Superstability and Finite Time Extinction For C_0-Semigroups
aa r X i v : . [ m a t h . F A ] S e p SUPERSTABILITY AND FINITE TIME EXTINCTION FOR C -SEMIGROUPS D. CREUTZ, M. MAZO, JR., AND C. PREDA
Abstract.
A new approach to superstability and finite time extinction of strongly continuous semigroups is pre-sented, unifying known results and providing new criteria for these conditions to hold analogous to the well-knownPazy condition for stability. That finite time extinction implies superstability which is in turn equivalent to several(both known and new) conditions follow from this new approach in a consistent fashion. Examples showing thatthe converse statements fail are constructed, in particular, an answer to a question of Balakrishnan on superstablesystems not exhibiting finite time extinction. Introduction
The study of C -semigroups as a means to understand systems, particularly systems modelled by(partial) differential equations, has a long and rich history. In the control of partial differential equations(PDE) the theory of C -semigroups plays a fundamental role allowing to apply similar ideas as thoseemployed in the case of ordinary (finite) differential equations.Consider the strongly continuous semigroup T ( t ) with generator A , arising from e.g. a PDE model.One can associate to this operator an alternative description in the form of a linear continuous-timesystem Σ( A, B, C ): x ′ ( t ) = Ax ( t ) + Bu ( t ) x (0) = 0 y ( t ) = Cx ( t ) , where B and C are bounded linear operators defined on a suitable function space. One can also define thetransfer function of such a system, i.e.: an analytic function G bounded on the open right-hand complexhalf-plane that satisfies the condition ˆ y ( λ ) = G ( λ )ˆ u ( λ ) , λ ∈ C + . It is well-known that the transfer function is given by G ( λ ) = C ( λI − A ) − B, λ ∈ C , Re ( λ ) > ω where ω is the growth bound of the semigroup generated by A . The representation of these systems bytransfer functions is the key to more sophisticated control theory.A central concept in the study of control systems is the notion of exponential stability (often referredto simply as stability). A related stronger condition known as superstability has been the focus of muchresearch (e.g. [Bal05], [Bal81], [NR92], [RW95], [Lum01]), in particular its connection with finite timeextinction. The importance of exponentially stable semigroups stems, in part, from the fact that suchsystems admit transfer functions. Furthermore, if the semigroup is superstable (i.e ω = −∞ ) then G ( λ )can be defined on the whole complex plane.The notion of superstability first appeared (in a very rough form) in the seminal work of Hille andPhillips [HP57] who were concerned primarily with the mathematical aspects of it and related properties,particularly the relationship between the spectrum of the infinitesimal generator and the stability of thesemigroup. Later work refined and extended this notion, applying more complicated machinery ([NR92],[RW95]). More recently Balakrishnan [Bal05] (and others [Udw05], [Udw12], [LX12], [SLX13]) havebecome interested in the superstability phenomena arising in the control theory of (physical) systems.Particularly interesting for control applications is the relationship between superstability and finite timeextinction (as opposed to asymptotic stability). In [Bal05], Balakrishnan poses the following question: Date : 12 June 2008. are there physical (i.e. differential operator) examples of superstability which are not of extinction-in-finite-time type? In this paper, we provide a positive answer to this question with a constructive example(in fact many similar examples can easily be obtained). We also collect, clarify and extend the existingresults in the field with a new unified approach. In practice, our results provide a clear methodology tocheck if a superstable system also exhibits finite time extinction. Thus our results also provide guidelinesfor the design of controllers to guarantee finite-time extinction.The key ingredient in our work is a new approach to the concepts of stability and superstability focusingon the “entry times” of the system into balls about the origin (in the Banach space). This approach,which has a certain probabilistic flavor though is not in itself probabilistic, allows us to unify the existingresults in the field with (largely) new proofs. More importantly, we obtain analogues of certain well-knownresults about stability for superstable and finite-time-extinction systems. In particular, an analogue ofPazy’s condition [Paz83] for stability is given for both superstability and finite time extinction.2.
Main Results
Our main result characterizing the types of stability is:
Theorem.
Let { T ( t ) } be a C -semigroup of bounded linear operators on a Banach space X . Define therelative entry time for each r ∈ N as u r = sup { t r +1 ( x ) − t r ( x ) : k x k ≤ } where t r ( x ) = inf { t ≥ k T ( t ′ ) x k ≤ e − r for all t ′ ≥ t } . Then (i) { T ( t ) } is stable if and only if lim sup r →∞ u r < ∞ ; (ii) { T ( t ) } is superstable if and only if lim u r = 0; (iii) { T ( t ) } has finite time extinction if and only if X r u r < ∞ . Our Pazy-type condition is:
Theorem.
