EQUIVALENT CONDITIONS FOR EXPONENTIAL STABILITY FOR A SPECIAL CLASS OF CONSERVATIVE LINEAR SYSTEMS

EQUIVALENT CONDITIONS FOR EXPONENTIAL STABILITY FOR A SPECIAL CLASS OF CONSERVATIVE LINEAR SYSTEMS G. WEISS Λ, M. Tucsnak y Λ Dept. of Electrical and Electronic Engineering Imperial College London Exhibition Road, London SW7 BT, UK e-mail: G.Weiss@ ic.ac.uk y Department of Mathematics University of Nancy-I, POB 39 Vandoeuvre les Nancy 5456, France e-mail: Marius.Tucsnak@ iecn.u-nancy.fr Keyords: Conservative linear system, exact controllability, exponential stability, optimizability, estimatability Abstract Let A be a possibly unbounded positive operator on the Hilbert space H, hich is boundedly invertible. Let C be a bounded operator from D(A ) (ith the norm kzk = ha z; zi) to another Hilbert space U. It is knon that the system of equations z(t) +A z(t) + CΛ C _z(t) = C Λ u(t); y(t) = C _z(t) +u(t); determines a ell-posed linear system ± ith input u and output y, input and output space U and state space X = D(A ) H. Moreover, ± is conservative, hich means that a certain energy balance equation is satisfied both by the trajectories of ± and by those of its dual system. In this paper e present various conditions hich are equivalent to the exponential stability of such a systems. Among the equivalent conditions are exact controllability and exact observability. Denoting V (s) = s I + s CΛC + A ; e also obtain that the system is exponentially stable if and only if s! A V (s) is a bounded L(H)-valued function on the imaginary axis. This is also equivalent to the condition that s! sv (s) is a bounded L(H)-valued function on the imaginary axis (or equivalently, on the open right half-plane). Introduction and main results In our paper [5] e have investigated a class of conservative linear systems ith a special structure, described by a second order differential equation (in a Hilbert space) and an output equation, see (.) and (.3) belo. Such systems occur frequently in applications, such as ave equations and beam equations, see for example [], [4], [4], [5] and the references therein. For this reason, it is useful to have criteria for their exponential stability. We recall the construction from the paper [5], in order to be able to state the ne results. Let H be a Hilbert space, and let A : D(A )! H be a self-adjoint, positive and boundedly invertible operator. We introduce the scale of Hilbert spaces H ff, ff R, as follos: for every ff, H ff = D(A ff ), ith the norm kzk ff = ka ff zk H. The space H ff is defined by duality ith respect to the pivot space H as follos: H ff = Hff Λ for ff>. Equivalently, H ff is the completion of H ith respect to the norm kzk ff =fl A ff zfl H. The operator A can be extended (or restricted) to each H ff, such that it becomes a bounded operator A : H ff! H ff 8 ff R : Let C be a bounded linear operator from H to U, here U is another Hilbert space. We identify U ith its dual, so that U = U Λ. We denote B = C Λ, so that B L(U; H ). The class of systems studied in [5] and also here is described by d dt z(t) +A z(t) + B d dt C z(t) =B u(t); (.) z() = z ; _z() = ; (.) y(t) = d dt C z(t) +u(t); (.3) here t [; ) is the time. The equation (.) is understood as an equation in H, i.e., all the terms are in H. Most of the linear equations modelling the damped vibrations of elastic structures can be ritten in the form (.), here z stands for d the displacement field and the term B dt C z(t), informally ritten as B C _z(t), represents a viscous feedback damping. The signal u(t) is an external input ith values in U (often a displacement, a force or a moment acting on the boundary) and the signal y(t) is the output (measurement) ith values in U as ell. The state x(t) of this system and its state space X are

