt [40] (see also Rabiger et al. [33, 35]) which shows exactly when a strongly continuous semigroup on F arises from a strongly continuous evolution family on X . We state a version of this result which will be used below; a more general version is proven in [35]. The set Cc1 (R+ ) consists of dierentiable functions on R+ that have compact support. Theorem 2.7. Let fT tgt0 be a strongly continuous semigroup generated by ? on F . The following are equivalent: (i) fT tgt0 is an evolution semigroup; i.e., there exists an exponentially bounded evolution family so that T t is de ned as in (5); (ii) there exists a core, C, of ? such that for all ' 2 Cc1 (R+ ), and f 2 C, it follows that 'f 2 Dom(?) and ?('f ) = ?'0 f + '?f . Moreover, there exists  2 (?) such that R(; ?) : F ! C00 (R+ ; X ) is continuous with dense range. Now let fU (t;  )gt be an evolution family on X and let ? be the generator of the corresponding evolution semigroup, fE tgt0 , as in (5). If D(
) 2 L1 (R+ ; Ls (X )), then the multiplication operator D is a bounded operator on F = Lp (R+ ; X ). Since a bounded perturbation of a generator of a strongly continuous semigroup is itself such a generator, the operator ?1 = ? + D generates a strongly continuous semigroup, fE1t gt0 on F (see, e.g., [28]). In fact, ?1 generates an evolution semigroup, see [33, 40]:

10

S. CLARK, Y. LATUSHKIN, S. MONTGOMERY-SMITH, AND T. RANDOLPH

Proposition 2.8. Let D(
) 2 L1(R+ ; Ls (X )):, and let fU (t;  )gt be an exponentially bounded evolution family. Then there exists a unique evolution family U1 = fU1 (t;  )gt which solves the integral equation (16). Moreover, U1 is exponentially stable if and only if ? + D is invertible. Proof. As already observed, ?1 = ? + D generates a strongly continuous semigroup, fE1t gt0 on F . To see that this is, in fact, an evolution semigroup, note that for  2 (?) \ (?1 ), Range(R(; ?)) = Dom(?) = Dom(? + D) = Range(R(; ?1 )) is dense in C0 (R+ ; X ). Also, if C is a core for ?, then it is a core for ?1 , and so for ' 2 Cc1 (R), f 2 C, ?1 ('f ) = ?('f ) + D('f ) = ?'0 f + '?f + 'Df = ?'0 f + '(? + D)f: Consequently, Corollary 2.7 shows that fE1t gt0 corresponds to an evolutionary family, fU1 (t;  )gt . Moreover, x(t) = U1 (t;  )x( ) is seen to de ne a mild solution to (14). Indeed,

(17)

E1t f = E t f +

Z

0

t

E (t? ) DE1 f d;

holds for all f 2 F . In particular, for x 2 X , and any ' 2 Cc1 (R), setting f = ' x in (17), where ' x(t) = '(t)x, and using a change of variables leads to Z

t

'()U1 (t; )x = '()U (t; )x + '() U (t;  )D( )U1 (; )x d:  Therefore, (16) holds for all x 2 X .

Finally, Theorem 2.6 shows that U1 is exponentially stable if and only if ?1 is invertible. The existence of mild solutions under bounded perturbations of this type is well known (see, e.g., [8]), but an immediate consequence of the approach given here is the property of robustness for the stability of fU (t;  )gt . Indeed, by continuity properties of the spectrum of an operator ?, there exists  > 0 such that ?1 is invertible whenever k?1 ? ?k < ; that is, fU1(t;  )gt is exponentially stable whenever kD(
)k1 < . Also, the type of proof presented here can be extended to address the case of unbounded perturbations. For an example of this, we refer to [35]. Finally, and most important to the present paper, is the fact that this approach provides insight into the concept of the stability radius. This topic is studied next. 3. Stability Radius The goal of this section is to use the previous development to study the (complex) stability radius of an exponentially stable system. Loosely speaking, this is a measurement on the size of the smallest operator under which the additively perturbed system looses exponential stability. This is an important concept for linear systems theory and was introduced by D. Hinrichsen and A. J. Pritchard as the basis for a state-space approach to studying robustness of linear time-invariant [14] and time-varying systems [13, 15, 30]. A systematic study of various stability radii in the spirit of the current paper has recently be given by A. Fischer and J. van Neerven [12]. 3.1. General estimates. In this subsection we give estimates for the stability radius of general nonautonomous systems on Banach spaces. The perturbations considered here are additive \structured" perturbations of output feedback type. That is, let U and Y be Banach spaces and let
(t) : Y ! U denote an unknown disturbance operator. The operators B (t) : U ! X and C (t) : X ! Y describe the structure of the perturbation in the following (formal) sense: if u(t) =
(t)y(t) is viewed as a feedback for the system x_ (t) = A(t)x(t) + B (t)u(t); x(s) = xs 2 Dom(A(s)); (18) y(t) = C (t)x(t); t  s  0; then the nominal system x_ (t) = A(t)x(t) is subject to the structured perturbation: (19) x_ (t) = (A(t) + B (t)
(t)C (t))x(t); t  0: In this section B and C do not represent input and output operators, rather they describe the structure of the uncertainty of the system. Also, systems considered throughout this paper are not

STABILITY IN BANACH SPACES

11

assumed to have dierentiable solutions and so (19) is to be interpreted in the mild sense as described in (16) where D(t) = B (t)
(t)C (t). Similarly, (18) is interpreted in the mild sense; that is, there exists a strongly continuous exponentially bounded evolution family fU (t;  )gt on a Banach space X which satis es Z

(20)

t

x(t) = U (t; s)x(s) + U (t;  )B ( )u( ) d; s y(t) = C (t)x(t); t  s  0:

In the case of time-invariant systems, equation (20) takes the form t

Z

e(t? )ABu( ) d; t  0; where fetAgt0 is a strongly continuous semigroup on X generated by A, x(0) = x0 2 Dom(A). x(t) = etA x0 + y(t) = Cx(t);

(21)

0

It should be emphasized that we will not address questions concerning the existence of solutions for a perturbed system (19) beyond the point already discussed in Proposition 2.8. In view of that proposition, we make the following assumptions: B , C and
are strongly measurable and essentially bounded functions of t; i.e., B (
) 2 L1 (R+ ; Ls (U; X )), C (
) 2 L1 (R+ ; Ls (X; Y )) and
(
) 2 L1(R+ ; Ls (Y; U )). As such, they induce bounded multiplication operators, B, C and
~ acting on the spaces Lp (R+ ; U ), Lp (R+ ; X ) and Lp (R+ ; Y ), respectively. Next, for an exponentially bounded evolution family fU (t;  )gt , de ne the \input-output" operator L on functions u : R+ ! U by the rule (Lu)(t) = C (t)

t

Z

0

U (t;  )B ( )u( ) d:

Using the above notation note that L = C G B. Much of the stability analysis that follows is based on this observation in combination with Theorem 2.5 which shows that the operator G completely characterizes stability of the corresponding evolution family. We now turn to the de nition of the stability radius. For this let U = fU (t;  )gt be an exponentially stable evolution family on X . Set D = B
~ C and let U
= fU
(t;  )gt denote the evolution family corresponding to solutions of the perturbed equation (14). That is, U
satis es

U
(t; s)x = U (t; s)x +

t

Z

s

U (t;  )B ( )
( )C ( )U
(; s)x d;

x 2 X:

De ne the (complex) stability radius for U with respect to the perturbation structure (B (
); C (
)) as the quantity rstab (U ; B; C ) = supfr  0 : k
(
)k1  r ) U
is exponentially stableg: This de nition applies to both nonautonomous and autonomous systems, though in the latter case the notation rstab (fetAg; B; C )) will be used to distinguish the case where all the operators except
(t) are independent of t. We will have occasion to consider the constant stability radius which is de ned for the case in which
(t) 
is constant; this will be denoted by rcstab (fetA g; B; C )) or rcstab (U ; B; C )), depending on the context. The above remarks concerning ? + D (see Proposition 2.8), when combined with Theorem 2.2, make it clear that (22) rstab (U ; B; C ) = supfr  0 : k
(
)k1  r ) ? + B
~ C is invertibleg: It is well known that for autonomous systems in which U and Y are Hilbert spaces and p = 2, the stability radius may be expressed in terms of the norm of the input-output operator or the transfer function: 1 = r (fetAg; B; C ) = 1 (23) stab 2 kLk sup kC (A ? is)?1 B k ; L(L )

s2R

see, e.g., [15, Theorem 3.5]. For nonautonomous equations, a scalar example given in Example 4.4 of [13] shows that, in general, a strict inequality 1=kLk < rstab (U ; B; C ) may hold. Moreover, even for autonomous systems, when Banach spaces are allowed or when p 6= 2, Example 3.13 and

12

S. CLARK, Y. LATUSHKIN, S. MONTGOMERY-SMITH, AND T. RANDOLPH

Example 3.15 below will show that neither of the equalities in (23) necessarily hold. Subsection 3.3 below focuses on autonomous equations and a primary objective there is to prove the following result. Theorem 3.1. For the general autonomous systems, 1  r (fetA g; B; C )  1 (24) stab kLkL(Lp ) sups2R kC (A ? is)?1 B k ; 1  p < 1: As seen next, the lower bound here holds for general nonautonomous systems and may be proven in a very direct way using the make-up of the operator L = C G B. This lower bound is also proven in [15, Theorem 3.2] using a completely dierent approach. Theorem 3.2. Assume U is an exponentially stable evolution family and let ? denote the generator of the corresponding evolution semigroup. If B (
) 2 L1(R+ ; Ls (U; X )) and C (
) 2 L1(R+ ; Ls (X; Y )); then L is a bounded operator from Lp (R+ ; U ) to Lp(R+ ; Y ), 1  p < 1, the formula L = C G B = ?C ??1 B holds, and 1  r (U ; B; C ): (25)

kLk

stab

In the \unstructured" case, where U = Y = X and B = C = I , one has 1  r (U ; I; I )  1 ; L = ???1 ; and k??1k stab r(??1 ) where r(
) denotes the spectral radius. Proof. Since U is exponentially stable, ? is invertible and ??1 = ?G . The required formula for L follows from (10). ~ To prove (25), let
(
) 2 L1(R+ ; Ls (Y; U )) and suppose that k
(
)k1 < Set H := ??1 B
. ~ 1=kLk. Then kL
k < 1, and hence I ? L
~ = I + C ??1 B
~ is invertible on Lp(R+ ; Y ). That is, I + CH is invertible on Lp(R+ ; Y ), and hence I + HC is invertible on Lp (R+ ; X ) (with inverse (I ? H(I + CH)?1 C )). Now, ? + B
~ C = ?(I + ??1 B
~ C ) = ?(I + HC ) and so ? + B
~ C is invertible. It follows from the expression (22) that 1=kLk  rstab (U ; B; C ). For the last assertion, suppose that rstab (U ; I; I ) > 1=r(??1 ). Then there exists  such that jj = r(??1 ) and  + ??1 is not invertible. But then setting
~  1 gives k
~ k = j1j < rstab (U ; I; I ), ~  + ??1 )? is invertible, a contradiction. and so ? +
~ =
( 3.2. The transfer function for nonautonomous systems. In this subsection we consider a time-varying version of equation (23) and then observe that the concept of a transfer function, or frequency-response function, arises naturally from these ideas. For this we assume in this subsection that X , U and Y are Hilbert spaces and p = 2. Let fU (t;  )gt be an uniformly exponentially stable evolution family and let fE tgt0 be the induced evolution semigroup with generator ? on the L2 (R+ ; X ). Recall that B and C denote multiplication operators, with respective multipliers B (
) and C (
), that act on the spaces L2 (R+ ; U ) and L2 (R+ ; X ), respectively. Let B~ and C~ denote operators of multiplication induced by B and C , respectively; e.g., (B~u)(t) = B(u(t)), for u : R+ ! L2(R+ ; U ). Now consider the operator G  as de ned in equation (12) and note that operator L := C~G  B~ may be viewed (formally) as an inputoutput operator for the \autonomized" system: f_ = ?f + Bu; g = C f , where the state space is L2 (R+ ; X ). It follows from the known Hilbert-space equalities in (23) that 1 1 t kL k = rstab (fE g; B; C ) = sups2R kC (? ? is)?1 Bk :

STABILITY IN BANACH SPACES

13

Note, however, that the rescaling identities (7) for ? imply that kL k = kC (? ? is)?1 Bk = kC ??1Bk = kLk; and so the stability radius for the evolution semigroup is also 1=kLk. In view of the above-mentioned nonautonomous scalar example for which 1=kLk < rstab (U ; B; C ) we see that even though the evolution semigroup (or its generator) completely determines the exponential stability of a system, it does not provide a formula for the stability radius. However, the operator C (? ? is)?1 B appearing above suggests that the transfer function for time-varying systems arises naturally when viewed in the context of evolution semigroups. Several authors have considered the concept of a tranasfer function for nonautonomous systems but the work of J. Ball, I. Gohberg, and M.A. Kaashoek [2] seems to be the most comprehensive in providing a system-theoretic input-output interpretation for the value of such a transfer function at a point. Their interpretation justi es the term frequency response function for time-varying nite-dimensional systems with \time-varying complex exponential inputs." Our remarks concerning the frequency response for time-varying in nite-dimensional systems will be restricted to inputs of the form u(t) = u0 et . For motivation, consider the input-output operator L associated with an autonomous system (21) where the nominal system is exponentially stable. The transfer function of L is the unique bounded analytic L(U; Y )-valued function, H , de ned on C + = f 2 C : Re  > 0g such that for any u 2 L2 (R+ ; U ), (Lcu )() = H ()^u();  2 C +; where b denotes the Laplace transform (see, e.g., [42]). In this autonomous setting, A generates a uniformly exponentially stable strongly continuous semigroup, and L = C G B where G is the operator of convolution with the semigroup operators etA (see (9)). Standard arguments show that (Lcu )() = C ( ? A)?1 B u^(); that is, H () = C ( ? A)?1 B . Now let L be the input-output operator for the nonautonomous system (20). We wish to identify the transfer function of L as the Laplace transform of the appropriate operator. We are guided by the fact that, just as ( ? A)?1 may be expressed as the Laplace transform of the semigroup generated by A, the operator ( ? ?)?1 is the Laplace transform of the evolution semigroup. For nonautonomous systems, L is again given by C G B, although now G from (10) is not, generally, a convolution operator. So instead recall the operator G  from (12) which is the operator of convolution with the evolution semigroup fE t gt0 . As noted above, the operator L := C~G  B~ may be viewed as an input-output operator for an autonomous system (where the state space is L2 (R+ ; X )). Therefore, the autonomous theory applies directly to show that, for u 2 L2 (R+ ; L2 (R+ ; U )), (26) (Ld  u)() = C ( ? ?)?1 Bu^ (): In other words, the transfer function for L is C ( ? ?)?1 B, where Z