Let { T ( t ) } be a C -semigroup. Then, if for some a > ,(i) Z ∞ a k T ( t ) k p dt < ∞ for some < p < ∞ then { T ( t ) } is stable ; (ii) Z ∞ a (cid:12)(cid:12) log k T ( t ) k (cid:12)(cid:12) − p dt < ∞ for some < p < ∞ then { T ( t ) } is stable ; (iii) Z ∞ a (cid:12)(cid:12) log k T ( t ) k (cid:12)(cid:12) − dt < ∞ then { T ( t ) } is superstable ; (iv) lim p ↓ Z ∞ a (cid:12)(cid:12) log k T ( t ) k (cid:12)(cid:12) − p dt < ∞ then { T ( t ) } has finite time extinction . Preliminaries
A family { T ( t ) } t ≥ of bounded linear operators on a Banach space X is called a strongly continuoussemigroup (or C -semigroup ) when T (0) = I, T ( t + s ) = T ( t ) T ( s ) for all t, s ≥
0, and lim t ↓ T ( t ) = I in the strong operator topology ( T ( t ) x → x as t ↓ x ∈ X ). As is well-known, this implies thatthe map t T ( t ) is (strongly) continuous.For a strongly continuous semigroup { T ( t ) } t ≥ , define D to be the set of all x ∈ X such thatlim t ↓ t − ( T ( t ) x − x ) exists. The infinitesimal generator of the semigroup { T ( t ) } t ≥ is the operator A on X , with the domain D ( A ) = D , given by Ax = lim t ↓ T ( t ) x − xt , x ∈ D ( A ) . The name “infinitesimal generator” is justified by the fact that Ax = d ( T ( t ) x ) dt (cid:12)(cid:12)(cid:12) t =0 , x ∈ D ( A ) . UPERSTABILITY AND FINITE TIME EXTINCTION FOR C -SEMIGROUPS 3 The pair ( A, D ) and the semigroup { T t } t ≥ uniquely determine one another.A strongly continuous semigroup { T ( t ) } t ≥ is called exponentially stable (or just stable ) when thereexists constants M > ρ > k T ( t ) k ≤ M e − ρt for all t ≥ . Equivalently, define the stability index to besup { ν > ∃ M > k T ( t ) k ≤ M e − νt for all t ≥ } , and stability is then the requirement that the index be positive.The growth characteristic is ω = lim t ↓ log k T ( t ) k t which, as is well-known, is equal to − ν where ν is the stability index (when the semigroup is stable).It is then natural to define superstability to be the condition that the growth characteristic is ω = −∞ . Alternatively, superstability can be defined as the equivalent condition that the operators T ( t ) bequasinilpotent (recall that an operator T is quasinilpotent when spec ( T ) = { } ).A system is said to have finite time extinction when there is some t ≥ T ( t ) x = 0 forall t ≥ t and all x ∈ X with k x k ≤ { T ( t ) } is nilpotent when there exists t such that T ( t ) = 0. The smallest possible choice t such that T ( t ′ ) = 0 for all t ′ > t is called the index of nilpotency for the semigroup.The reader should note that in what follows we often defer the proofs until after all results are stated, thepurpose being to stress the similarities among the theorems characterizing these three concepts (perhapsthe most useful aspect of our approach).4. Final Entry Times
Definition 1.
Let { T ( t ) } be a C -semigroup on a Banach space X . For each x ∈ X and r ∈ N the finalentry time of x into the e − r -ball is t r ( x ) := inf { t ≥ k T ( t ′ ) x k ≤ e − r for all t ′ ≥ t } , and the final entry time of the -ball into the e − r -ball (referred to from here on as just the final entry timeof the e − r -ball) is t r := sup { t r ( x ) : k x k ≤ } , where we adopt the (usual) convention that the infimum of the empty set is ∞ (that is, t r ( x ) = ∞ whenno such time exists).The relative entry time of x into the e − r -ball is u r ( x ) := ( t r +1 ( x ) − t r ( x ) when t r +1 ( x ) < ∞∞ when t r +1 ( x ) = ∞ and the relative entry time of the -ball into the e − r -ball (referred to from here on as just the finalentry time of the e − r -ball) is u r := sup { u r ( x ) : k x k ≤ } . Lemma 2.