defined by x(t) = z(t) ; X = H _z(t) H: We use the standard notation for certain function spaces, such as H n (; ; W ), Hloc n (; ; W ), Cn (; ; W ) and BC n (; ; W ) (ith n f; ; ;:::g). BC stands for bounded and continuous. We rite C instead of C. We assume that the reader understands the concepts of a ellposed linear system and of a conservative linear system, see for example [], [4], [5, Sections,3,4]. The first main result of [5] has been the folloing: Theorem. With the above assumptions, the equations (.)(.3) determine a conservative linear system ±, in the folloing sense: There exists a conservative linear system ± hose input and output spaces are both U and hose state space is X. Ifu z L ([; );U) is the input function, x = X is the z initial state, x = is the corresponding state trajectory and y is the corresponding output function, then () z BC(; ; H ) BC (; ; H) Hloc (; ; H ): () The to components of x are related by = _z. (3) C z H (; ; U ) and the equations (.) (in H ) and (.3) (in U) hold for almost every t (hence, y L ([; );U)). If _z is a continuous function of t, ith values in H (see Theorems. and.4 in [5] for sufficient conditions for this), then (.) and (.3) can be reritten as z(t) +A z(t) + B C _z(t) = B u(t); (.4) y(t) = C _z(t) +u(t): (.5) We introduce the space Z = H + A B U, hich is a Hilbert space if e define on it a suitable norm, see [5, Theorem.]. We can rerite the equations (.4), (.5) as a first order system as follos: ρ _x(t) = Ax(t) +Bu(t); (.6) y(t) = Cx(t) +u(t); here I A = A B ; B = ; (.7) C B ρ z ff D(A) = H H fi A z + B C H ; (.8) C : Z H! U; C = [ C ] : (.9) We denote by C the restriction of C to D(A). A is the generator of a strongly continuous semigroup of contractions on X, denoted T =(Tt) t. For the concepts of semigroup generator, control operator, observation operator and transfer function of a ell-posed linear system, e refer to [], []. We denote by C! the open right half-plane in C here Re s>!. We kno from [5, Proposition 5.3] that for any s ρ(a) (in particular, for any s C ), the operator s I + A + s B C L(H ;H ) has a bounded inverse denoted V (s): V (s) = s I + A + s B C L(H ;H ): (.) The folloing proposition is a restatement of a part of Theorem.3 in [5]. Proposition. With the notation of Theorem. and (.7) (.), the semigroup generator of ± is A, its control operator is B and its observation operator is C. The transfer function of ± is given for all s C by G(s) =C(sI A) B + I = I C sv (s)b and e have kg(s)k for all s C. No e have all the necessary ingredients to state the ne results of this paper. The folloing theorems use various controllability, observability and stability concepts. The precise definition of these concepts is given in Section. Theorem.3 With the above notation, the folloing assertions are equivalent: () The pair (A; B) is exactly controllable (in some finite time). () The pair (A; C) is exactly observable (in some finite time). (3) The semigroup T is exponentially stable. (4) The pair (A; B) is optimizable. (5) The pair (A; C) is estimatable. (6) We have sup ka V (s)k sc L(H) <. (7) We have sup ksv (s)k sc L(H) <. (8) For a dense subset of R, denoted E, e have ie ρ ρ(a) and sup ka V (i!)k L(H) < :!E (9) For a dense subset of R, denoted E, e have ie ρ ρ(a) and sup k!v (i!)k L(H) < :!E The equivalence of ()(5) remains valid for every conservative system. This fact, the theorem above, the other ne theorems

stated in this paper and various other results ill be proved in the journal version of this paper, see [9]. By a ell-knon theorem of Jan Prüss and Huang Falun, an operator semigroup T ith generator A is exponentially stable if and only if (si A) is uniformly bounded on C. In the specific case of the semigroup generated by A from (.7)-(.8), the resolvent (si A) can be ritten as a matrix of operators: (si A) = s [I V (s)a ] V (s) V (s)a sv (s) Thus, to verify that the stability condition (3) in Theorem.3 holds, e ould have to verify that the four entries of this matrix are all uniformly bounded on C. Hoever, conditions (6) and (7) in Theorem.3 tell us that, in fact, e only have to verify one entry: one of those in the second column in the matrix. Conditions (8) and (9) tell us that, in fact, the boundedness of one of these to entries