1

C ( ? ?)?1 Bu = C e? E  Bu d; u 2 L2(R+ ; U ): 0 Evaluating these expressions at t 2 R+ gives Z t (27) [C ( ? ?)?1 Bu](t) = C (t)U (t;  )B ( )u( )e?(t? ) d: 0 ? 1 It is natural to call C ( ? ?) B the transfer function for the nonautonomous system. Moreover, the

following remarks show that, by looking at the right-hand side of (27), this gives a natural \frequency response" function for nonautonomous systems. To see this, we rst consider autonomous systems and note that the de nition of the transfer function for an autonomous system can be extended to allow for a class of \Laplace transformable" functions that are in L2loc(R+ ; U ) (see, e.g., [42]). This class includes constant functions of the form v0 (t) = u0, t  0, for a given u0 2 U . If a periodic input signal of the form u(t) = u0 ei!t , t  0, (for some u0 2 U and ! 2 R) is fed into an autonomous system with initial condition x(0) = x0 , then, by de nition of the input-output operator, we have (Lu)(t) = C (i! ? A)?1 Bu0
ei!t ? CetA x0 ; (Lu)(t) = C

t

Z

0

e(t?s)A Bu(s) ds:

14

S. CLARK, Y. LATUSHKIN, S. MONTGOMERY-SMITH, AND T. RANDOLPH

Thus, the output

y(t; u(
); x0 ) = (Lu)(t) + CetA x0 = C (i! ? A)?1 Bu0
ei!t has the same frequency as the input. In view of this, the function C (i! ? A)?1 B is sometimes called the frequency response function. Now recall that the semigroup fetAgt0 is stable and so limt!1 kCetAx0 k = 0. On the other hand, consider v(t) = u0 and (formally) apply C (i! ? ?)?1 B to this v. For x0 = (i! ? A)?1 Bu0 , a calculation based on the Laplace transform formula for the resolvent of the generator (applied to the evolution semigroup fE t gt0 ) yields the identity   C (i! ? ?)?1 Bu0 (t) = C (i! ? A)?1 Bu0 ? CetA x0
e?i!t : Let us consider this expression [C (i! ? ?)?1 Bu0](t) in the nonautonomous case. By equation (27), this coincides with the frequency response function for time-varying systems which is de ned in [2, Corollary 3.2] by the formula Z

0

t

C (t)U (t;  )B ( )u0 ei!( ?t) d:

Also, as noted in this reference, the result of our derivation agrees the Arveson frequency response function as it appears in [41]. We recover it here explicitly as the Laplace transform of an inputoutput operator (see equation (26)). 3.3. Autonomous systems. In this subsection we give the proof of (24) when X , U and Y are Banach spaces. In the process, however, we also consider two other \stability radii": a pointwise stability radius and a dichotomy radius. First, we give a generalization to Banach spaces of Theorem 1.1 (cf. [20]). Here, Fper denotes the Banach space Lp ([0; 2]; X ), 1  p < 1. If fetA gt0 is a strongly continuous semigroup on X , t gt0 will denote the evolution semigroup de ned on Fper by the rule E t f (s) = etA f ([s ? fEper per t](mod 2)); its generator will be denoted by ?per . The symbol  will be used to denote the set of all nite sequences fvk gNk=?N in D(A), or fuk gNk=?N in U . Theorem 3.3. Let A generate a C0 semigroup fetAgt0 on X . Let B and C be as above, and
2 (Y; U ). Let fet(A+B
C )gt0 be the strongly continuous semigroup generated by A + B
C . Then the following are equivalent: (i) 1 2 (e2(A+B
C )); (ii) iZ  (A + B
C ) and P + B
C )?1 vk eik(
) kFper k k (A ? ik P sup < 1; ik(
) (iii) iZ  (A + B
C )

fvk g2

and

k

k vk e

kFper

k Pk (A ?Pik + B
C )vk eik(
) kFper > 0: fvk g2 k k vk eik(
) kFper denotes the generator of the evolution semigroup on Fper , as above, and if 1 2 inf

Further, if ?per

(e2A ), then ?per is invertible and (28)

P ?1 ik(
) k k C (AP? ik ) Buk e kLp ([0;2];Y ) ? 1 kC ?per Bk = sup k k uk eik(
) kLp([0;2];U ) fuk g2

where C ??per1 B 2 L(Lp ([0; 2]; U ); Lp ([0; 2]; Y )). Proof. The equivalence of (i){(iii) follows as in Theorem 2.3 of [20]. For the last statement, let fuk g be a nite set in U and consider functions f and g of the form

f (s) =

X

X

k

k

(A ? ik)?1 Buk eiks ; and g(s) =

Buk eiks :

STABILITY IN BANACH SPACES

Then f = ??per1 g. For,



d (?per f )(s) = dt =

X

k

t=0

15

etA f ([s ? t]mod2)

[A(A ? ik)?1 Buk eiks ? ik(A ? ik)?1Buk eiks ] = g(s): P

For functions of the form h(s) = k uk eiks , where fuk gk is a nite set in U , we have C ??1 Bh = P ?1 ik(
) k C (A ? ik ) Buk e . Taking the supremum over all such functions gives: ?1 Bhk kC ??per1 Bk = sup kC ?kper hk h

P ?1 ik(
) k k C (AP? ik ) Buk e kLp ([0;2];Y ) : = sup k k uk eik(
) kLp([0;2];U ) fuk g2

In view of these facts we introduce a \pointwise" variant of the constant stability radius: for

t0 > 0 and  2 (et0 A ), de ne the pointwise stability radius rcstab (et0 A ; B; C ) := supfr > 0 : k
kL(Y;U )  r )  2 (et0 (A+B
C ))g: By rescaling, the study of this quantity can be reduced to the case of  = 1 and t0 = 2. Indeed, rcstab (et0 A ; B; C ) = 2t rcstab (e2A0 ; B; C ); where A0 = 2t0 A: 0

Also, after writing  = jjei ( 2 R), note that

rcstab (e2A ; B; C ) = rc1stab (e2A00 ; B; C ); Therefore, for

for A00 = A ? 21 (ln jj + i):

rcstab (et0 A ; B; C ) = 2t rc1stab (e2A000 ; B; C ) 0

A000 = 21 (t0 A ? ln jj ? i): In the following theorem we estimate rc1stab (e2A ; B; C ). The idea for the proof goes back to [15].