For any r we have u r = t r +1 − t r (meaning when either or both t r , t r +1 = ∞ then u r = ∞ ).Proof. First note that u r = ∞ if and only if t r +1 = ∞ and that t r = ∞ implies t r +1 = ∞ so we need onlyhandle the case that all three are finite. By definition, u r + t r = sup { u r ( x ) : k x k ≤ } + sup { t r ( x ) : k x k ≤ } ≥ t r +1 . On the other hand, there exists a sequence x n such that t r ( x n ) → t r . So for any ǫ > n we have t r ( x n ) > t r − ǫ and so t r < t r ( x n ) + ǫ . By definition t r +1 ≥ t r +1 ( x ) for all x , in particularfor the x n . Then t r +1 − t r ≥ t r +1 ( x n ) − t r ( x n ) − ǫ ≥ u r − ǫ. As ǫ is arbitrary, the claim follows. (cid:3) D. CREUTZ, M. MAZO, JR., AND C. PREDA
Lemma 3.
For any x and r with k x k > e − r we have k T ( t r ( x )) x k = e − r . Moreover, k T ( t r ) k = e − r .Proof. This follows directly from the strong continuity of T ( t ) (which is automatic for right continuityand follows from the uniform boundedness principle for left continuity). (cid:3) Lemma 4.
The sequence u r is nonincreasing in r : u r +1 ≤ u r . Hence the limit lim r →∞ u r always exists.Proof. First note that if u r ever attains ∞ then in fact t r +1 = ∞ at that point and for all subsequent t r meaning that u r remains ∞ from then on. So we need only handle the case when u r < ∞ for all r (andtherefore assume that t r < ∞ for all r ).Now suppose that u r > u r − for some r . Then there is some x such that u r ( x ) > u r − (since u r is thesupremum over all x ). Set y = eT ( t r ( x )) x so that T ( t ) y = eT ( t + t r ( x )) x for all t ≥ . By definition of t r +1 ( x ) we have that k T ( t + t r ( x )) x k ≤ e − r − if and only if t + t r ( x ) ≥ t r +1 ( x ) . Then k T ( t ) y k = e k T ( t + t r ( x )) x k ≤ e − r if and only if t ≥ t r +1 ( x ) − t r ( x ) = u r ( x ) , which means that t r ( y ) = u r ( x ). Now t r − ( y ) = 0 since k T ( t ′ + t r ( x )) x k ≤ e − r by definition of t r ( x ) andso k T ( t ′ ) y k = e k T ( t ′ + t r ( x )) x k ≤ ee − r = e − ( r − for all t ′ ≥ . Hence u r − ( y ) = u r ( x ). But this means that u r ( x ) > u r − ≥ u r − ( y ) = u r ( x ) , which is a contradiction. Therefore u r ≤ u r − for all r as claimed. (cid:3) Lemma 5.
If the T ( t ) are (not necessarily proper) contractions (i.e., k T ( t ) k ≤ ) then t r ( x ) = sup { t ≥ k T ( t ) x k ≥ e − r } , which is to say the t r are “stopping times”.Proof. The fact that T ( t ) are contractions forces that for any q > k T ( t + q ) x k = k T ( q ) T ( t ) x k ≤ k T ( t ) x kk T ( q ) k ≤ k T ( t ) x k , by the semigroup property. (cid:3) The Entry Time Growth Characteristic ω ( ET )0 Definition 6.
The entry time growth characteristic of a C -semigroup { T ( t ) } is defined by ω ( ET )0 := − (cid:0) lim r →∞ u r (cid:1) − , which always exists by Lemma 4. We now show that the entry time growth characteristic is equal to the usual growth characteristic ω defined previously for stable semigroups. Note that if the semigroup is not stable then ω ( ET )0 = 0. Theorem 7.
For a stable C semigroup T ( t ) , ω ( ET )0 = ω = lim t ↓ log k T ( t ) k t = inf t ≥ log k T ( t ) k t = lim t →∞ log k T ( t ) k t . Proof.