of (si A) has to be verified only on a dense part of the imaginary axis, and e can still conclude exponential stability. The version of this theorem corresponding to bounded B and C, i.e., ith C L(H; U), is in Liu [5, Sections -3] (ithout conditions (4)(7) and (9)). Using the boundedness of C (and hence also of B ), Liu as able to give in [5, Theorem 3.4] also other, Hautus-type conditions hich are equivalent to the exponential stability of T. For unbounded C, e ere only able to obtain a Hautus-type estimate as a necessary condition for exponential stability, see Proposition.7. We mention that semigroups of the type discussed in this paper do not necessarily satisfy the spectrum determined groth condition. For a counterexample (a damped ave equation on a compact manifold) see Lebeau [4]. Recall that kzk denotes the norm of z in H. In the proof of Theorem.3 (more precisely, to sho that (6)=)(3)) e use the folloing proposition, hich is of independent interest. For bounded C, this proposition follos easily from [5, Theorem 3.4], but for unbounded C the proof is more delicate (it ill be given in the journal version of this paper). Related results for a bounded (but not necessarily positive) operator in place of C Λ C ere given in Liu, Liu and Rao [6]. Proposition.4 With the above notation, if C is bounded from belo, in the sense that there exists a c>such that kc zk U ckzk 8 z H ; then T is exponentially stable. Some background concerning the concepts used in the first section In this section e recall some controllability, observability and stability concepts, quoting the relevant literature. We also list the more interesting ne results hich are needed as intermediate steps in the proofs of our main results. : Throughout this section, U, X and Y are Hilbert spaces and A : D(A)! X is the generator of a strongly continuous semigroup T = (Tt) t on X. The space X is D(A) ith the norm kzk = k(fii A)zk, here fi ρ(a) is fixed, hile X is the completion of X ith respect to the norm kzk = k(fii A) zk. We assume that the reader understands the concept of an admissible (in particular, infinite-time admissible) control operator for T. IfB L(U; X ) is admissible, then for every fi e denote by Φ fi the operator Φ fi u = Z fi TtffBu(ff)dff: (.) We have Φ fi L(L ([; );U);X). IfB is admissible, then for every x X and every u L ([; );U), the function x(t) = Ttx +Φ t u is called the state trajectory corresponding to the initial state x and the input function u. We have x Hloc (; ; X) and _x(t) = Ax(t) +Bu(t) (equality in X ) for almost every t. If, moreover, B is infinite-time admissible, then e denote ~Φu = lim fi! Z fi TtBu(t)dt; (.) and e have Φ ~ L(L ([; );U);X). Similarly, e assume that the reader understands the concepts of an admissible (in particular, infinite-time admissible) observation operator for T, also presented in Section I.. If C L(X ;Y) is admissible, then e denote by Ψ the unique continuous operator from X to L loc ([; );Y) such that (Ψx )(t) =C Tt x 8 x D(A): (.3) In particular, if C is infinite-time admissible, then Ψ L(X; L ([; );Y)). Recall that B is an (infinite-time) admissible control operator for T if and only if B Λ is an (infinitetime) admissible observation operator for T Λ. Definition. Let A be the generator of a strongly continuous semigroup T on X and let B L(U; X ) be an admissible control operator for T. The pair (A; B) is exactly controllable in time T >, iffor every x X there exists a u L ([;T];U) such that Φ T u = x. (A; B) is exactly controllable if the above property holds for some T >. (A; B) is optimizable if for any x X, there exists u L ([; );U) such that the state trajectory corresponding to x and u is in L ([; );X). Clearly, exact controllability implies optimizability. Optimizability is one possible generalization of the concept of stabilizability, as knon from finite-dimensional control theory. We refer to [3] for details on optimizability and estimatability. No e introduce the corresponding observability concepts via duality.