See also further developments in [12]. Theorem 3.4. Let fetAgt0 be a strongly continuous semigroup generated by A on X , and assume 1 2 (e2A ). Let ?per denote the generator of the induced evolution semigroup on Fper . Let B 2 L(U; X ), and C 2 L(X; Y ). Then 1 1 1 2A (29) kC ??per1 Bk  rcstab (e ; B; C )  supk2ZkC (A ? ik)?1 B k : If U and Y are a Hilbert spaces and p = 2, then equalities hold in (29). Proof. The rst inequality follows from an argument as in Theorem 3.2. For the second inequality, let  > 0, and choose u 2 U with kuk = 1 and k0 2 Z such that kC (A ? ik0 )?1 B ukY  sup kC (A ? ik)?1 B k ?  > 0: k2Z  Using the Hahn-Banach Theorem, choose y 2 Y  with kyk  1 such that  ?1 B u  C ( A ? ik ) 0  y ; kC (A ? ik )?1 B uk = 1: 0 Y

De ne
2 L(Y; U ) by




y = ? kC (A ?hyik; y)?i 1 B uk u; y 2 Y: 0

Y

16

We note that (30) and (31)

S. CLARK, Y. LATUSHKIN, S. MONTGOMERY-SMITH, AND T. RANDOLPH

 A ? ik0 )?1 B ui
C (A ? ik0 )?1 B u = ? hkyC;(CA(? ik )?1 B uk u = ?u; 0

Y

k
k  kC (A ? ik1 )?1 B uk  sup kC (A ? 1ik )?1 B uk ?  : 0 Y 0 Y k2Z ? 1 Now set v := (A ? ik0 ) B u in X . By (30),
C v = ?u, and so (A ? ik0 + B
C )v = (A ? ik0 )v + B
C v = B u ? B u = 0: Therefore,

P ik + B
C )vk eik(
) kFper k(A ? ik0 + B
C )v eik0 (
) kFper k k (A ?P  = 0: inf fvk g2 k k uk eik(
) kFper kveik0 (
)kFper By Theorem 3.3, 1 2= (e2(A+B
C )). This shows that rc1stab (e2A ; B; C )  k
k. To nish the proof, suppose that rc1stab (e2A ; B; C ) > (supk2ZkC (A ? ik)?1 B k)?1 . Then with r := (supk2ZkC (A ? ik)?1 B ukY ? )?1 , and  > 0 chosen to be suciently small, one has

1 1 2A supk2ZkC (A ? ik)?1B ukY < r < rcstab (e ; B; C ): But then by (31), k
k  r < rc1stab (e2A ; B; C ), which is a contradiction. For the last statement of the theorem, note that Parseval's formula applied to (28) gives
?P k C (A ? ik)?1 Buk k2Y 1=2 k ? 1 (32)  sup kC (A ? ik)?1 B k: kC ?per Bk = sup P k2Z fuk g2 ( k kuk k2U )1=2 Therefore, 1 1  ? 1 sup k C ( A ? ik)?1B k kC ?per Bk k2Z and hence equalities hold in (29). Next we consider the following \hyperbolic" variant of the constant stability radius. Recall, that a strongly continuous semigroup fetA gt0 on X is called hyperbolic if (etA ) \ T = ;; where T = fz 2 C : jz j = 1g; for some (and, hence, for all) t > 0 (see, e.g., [27]). The hyperbolic semigroups are those for which the dierential equation x_ = Ax has exponential dichotomy (see, e.g., [10]) with the dichotomy projection P being the Riesz projection corresponding to the part of spectrum of eA that lies in the open unit disc. For a given hyperbolic semigroup fetAgt0 and operators B , C we de ne the constant dichotomy radius as: rcdich (fetA g; B; C ) := supfr  0 : k
kL(Y;U )  r implies (et(A+B
C ) ) \ T = ; for all t > 0g: The dichotomy radius measures the size of the smallest
2 L(Y; U ) for which the perturbed equation x_ = [A + B
C ]x looses the exponential dichotomy. Now for any  2 [0; 1], consider the rescaled semigroup generated by A := A ? i consisting of operators etA = e?it etA , t  0. The pointwise stability radius can be related to the dichotomy radius as follows. Lemma 3.5. Let fetAgt0 be a hyperbolic semigroup. Then rcdich (fetAg; B; C ) = inf rc1stab (e2A ; B; C ): 2[0;1]

STABILITY IN BANACH SPACES

17

Proof. Denote the left-hand side by  and the right-hand side by . First x r < . Let  2 [0; 1]. If k
k  r, then 1 2 (e2(A +B
C ) ) and so ei2 2 (e2(A+B
C )) for all  2 [0; 1]. That is, eis 2 (e2(A+B
C )) for all s 2 R, and so (e2(A+B
C )) \ T = ;. This shows that r  , and so  . Now suppose r < . If k
k  r, then (fet(A+B
C )g) \ T = ;, and so eit 2 (et(A+B
C )) for all  2 [0; 1]; t 2 R. That is, 1 2 (et(A +B
C )). This says r  and so   . Under the additional assumption that the semigroup fetAgt0 is exponentially stable (that is, hyperbolic with a trivial dichotomy projection P = I ), Lemma 3.5 gives, in fact, a formula for the constant stability radius. Indeed, the following simple proposition holds. Proposition 3.6. Let fetAgt0 be an exponentially stable semigroup. Then rcdich (fetA g; B; C ) = rcstab (fetAg; B; C ): Proof. Denote the left-hand side by  and the right-hand side by . Take r < and any
with k
k  r. By de nition of the constant stability radius, !0 (fet(A+B
C )g) < 0. In particular, (et(A+B
C ) ) \ T = ;, and r   shows that  . Suppose that < r <  for some r. By the de nition of the stability radius , there exists a
with k
k 2 ( ; r) such that the semigroup fet(A+B
C )gt0 is not stable. For any  2 [0; 1] one has k
k  r < . By the de nition of the dichotomy radius  it follows that the semigroup fet(A+B
C )gt0 is hyperbolic for each  2 [0; 1]. Now consider its dichotomy projection

P ( ) = (2i)?1

Z

T

?


 ? eA+B
C ?1 d;

which is the Riesz projection corresponding to the part of (eA+B
C ) located inside of the open unit disk. The function  7! P ( ) is norm continuous. Indeed, since the bounded perturbation B
C of the generator A is continuous in  , the operators et(A+B
C ), t  0, depend on  continuously (see, e.g., [28, Corollary 3.1.3]); this implies the continuity of P (
) (see, e.g., [10, Theorem I.2.2]). By assumption fetA gt0 is exponentially stable, so P (0) = I . Also, P (1) 6= I since the semigroup t fe (A+B
C )gt0 with k
k  r <  is hyperbolic but not stable. Since either kI ? P ( )k = 0 or kI ? P ( )k  1, this contradicts the continuity of kP (
)k. A review of the above development shows that the inequality claimed in (24) of Theorem 3.1 can now be proved. Proof. (of Theorem 3.1) Indeed, rstab (fetA g; B; C )  rcstab (fetA g; B; C ), and so 1  r (fetA g; B; C )  rc (fetA g; B; C ) (Theorem 3.2)

kLk

stab

stab

 rcdich (fetAg; B; C )  2inf rc1 (e2A ; B; C ) [0;1] stab  2inf [0;1] sup = sup

1

k2ZkC (A ? ik )?1 B k

(Proposition 3.6) (Lemma 3.5) (Theorem 3.4)

1

s2R kC (A ? is)?1 B k

We will need below the following simple corollary that holds for bounded generators A. (In fact, as shown in [12, Cor. 2.5], formula (33) below holds provided A generates a semigroup fetAgt0 that is uniformly continuous just for t > 0.) Corollary 3.7. Assume A 2 L(X ) generates a (uniformly continuous) stable semigroup on a Banach space X . Then (33) rcstab (fetA g; B; C ) = sup kC (A1 ? is)?1 B k : s2R