That ω (the usual growth characteristic), which is defined as the limit when t goes down to zeroabove, is equal to the infimum is well-known.Define ω ( t ) = t log k T ( t ) k . Then,( t + s ) ω ( t + s ) = log k T ( t + s ) k ≤ log k T ( t ) kk T ( s ) k = log k T ( t ) k + log k T ( s ) k = tω ( t ) + sω ( s ) UPERSTABILITY AND FINITE TIME EXTINCTION FOR C -SEMIGROUPS 5 is a subadditive sequence and therefore lim t →∞ tω ( t ) t = lim t →∞ ω ( t )exists. Call this limit ω . Set w r = ω ( t r ). By Lemma 3 we know that w r = − rt r . Now lim r →∞ w r = ω and − r + 1 t r +1 = − rt r t r t r +1 − t r +1 , and so w r +1 = w r t r t r +1 − t r +1 , which means that w r +1 t r +1 − w r t r = − . Consider the case when ω > −∞ . Taking limits in the above, lim r →∞ ωu r = − ω ( ET )0 = ω .Now consider the case when ω = −∞ . Suppose that ω ( ET )0 > −∞ . Then lim r →∞ u r = c > − t r +1 w r +1 − t r w r = t r ( w r +1 − w r ) + u r w r +1 ≤ u r w r +1 , since w r +1 − w r ≤
0. But then − ≤ lim r →∞ u r w r +1 = c ( −∞ ) = −∞ , contradicting that ω ( ET )0 > −∞ . Therefore in either case ω ( ET )0 = ω .Now we show that ω = ω . Since we know ω = inf t ≥ ω ( t ) and ω = lim t →∞ ω ( t ) we already have that ω ≤ ω . For any positive integer n and any s ≥ ω ( ns ) = 1 ns k T ( s ) n k ≤ ns log k T ( s ) k n = ω ( s ) , and therefore lim t →∞ ω ( t ) = lim n →∞ ω ( ns ) ≤ ω ( s ) , which means that ω = lim t →∞ ω ( t ) ≤ inf t ≥ ω ( t ) = ω . Therefore ω = ω and the proof is complete. (cid:3) Equivalence of Stability Notions
We now present the main theorems characterizing the various notions of stability. One of our aimsin this paper is to collect and clarify the various characterizations of these notions. To this end weinclude several known characterizations and provide new proofs using our techniques. Specifically, theequivalences in this section, excepting the conditions which involve the relative entry times u r , are known(see e.g. [Bal05]). Theorem 8.
For a C -semigroup { T ( t ) } and ν > , the following are equivalent:(i) { T ( t ) } is stable with stability index ν : ( ∀ ρ < ν )( ∃ M > ∀ t ≥ k T ( t ) k ≤ M e − ρt ; (ii) lim r →∞ u r = ν − < ∞ ; (iii) spec ( T ( t )) ⊆ { λ ∈ C : | λ | ≤ e − tν } ; (iv) ω = − ν. Theorem 9. If { T ( t ) } is a stable C -semigroup with stability index ν and A is the generator of T ( t ) withdomain D then spec ( A ) ⊆ { λ ∈ C : Re λ < − ν } . The converse does not hold. Remark.
The spectrum of the generator, spec ( A ) , depends very delicately on the domain of definition(see the counterexamples below). In particular, the operator A treated as having full domain may wellhave spectrum much larger than spec ( A ) . D. CREUTZ, M. MAZO, JR., AND C. PREDA
Theorem 10.
For a C -semigroup { T ( t ) } , the following are equivalent:(i) { T ( t ) } is superstable : { T ( t ) } is stable with stability index ∞ ; (ii) lim r →∞ u r = 0; (iii) ( ∀ ν > ∃ M ν > ∀ t ≥ k T ( t ) k ≤ M ν e − νt ; (iv) T ( t ) are all quasinilpotent: spec ( T ( t )) = { } ; (v) ω = −∞ . Remark.
The constants M ν in condition (iii) must tend to infinity as ν → ∞ since otherwise thesemigroup will be identically . Theorem 11. If { T ( t ) } is a superstable C -semigroup with generator A and domain D then spec ( A ) = ∅ .The converse does not hold. Theorem 12.
For a C -semigroup { T ( t ) } on a Banach space X with generator A having domain D , and ≤ k < ∞ the following are equivalent:(i) { T ( t ) } has finite time extinction at time k :( ∀ x ∈ X )( ∃ t ∞ ( x ) ≥ ∀ t ≥ t ∞ ( x )) T ( t ) x = 0 and sup { t ∞ ( x ) : k x k ≤ } = k ; (ii) X r u r = k < ∞ ; (iii) ( ∀ ν > ∃ M ν > ∀ t ≥ k T ( t ) k ≤ M ν e − νt and sup ν> log M ν ν = k ; (iv) ( ∃ M > ∀ ν ≥ ∀ t ≥ k T ( t ) k ≤ M e − ν ( t − k ) ; (v) T ( t ) is nilpotent with nilpotency index k : T ( q ) = 0 for q > k and T ( q ) = 0 for q < k ; (vi) the resolvent function R ( λ, A ) is entireand (cid:12)(cid:12) R ( λ, A ) (cid:12)(cid:12) ≤ C (1 + | λ | ) − N e k | Reλ | for some constants C, N .