Definition. Suppose that C L(X ;Y) is an admissible observation operator for T (equivalently, C Λ is an admissible control operator for the adjoint semigroup T Λ ). We say that (A; C) is exactly observable (in time T )if(a Λ ;C Λ ) is exactly controllable (in time T ). (A; C) is called estimatable if (A Λ ;C Λ ) is optimizable. Let Ψ be the operator defined in (.3) and for every fi put Ψ fi = P fi Ψ. Then (A; C) is exactly observable in time T > if and only if Ψ T is bounded from belo. Recall that the groth bound of a strongly continuous semigroup T is! (T) = lim t! t log kttk =inf t> t log kttk, see for example Pazy [7]. The semigroup T is exponentially stable if its groth bound is negative:! (T) <. Let T be a strongly continuous semigroup on X, ith generator A. A ell-knon spectral mapping result of Prüss [8, p. 85] implies that if the function k(si A) k is bounded on C, then T is exponentially stable. A little later and independently, this result as explicitly stated and proved by Huang Falun []. A short proof as given in Weiss [, Section 4]. Here e need a result hich is closely related to the one just mentioned, ithout being an obvious consequence of it. The result is very slightly more general than another result of Huang Falun, see [, Theorem 3]. Moreover, the proposition belo gives an estimate for the groth bound! (T). Proposition.3 Let T be a strongly continuous semigroup on X ith generator A. Assume that! (T) and E is a dense subset of R such that ie ρ ρ(a) and k(i!i A) km 8! E; for some M >. Then T is exponentially stable, more precisely,! (T) M. Proposition.4 Suppose that X, T, A, U and B are as in Definition.. Then the folloing three statements are equivalent: () T is exponentially stable. () (A; B) is optimizable, C ρ ρ(a) and, for some M>, fl (si A) Bfl L(U;X ) M 8 s C : (3) (A; B) is optimizable,! (T), there exists a dense subset of R, denoted E, such that ie ρ ρ(a) and, for some M >, fl (i!i A) Bfl L(U;X ) M 8! E: Recall that for any ell-posed linear system ± ith input function u, state trajectory x and output function y, x(fi ) P fi y =± fi x() P fi u ; (.4) here P fi denotes the truncation of a function to [;fi] and Tfi Φ fi ± fi = : (.5) Ψ fi F fi We denote the input, state and output spaces of ± by U, X and Y, respectively. Then the operators ± fi appearing above are bounded from X L ([;fi];u) to X L ([;fi];y), hich means that for some c fi Z fi Z fi kx(fi )k + ky(t)k dt c fi kx()k + ku(t)k dt : As explained in Section I., the system ± is conservative if the operators ± fi are unitary, from X L ([;fi];u) to X L ([;fi];y). This implies that for any input function u H (; ; U ) and any initial state x() = x X ith Ax + Bu() X, the function kx(t)k is in C [; ) and d dt kx(t)k = ku(t)k ky(t)k 8 t ; (.6) see [5, Proposition 4.3]. Conversely, if (.6) holds for both the system ± and for its dual system ± d, then ± is conservative, see [5, Corollary 4.4]. Proposition.5 Let ± be a conservative linear system ith input space U, state space X, output space Y, semigroup T, control operator B, observation operator C and transfer function G. Then the folloing statements are true: () T is a semigroup of contractions. () B is infinite-time admissible. (3) C is infinite-time admissible. (4) kg(s)k for all s C. Proposition.6 With the notation of Proposition.5 and denoting the generator of T by A, for each fi >, the folloing statements are equivalent: () The pair (A; B) is exactly controllable in time fi. () The pair (A; C) is exactly observable in time fi. (3) ktfi k < (in particular, T is exponentially stable). Proposition.7 With the above notation, if T is exponentially stable, then denoting M = sup!r k(i!i A) k L(X ) e have, for every z H and every! [; ), k(! I A )zk +! kb C zk M kzk : Acknoledgments. This research as supported by EPSRC (from the UK) under grant number GR/R548/ and by CNRS (from France) under grant number 94376 DRCI. We have had useful discussions on this research ith Olof Staffans, Arjan van der Schaft and Peng-Fei Yao.

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