18

S. CLARK, Y. LATUSHKIN, S. MONTGOMERY-SMITH, AND T. RANDOLPH

Proof. By Theorem 3.1, it remains to prove only the inequality \". Fix
with k
k strictly less than the right-hand side of (33). Since A + B
C 2 L(X ), it suces to show that A + B
C ?  = (A ? )(I + (A ? )?1 B
C ) is invertible for each  with Re   0. By the analyticity of resolvent, supRe 0 kC (A ? )?1 B k  sups2R kC (A ? is)?1 B k. Thus, k
k < sup kC (A1 ? is)?1 B k s2R 1 1  sup kC (A ? )?1 B k  kC (A ? )?1 B k ; Re   0 Re 0

implies that I + C (A ? )?1 B
is invertible. Therefore (cf. the proof of Theorem 3.2), I + (A ? )?1 B
C is invertible. 3.4. The norm of the input-output operator. Since the lower bound on the stability radius is given by the norm of the input-output operator, which is de ned by way of the solution operators, it is of interest to express this quantity in terms of the operators A, B and C . In this subsection it is shown that for autonomous systems this quantity can, in fact, be expressed explicitly in terms of the transfer function: R ?1 is(
) dskLp (R;Y ) k R C (A R? is) Bu(s)e kLk = sup (34) : k R u(s)eis(
) dskLp(R;U ) u2S (R;U ) Here we use S (R ; X ) to denote the Schwartz class of rapidly decreasing X -valued functions de ned on R: fv : R ! X sups2R ksm v (n) (s)k < 1; n; m 2 N g. As noted in (23), kLk equals sups2R kC (A ? is)?1 B k if U and Y are Hilbert spaces and p = 2. The section concludes by providing a similar expression, involving sums, which serves as a lower bound for the constant stability radius. The current focus is on autonomous systems so let fetAgt0 be a strongly continuous semigroup generated by A and consider the evolution semigroups fERt gt0 de ned on functions on the entire real line as in (2), and fE t gt0 de ned for functions on the half-line as in (8). As before, ?R and ? will denote the generators of these semigroups on Lp (R; X ) and Lp(R+ ; X ), respectively. Both semigroups will be used as we rst show that kC ??R 1Bk equals the expression in (34) and then check that kLk  kC ??1Bk = kC ??R 1Bk. Given v 2 S (R; X ), let gv denote the function Z gv ( ) = 21 v(s)eis ds;  2 R; R and set G = fgv : v 2 S (R; X )g: Assuming sups2R k(A ? is)?1 k < 1, de ne, for a given v 2 S (R; X ), the function Z fv ( ) = 21 (A ? is)?1 v(s)eis ds;  2 R; R and set F = ffv : v 2 S (R; X )g. Proposition 3.8. Assume sups2R k(A ? is)?1k < 1. Then (i) G consists of dierentiable functions, and is dense in Lp (R; X ); (ii) F is dense in Dom(?R); (iii) if v 2 S (R; X ) then ?Rfv = gv . Proof. For g 2 L1 (R; X ), denote the Fourier transform by Z 1 g^( ) = 2 e?is g(s) ds: R Note that G = fg : R ! X : 9 v 2 S (R; X ) so that g^ = vg, and so G contains the set fg 2 L1 (R; X ) : g^ 2 S (R; X )g. Since the latter set is dense in Lp (R; X ), property (i) follows. G consists of dierentiable functions since for v 2 S (R; X ), the integral de ning gv converges absolutely. Moreover, for v 2 S (R; X ), the function w(s) = (A ? is)?1 v(s), s 2 R, is also in S (R; X ), since sups2R k(A ? is)?1 k < 1. Hence fv is dierentiable with derivative Z Z fv0 ( ) = 21 is(A ? is)?1 v(s)eis ds = 21 isw(s)eis ds: R R

STABILITY IN BANACH SPACES

19

So fv0 2 Lp (R; X ), and hence F is dense in Dom(?d=dt + A). Property (iii) follows from the following calculation: Z (?fv )( ) = 21 [?is(A ? is)?1 v(s)eis + A(A ? is)?1 v(s)eis ] ds ZR 1 = 2 (A ? is)(A ? is)?1 v(s)eis ds = gv ( ): R

Set S = fv 2 S (R; X ) : v(s) 2 Dom(A) for s 2 R; Av 2 S (R; X )g. Proposition 3.9. Let fetAgt0 be a strongly continuous semigroup generated by A. Let ? and ?R be the generators of the evolution semigroups on Lp (R+ ; X ) and Lp(R; X ) , as de ned in (8) and (2), respectively. Then the following assertions hold: (i) if (A) \ iR = ; and sups2R k(A ? is)?1 k < 1 then R ? is)v(s)eis(
) dskLp(R;X ) ; k R(A R k?Rk;Lp(R;X ) = vinf 2S k R v(s)eis(
) dskLp(R;X ) (ii) if ?R is invertible on Lp (R; X ) , then fetA gt0 is hyperbolic and R ?1 eis(
) dskLp (R;X ) k??R 1kL(Lp(R;X )) = sup k R(Ak R? isv()s)eisv((s
))ds kLp(R;X ) ; v2S (R;X ) R

(iii) if ? is invertible on Lp (R+ ; X ), then fetA gt0 is exponentially stable and k??1kL(Lp(R+;X )) = k??R 1 kL(Lp(R;X )) : Proof. To show (i) let v 2 S (R; X ). Since sups2R k(A ? is)?1 k < 1, the formula w(s) = (A ? is)?1 v(s); s 2 R, de nes a function, w, in S . Now, Z Z 1 1 ? 1 is gv ( ) = 2 (A ? is)(A ? is) v(s)e ds = 2 (A ? is)w(s)eis ds R R and Z fv ( ) = 21 w(s)eis ds: R However, from Proposition 3.8, R ? is)w(s)eis(
) dsk : k k g k k ? v Rfv k R(A R = inf = inf k?Rk = finf v2S (R;X ) kfv k w2S k R w(s)eis(
) dsk v 2F kfv k To see (ii) note that ?1  k ? Rfv k ? 1 ? 1 k? k = k? k = inf = sup kfv k : 

v2S (R;X ) kgv k For (iii) note that k?Rk;Lp(R;X )  k?k;Lp(R+;X ). Indeed, let f 2 Lp (R; X ) with supp f  R+ . If f 2 Dom(?R), then supp ?Rf  R+ and k?Rf kLp(R;X ) = k?f kLp(R+;X ) . To see that k?Rk  k?k , let  > 0 and choose f 2 Dom(?R) with compact support such that kf kLp(R;X ) = 1 and k?Rk  k?Rf k ? . Now choose  2 R such that f (s) := f (s ?  ), s 2 R, de nes a function, f 2 Lp (R; X ), with supp f  R+ . Let f denote the element of Lp (R+ ; X ) which coincides with f on R+ . Then kf k = kf k and ?f = ?d=dt f (
?  ) + Af (
?  ) = (?Rf ) . Therefore, k?Rk  k?Rf k ?  = k(?Rf ) k ?  = k?Rf k ?   k?k ? . Proposition 3.10. The set GU = fgu : u 2 S (R; U )g is dense in Lp(R; U ). If u 2 S (R; U ) and B 2 L(U; X ) then Bu 2 S (R; X ) and ?RfBu = Bgu. R