Remark.
In condition (i), the definition of finite time extinction, the t ∞ can be chosen uniformly onbounded sets, in particular on balls around the origin of finite radius, however, t ∞ cannot be chosenuniformly over x unless the underlying space of the Banach space is compact (i.e., L [0 , not L [0 , ∞ ) ). Theorem 13.
Extinction in finite time implies superstability and superstability implies stability. Theconverses of both statements are false.
Theorem 14.
Let { T t } be a C -semigroup. Then, if for some a > ,(i) Z ∞ a k T ( t ) k p dt < ∞ for some < p < ∞ then { T ( t ) } is stable ; (ii) Z ∞ a (cid:12)(cid:12) log k T ( t ) k (cid:12)(cid:12) − p dt < ∞ for some < p < ∞ then { T ( t ) } is stable ; (iii) Z ∞ a (cid:12)(cid:12) log k T ( t ) k (cid:12)(cid:12) − dt < ∞ then { T ( t ) } is superstable ; (iv) lim p ↓ Z ∞ a (cid:12)(cid:12) log k T ( t ) k (cid:12)(cid:12) − p dt < ∞ then { T ( t ) } has finite time extinction . Remark.
Stability can only occur when k T ( t ) k is eventually bounded by (that is, t < ∞ ). Taking a = t will cause the integrals to converge whenever there is some value of a that causes convergence. UPERSTABILITY AND FINITE TIME EXTINCTION FOR C -SEMIGROUPS 7 Proofs of Equivalences
As usual, let { T ( t ) } be a C -semigroup with generator A having dense domain D on the Banach space X . Proof. (of Theorem 8). Assume condition (ii) holds:lim r →∞ u r = ν − < ∞ . Fix ǫ >
0. Then there exists r ∗ such that u r < ν − + ǫ for all r ≥ r ∗ . Hence for r ≥ r ∗ , t r − t r ∗ ≤ ( r − r ∗ )( ν − + ǫ ) . For any t ≥ r ∗ ( ν − + ǫ ) pick r ≥ r ∗ such that r ( ν − + ǫ ) + t r ∗ ≤ t < ( r + 1)( ν − + ǫ ) + t r ∗ . Since t ≥ t r (as r ( ν − + ǫ >
0) we have that k T ( t ) k ≤ e − r and t < ( r + 1)( ν − + ǫ ) + t r ∗ implies r > t − t r ∗ ν − + ǫ − . Then k T ( t ) k ≤ e − r < e − t − tr ∗ ν − ǫ = e tr ∗ ν − ǫ e − t ν − ǫ . So { T ( t ) } has stability index less than ν − + ǫ . Since ǫ was arbitrary, condition (i) holds.Conversely, assume (i) holds. Then for any ρ < ν there is M such that k T ( t ) k ≤ M e − ρt . For t ≥ r − log Mρ we then have that k T ( t ) k ≤ e − r . Hence t r ≤ r − log Mρ .