R

v2S (R;X )

kfv k

20

S. CLARK, Y. LATUSHKIN, S. MONTGOMERY-SMITH, AND T. RANDOLPH

Proof. The rst statement is clear, as in Proposition 3.8. The second follows from the properties of Schwartz functions, and from the calculation: Z Z 1 1 is ?RfBu = gBu ( ) = 2 Bu(s)e ds = B 2 u(s)eis ds: R R

Recall, see Theorem 1.4 and Theorem 2.2, that fetAgt0 is hyperbolic (resp., stable) if and only if ?R (resp., ?) is invertible on Lp (R; X ) (resp., Lp (R+ ; X )). Theorem 3.11. If ?R is invertible on Lp(R; X ) , then R ?1 is(
) dskLp (R;Y ) k ? 1 R C (A R? is) Bu(s)e (35) : kC ?R Bk = sup k R u(s)eis(
) dskLp(R;U ) u2S (R;U ) If ? is invertible on Lp (R+ ; X ), then the norm of L = C ??1B, as an operator from Lp (R+ ; U ) to Lp (R+ ; Y ), is given by the above formula: (36) kLk = kC ??R 1Bk: If, in addition, U and Y are Hilbert spaces and p = 2, then (37) kLk = sup kC (A ? is)?1 B kL(U;Y ) : s2R

Proof. For u 2 S (R; U ), consider functions fBu and gu . Proposition 3.10 gives fBu = ??R 1 Bgu and ?1 kC ??R 1Bk = sup kC ?R BgupkLp(R;Y ) = sup kC fBuk

kgu kL (R;U ) gu 2GU kgu k R ? 1 k R C (AR? is) Bu(s)eis(
) dskLp(R;Y ) ; = sup k R u(s)eis(
) dskLp(R;U ) u2S (R;U ) gu 2GU

which proves (35). Now, if ? is invertible on Lp (R+ ; X ), then fetAgt0 is exponentially stable by Corollary 2.6. Hence, ?R is invertible on Lp (R; X ). Moreover, for the case of the stable semigroup fetAgt0 , the formula for ??R 1 (see, e.g., [22]) takes the form (??R 1 f )(t) = If supp f  (0; 1), then (38)

1

Z

(??R 1 f )(t) =

0 Z

esA f (t ? s) ds =

t

?1

t

Z

e(t?s)A f (s) ds =

?1 t

Z

0

e(t?s)A f (s) ds:

e(t?s)A f (s) ds:

For a function h 2 Lp(R+ ; X ), de ne an extension h~ 2 Lp (R; X ) by h~ (t) = h(t) for t  0 and h~ (t) = 0 for t < 0. Then (38) shows that ??R 1 h~ = (??1 h) . In particular, for u 2 Lp (R+ ; U ), 1 ~. Therefore, ?1Bu = C ?? Lf u = C ?^ R Bu kLukLp (R+;Y ) = kLfu kLp(R;Y ) = kC ??R 1 Bu~kLp(R;Y )  kC ??R 1Bk
ku~kLp(R;U ) = kC ??R 1Bk
kukLp(R+;U ) : This shows that kLk  kC ??R 1Bk. To prove that equality holds in (36), let  > 0 and choose u 2 Lp (R; U ), kuk = 1, such that kC ?R?1BukLp(R;Y )  kC ??R 1Bk ? . Without loss of generality, u may be assumed to have compact support. Now choose r such that supp u(
? r)  (0; 1) and set w(
) := u(
? r). Then w 2 Lp (R; U ) with supp w  (0; 1). Let w denote the element of Lp (R+ ; U ) that coincides with w on R+ . As in (38) we have

C ??R 1Bw(t) = C

t

Z

0

e(t?s)A Bw(s) ds = C

Z

t

?1

e(t?s)A Bw(s) ds:

STABILITY IN BANACH SPACES

21

Since kwkLp(R+;U ) = kwkLp(R;U ) = kukLp(R;U ) = 1, it follows that kLk  kLwkLp(R+;Y ) = kLfwkLp(R;Y ) = kLw~kLp(R;Y ) = kC ??R 1BwkLp(R;Y ) Z




e(
? )ABu( ) d kLp (R;Y ) ?1 = kC ??1BukLp(R;Y )  kC ??1Bk ? :

= kC

R

R

This con rms (36). Parseval's formula and (24) give (37). If fetAgt0 is exponentially stable then the inequalities in (24) give lower and upper bounds on the stability radius in terms of L and C (A ? is)?1 B , respectively. The previous theorem shows that kLk can be explicitly expressed in terms of an integral involving C (A ? is)?1 B . We conclude by observing that a lower bound for the constant stability radius can be expressed by a similar formula involving a sum. For this, let  2 [0; 1] and set P ?1 ik(
) k k C (A ?Pi ? ik ) Buk e kLp ([0;2];Y ) : S := sup ik(
) p

k

fuk g2

k uk e

kL ([0;2];U )

We note that S is computed as in equation (28) with A replaced by A = A ? i . Corollary 3.12. Let fetAgt0 be an exponentially stable semigroup generated by A. Then 1 1 tA g; B; C )   rc ( f e stab sup2[0;1] S sups2R kC (A ? is)?1 B k : Proof. Fix  2 [0; 1], and let ?per; denote the generator on Lp ([0; 2]; X ) of the evolution semigroup 1 Bk = S , and so by Theorem 3.4, induced by fetA gt0 . By Theorem 3.3, kC ??per;  1 1 1 2A S  rcstab (e ; B; C )  supk2ZkC (A ? ik)?1 B k : By Proposition 3.6, taking the in mum over  2 [0; 1] gives 1 rc1 (e2A ; B; C ) = rcstab (fetAg; B; C ) [0;1] stab sup2[0;1] S  2inf 1  2inf [0;1] supk2ZkC (A ? ik )?1 B k = sup kC (A1 ? is)?1 B k : s2R

3.5. Two counterexamples. In contrast to the Hilbert space setting, the following Banach space examples show that either inequality in (24) may be strict. We start with the example where the second inequality in (24) is strict. Example 3.13. An example due to W. Arendt (see, e.g., [27], Example 1.4.5) exhibits a (positive) strongly continuous semigroup fetAgt0 on a Banach space X with the property that s0 (A) < !0 (A) < 0 for the abscissa of uniform boundedness of the resolvent and the growth bound. Now, for  such that 0    ?!0 (A), consider a rescaled semigroup generated by A + , and denote by ?A+ the generator of the induced evolution semigroup on Lp (R+ ; X ). The following relationships hold: for 0   < ?!0 (A); s0 (A + ) = s0 (A) +  < !0 (A) +  = !0 (A + ) < 0; for 0 := ?!0 (A); s0 (A + 0 ) < !0 (A + 0 ) = 0: This says that s0 (A + ) < 0 for all  2 [0; 0] and hence M := sup sup k(A +  ? is)?1k < 1: 2[0;0 ] s2R

22

S. CLARK, Y. LATUSHKIN, S. MONTGOMERY-SMITH, AND T. RANDOLPH

Now note (see Corollary 2.6) that !0 (A + ) < 0 if and only if k??A+1 k < 1. Since !0 (A + ) ! 0 as  ! 0 , we conclude that k??A+1  k ! 1 as  ! 0 . Since  7! k??A+1  k is a continuous function of  on [0; 0 ), there exists 1 2 [0; 0 ) such that k?A?+1 1 k > M , and so the following inequality is strict: 1 1 1 kLk = k??A+1 1 k < sups2R k(A + 1 ? is)?1 k : Also, we claim that there exists 2 2 [0; 0 ) such that the following inequality is strict: rcstab (fet(A+2 ) g; I; I ) < sup k(A +1  ? is)?1 k : 2 s2R To see this, let us suppose that for each  2 [0; 0 ) one has rcstab (fet(A+) g; I; I )  1=(2M ). Again, using that !0 (A + ) ! 0 as  ! 0 , nd  2 [0; 0) such that j!0 (A + )j < 1=(2M ). Let
= !0 (A + )I . Since k
k = j!0 (A + )j, by the de nition of stability radius one has: 0 > !0 (A +  +
) = !0 (A + ) ? !0 (A + ) = 0; a contradiction. Thus, there exists 2 2 [0; 0) such that 1 1 < 1  rcstab (fet(A+2 ) g; I; I )  2M M sups2R k(A + 2 ? is)?1 k ; as claimed. } This example shows that the second inequality in (24) can be strict due to the Banach-space pathologies related to the failure of Gearhart's Theorem 1.1. Another example, given below, shows that the rst inequality in (24) could be strict due to the lack of Parseval's formula (see (32) in the proof of Theorem 3.4): That is, the choice of p = 2 in (23) is as important as the fact that X in (23) is a Hilbert space. First, we need a formula for the norm of the input-output operator on L1 (R+ ; X ). Proposition 3.14. Assume fetAgt0 is an exponentially stable C0 semigroup on a Banach space X . The norm of the operator L = ??1 on L1 (R+ ; X ) is Z1 ? 1 k? kL(L1(R+;X )) = sup ketAxk dt: kxk=1 0

(39) Proof. Recall, see (9), that

??1 f (t) = ?

Zt

0

eAf (t ?  ) d; t 2 R+ ; f 2 L1 (R+ ; X )

is the convolution operator. Choose positive n 2 L1 (R+ ; R) with kn kL1 = 1 such that kg  n ? gkL1(R+;X ) ! 0 as n ! 1 for each g 2 L1 (R+ ; X ): Fix x 2 X , kxk = 1, let f = n x 2 L1 (R+ ; X ) and note that ??1 f (t) = ?

Zt

0

eAx n (t ?  ) d = ?(g  n )(t) for g(t) = etA x; t 2 R+ :

This implies \" in (39). To see \", take f = Ni=1 P i xi with i 2 L1 (R+ ; R) having disjoint supports and kxi k = 1, i = 1; : : : ; N . Now kf kL1(R+;X ) = i ki kL1 and for fi (t) = etA xi one has ??1 f (t) = ?

P

Zt X

0

i

eAxi i (t ?  ) d = ?

X

i

(fi  i )(t):

STABILITY IN BANACH SPACES

Using Young's inequality,

k??1 f kL1(R+;X ) 

X



X

i i

23

kfi  i kL1(R+;X ) kfi kL1(R+;X ) ki kL1  sup

Z1

kxk=1 0

ketAxk dt

X

i

ki kL1 :

Example 3.15. Take X = C 2 with the `1 norm. Let     1 1 tA = e?t cos(t) e?t sin(t) ; A= ? so that e ?1 ?1 ?e?t sin(t) e?t cos(t) and





1 ?1 ? is ?1 : 1 ?1 ? is (1 + is)2 + 1 Since the extreme points of X are ei e1 and ei e2 ( 2 R), where e1 and e2 are the unit vectors of C 2 , we see that k(A ? is)?1 k = j(1j1++isis)j2++11j : It may be numerically established that sup k(A ? isI )?1 k  1:087494476: (A ? is)?1 =

s2R

By Corollary 3.7, the reciprocal to the last expression is equal to rcstab (fetA g; I; I ). On the other hand, using Proposition 3.14,

kLk = k??A1k = sup =

Z

0

1

Z

kxk=1 0

1

ketA xk dt

je?t cos(t)j + je?t sin(t)j dt  1:262434309:

Therefore, the rst inequality in (24) may be strict. The following example shows that the norm of the input-output operator depends on p. Example 3.16. Let   9 = 2 ? 5 = 2 A = 25=2 ?13=2 ; acting on C 2 with the Euclidean norm. Thus   cos t + (11 = 2) sin t ? (5 = 2) sin t tA ? t e =e (25=2) sin t cos t ? (11=2) sin t : Then Z 1 k??1 kL1!L1  ketA e1k dt  7:748310791; whereas

}

0

k??1 kL2!L2 = sup k(A ? is)?1 k  2:732492852: s2R

}

24

S. CLARK, Y. LATUSHKIN, S. MONTGOMERY-SMITH, AND T. RANDOLPH

4. Internal and External Stability Work aimed at properties of stability and robustness of linear time-invariant systems is often based on transform techniques. More speci cally, if the transfer function H () = C (A ? )?1 B is a bounded analytic function of  in the right half-plane C + = f 2 C : Re  > 0g, then the autonomous system (21) is said to be externally stable. This property is often used to deduce internal stability of the system, i.e., the uniform exponential stability of the nominal system x_ = Ax. The relationship between internal and external stability has been studied extensively; see, e.g., [1, 7, 6, 18, 23, 37, 38] and the references therein. In this section we examine the extent to which these techniques apply to Banach-space settings and time-varying systems. For this, input-output stability of the system (20) will refer to the property that the input-output operator L is bounded from Lp (R+ ; U ) to Lp (R+ ; Y ). If internal stability is assumed initially, then the inequalities in (24) exhibit a relationship between these concepts of stability. The next two theorems look at these relationships more closely and show, in particular, when internal stability may be deduced from one of the \external" stability conditions. Therefore, throughout this section fU (t;  )gt will denote a strongly continuous exponentially bounded evolution family that is not assumed to be exponentially stable. 4.1. The nonautonomous case. In this subsection we give a very short proof of the fact that for general nonautonomous systems on Banach spaces, internal stability is equivalent to stabilizability, detectability and input-output stability. Before proceeding, it is worth reviewing some properties of time-invariant systems. For this, let fetAgt0 be a strongly continuous semigroup generated by A on X , and let H+1 (L(X )) denote the space of operator-valued functions G : C ! L(X ) which are analytic on C + and sup2C + kG()k < 1. If X is a Hilbert space, it is well known that fetAgt0 is exponentially stable if and only if  7! ( ? A)?1 is an element of H+1 (L(X )); see, e.g., [9], Theorem 5.1.5. This is a consequence of the fact that when X is a Hilbert space, s0 (A) = !0 (etA ) (see [27] or Theorem 1.1). If X is a Banach space, then strict inequality s0 (A) < !0 (etA ) can hold, and so exponential stability is no longer determined by the operator G() = ( ? A)?1 . Extending these ideas to address systems (21), one considers H () = C ( ? A)?1 B : it can be shown that if U and Y are Hilbert spaces, then (21) is internally stable if and only if it is stabilizable, detectable and externally stable (i.e., H (
) 2 H+1 (L(U; Y ))). See R. Rebarber [37] for a general result of this type. It should be pointed out that this work of Rebarber and others more recently allows for a certain degree of unboundedness of the operators B and C . Such \regular" systems (see [42]), and their time-varying generalizations, might be addressed by combining the techniques of the present paper (including the characterization of generation of evolution semigroups as found in [35]) along and with those of [15] and [17]. This will not be done here. If one allows for Banach spaces, the conditions of stabilizability and detectability are not sucient to ensure that external stability implies internal stability. Indeed, let A generate a semigroup for which s0 (A) < !0 (etA ) = 0 (see Example 3.13). Then the system (21) with B = I and C = I is trivially stabilizable and detectable and externally stable. But since !0 (etA ) = 0, it is not internally stable. Since the above italicized statement concerning external stability fails for Banach-space systems (21) and does not apply to time-varying systems (20), we aim to prove the following extension of this. Theorem 4.1. The system (20) is internally stable if and only if it is stabilizable, detectable and input-output stable. This theorem appears as part of Theorem 4.3 below. A version of it for nite-dimensional timevarying systems was proven by B. D. O. Anderson in [1]. The fact that Theorem 4.1 actually extends the Hilbert-space statement above follows from the fact that the Banach-space inequality sup2C + kH ()k  kLk (see [43]) which relates the operators that de ne external and input-output stability is actually an equality for Hilbert-space systems (see also [42]). In Theorems 4.1 and 4.3, below, the following de nitions are used. De nition 4.2. The nonautonomous system (20) is said to be

STABILITY IN BANACH SPACES

25

(a) stabilizable if there exists F (
) 2 L1 (R+ ; Ls (X; U )) and a corresponding exponentially stable evolution family fUBF (t;  )gt such that, for t  s and x 2 X , one has: (40)

UBF (t; s)x = U (t; s)x +

t

Z

s

U (t;  )B ( )F ( )UBF (; s)x d ;

(b) detectable if there exists K (
) 2 L1 (R+ ; Ls (Y; X )) and a corresponding exponentially stable evolution family fUKC (t;  )gt such that, for t  s and x 2 X , one has: (41)

UKC (t; s)x = U (t; s)x +

t

Z

s

UKC (t;  )K ( )C ( )U (; s)x d:

An autonomous control system is called stabilizable if there is an operator F 2 L(X; U ) such that A + BF generates a uniformly exponentially stable semigroup; that is, !0 (A + BF ) < 0. Such a system is detectable if there is an operator K 2 L(Y; X ) such that A + KC generates a uniformly exponentially stable semigroup. Using Theorem 2.5 to characterize exponential stability in terms of the operator G as in (10) makes the proof of the following theorem a straightforward manipulation of the appropriate operators. Theorem 4.3. The following are equivalent for a strongly continuous exponentially bounded evolution family of operators U = fU (t;  )gt on a Banach space X . (i) U is exponentially stable on X ; (ii) G is a bounded operator on Lp (R+ ; X ); (iii) system (20) is stabilizable and G B is a bounded operator from Lp (R+ ; U ) to Lp(R+ ; X ); (iv) system (20) is detectable and C G is a bounded operator from Lp (R+ ; X ) to Lp (R+ ; Y ); (v) system (20) is stabilizable and detectable and L = C G B is a bounded operator from Lp (R+ ; U ) to Lp (R+ ; Y ). Proof. The equivalence of (i) and (ii) is the equivalence of (i) and (ii) in Theorem 2.5. To see that (ii) implies (iii), (iv), and (v), note that B and C are bounded, and thus L is bounded when G is bounded. So when (ii) holds, the exponential stability of U , together with boundedness of B (
), C (
), F (
) and K (
), assure the existence of the evolution families fUBF (t;  )gt and fUKC (t;  )gt as solutions of the integral equations in De nition 4.2; thereby showing that (iii), (iv), and (v) hold. To see that (iii) ) (ii), rst note that the assumption of stabilizability assures the existence of an exponentially stable evolution family UBF = fUBF (t;  )gt satisfying equation (40) for some F (
) 2 L1 (R+ ; Ls (X; U )). Given this exponentially stable family, we de ne the operator G BF by

(42)

G BF f (s) :=

s

Z

0

UBF (s;  )f ( ) d =

1

Z

0

 f )(s) d (EBF

 f gt0 is the semigroup induced by the evolution family UBF as described in equation (5). where fEBF G BF is a bounded operator on Lp (R+ ; X ) by the equivalence of (i) and (ii). For f (
) 2 Lp(R+ ; X ) and s 2 R+ , take x = f (s) in equation (40). Then, let  =  ? s, to obtain

UBF (t; s)f (s) = U (t; s)f (s) +

t?s

Z

0

U (t;  + s)B ( + s)F ( + s)UBF ( + s; s)f (s) d:

t gt0 we obtain From this equation and from the de nition of the semigroups fE t gt0 , and fEBF t?s f )(t) = (E t?s f )(t) + (EBF

and hence for 0  r and 0   that r f )( ) = (E r f )( ) + (EBF

Integrate from 0 to 1 to obtain (G BF f )() = (G f )() +

Z

0

t?s

Z

0

r

Z

0

 f )( ) d: (E r? BF EBF

1Z r 0

 f )(t) d (E t?s? BF EBF

 f )( ) d dr: (E r? BF EBF

26

S. CLARK, Y. LATUSHKIN, S. MONTGOMERY-SMITH, AND T. RANDOLPH

Let r =  +  and  =  to obtain

Z

1Z 1

 f )( ) d d (E  BF EBF (G BF f )() = (G f )( ) + 0 0 = (G f )( ) + (G BF G BF f )(): That G is bounded now follows from equation (43), the boundedness of G B, and the boundedness of G BF and F . To see that (iv) ) (ii), rst note that the assumption of detectability assures the existence of an exponentially stable evolution family UKC = fUKC (t;  )gt satisfying equation (41) for some K (
) 2 L1 (R+ ; Ls (Y; X )). Given this exponentially stable family, the operator G KC , de ned in a manner analogous to G BF in equation (42), is a bounded operator on Lp (R+ ; X ). A derivation beginning with equation (41), and similar to that which gave equation (43), now gives G KC = G + G KC KC G . This equation, together with the assumed boundedness of G KC , K, and C G , gives the boundedness of G . Finally, to see that (v) ) (ii), again note that the assumption of detectability yields an exponentially stable evolution family UKC and an associated bounded operator G KC . For u(
) 2 Lp (R+ ; U ), and s 2 R+ take x = B (s)u(s) in equation (41). A calculation similar to that which gave equation (43) now gives G KC B = G B + G KC KC G B. The assumed boundedness of L = C G B, K, and G KC , now yields the boundedness of G B . The boundedness of G B together with the assumption of stabilizability implies that G is bounded by the equivalence of (iii) and (ii). 4.2. The autonomous case. The main result of this subsection is Theorem 4.4 which builds on Theorem 3.11 and parallels Theorem 4.3 for autonomous systems of the form (21). The main point is to provide explicit conditions, in terms of the operators A, B and C , which imply internal stability. Let A := A ? I denote the generator of the rescaled semigroup fe?tetA gt0 . Theorem 4.4. Let fetAgt0 be a strongly continuous semigroup on a Banach space X generated by A. Let U and Y be Banach spaces and assume B 2 L(U; X ) and C 2 L(X; Y ). Then the following are equivalent. (i) fetAgt0 is exponentially stable;

(43)

(ii)

G

is a bounded operator on Lp (R+ ; X ); R

k R(AR? is)?1 v(s)eis(
) dskLp(R;X ) < 1 (iii) (A) \ C + = ; and sup k R v(s)eis(
) dskLp(R;X ) v2S (R;X ) for all   0; R is)?1 Bu(s)eis(
) dskLp(R;X ) k R(A ? R (iv) (A) \ C + = ;, sup