Suppose lim u r > ρ + 2 δ for some δ >
0. Then for sufficiently large r ′ we have u r > ρ + δ for r ≥ r ′ so t r +1 = t r ′ + u r ′ + · · · + u r > t r ′ + ( r − r ′ ) 1 ρ + ( r − r ′ ) δ, and therefore r + 1 − log Mρ − ( r − r ′ ) 1 ρ > ( r − r ′ ) δ, so r ′ + 1 − log Mρ + r ′ δ > rδ, but the left-hand side is constant and the right hand tends to ∞ as r → ∞ . This means that lim u r ≤ ρ .Since ρ < ν is arbitrary we have (ii).Now assume (iv) holds. Then by Gelfand’s spectral radius formula,sup (cid:12)(cid:12) spec ( T ( t )) (cid:12)(cid:12) = lim t k T ( t ) k t = e ω t = e − tν . Hence (iii) holds. Likewise, if (iii) holds then by Gelfand’s formula, ω = − ν so (iv) holds.The equivalence of (i) and (ii) with (iv) is a direct consequence of Theorem 7. This completes the proofthat (i) through (iv) are equivalent. (cid:3) Proof. (of Theorem 9). This is well-known. (cid:3)
Proof. (of Theorem 10). The equivalences follow from identical arguments to those for the case of stablesemigroups (simply replace ν − by 0). (cid:3) Proof. (of Theorem 11). This follows from Theorem 9: if z ∈ spec ( A ) then the stability index ν is at least − Re ( z ) but a superstable semigroup has stability index ∞ . (cid:3) D. CREUTZ, M. MAZO, JR., AND C. PREDA
Proof. (of Theorem 12). Assume that (ii) holds. Then for any x with k x k ≤ t r +1 ( x ) − t ( x ) = r X j =0 u r ( x ) ≤ r X j =0 u r → k, hence (i) holds.Conversely, assume (i) holds and suppose (ii) fails. If P r u r = ℓ < k then t ∞ ( x ) ≤ ℓ < k for all x, contradicting (i). So it must be that P r u r = ℓ > k . By Lemma 2 we have X r u r = X r t r +1 − t r = t ∞ − t , and there is then a sequence x n such that t ∞ ( x n ) → ℓ > k contradicting (i).Assume (ii) holds. For ν > r ∗ such thatsup r ≥ r ∗ u r < ν − and set M ν = e νt r ∗ . Then, as in the proof of stability, we have rν − ≤ t − t r ∗ < ( r + 1) ν − implies k T ( t ) k ≤ M ν e − tν . Now log M ν ν = 1 ν + t r ∗ ( ν ) and r ∗ ( ν ) → ∞ as ν → ∞ , hence log M ν ν → P r u r = k . So (iii) holds.Assume (iii) holds. Then M ν ≤ e kν for all ν, hence (iv) holds with M = 1.Assume (iv) holds. Then for t > k we have k T ( t ) k ≤ M e − ν ( t − k ) for all ν and t − k > , so lim ν e − ν ( t − k ) = 0 hence k T ( t ) k = 0. Thus (i) holds.The equivalence of (i) and (v) is trivial.To see that (i) and (vi) are equivalent, note that (i) and (vi) both imply spec ( A ) = ∅ . Then R ( λ, A ) = Z ∞ e − λt T ( t ) dt for all λ (in general for Re λ > ω , see, e.g,. [Bal81]).Hence R ( iλ, A ) is the Fourier Transform of T ( t ). By the Paley-Wiener Theorem, R ( iλ, A ) is the FourierTransform of a compactly supported function (i.e. T ( t ) = 0 for all t ≥ k ) if and only if (vi) holds. Thisargument first appeared in [GK70]. (cid:3) Proof of the Pazy-type Criteria
Lemma 15.
Suppose that k T ( t ) k ≤ for all t ≥ (that is, t = 0 ). Let F : R ∪ {±∞} → [0 , ∞ ] be adecreasing function such that F ( ∞ ) = 0 . Then ∞ X r =0 u r F ( r + 1) ≤ Z ∞ F ( − log k T ( t ) k ) dt ≤ ∞ X r =0 u r F ( r ) . Proof.
Observe that Z ∞ F ( − log k T ( t ) k ) dt = Z t F ( − log k T ( t ) k ) dt + ∞ X r =0 Z t r +1 t r F ( − log k T ( t ) k ) dt + Z ∞ t ∞ F ( − log k T ( t ) k ) dt. Now t = 0 so the first term on the right is 0. For t ≥ t ∞ we have that k T ( t ) k = 0 so F ( − log k T ( t ) k ) = F ( ∞ ) = 0 meaning that the third term on the right is zero. UPERSTABILITY AND FINITE TIME EXTINCTION FOR C -SEMIGROUPS 9 For the middle terms, note that for t < t ′ , k T ( t ′ ) k = k T ( t ) T ( t ′ − t ) k ≤ k T ( t ) kk T ( t ′ − t ) k ≤ k T ( t ) k , since k T ( t ′ − t ) k ≤ k T ( t r +1 ) k ≤ k T ( t ) k ≤ k T ( t r ) k for t r ≤ t ≤ t r +1 , and therefore, by Lemma 3, − r − k T ( t r +1 ) k ≤ log k T ( t ) k ≤ log k T ( t r ) k = − r for t r ≤ t ≤ t r +1 , and so since F is decreasing F ( r + 1) ≤ F ( − log k T ( t ) k ) ≤ F ( r ) for t r ≤ t ≤ t r +1 . So for each term in the sum, u r F ( r + 1) = Z t r +1 t r F ( r + 1) dt ≤ Z t r +1 t r F ( − log k T ( t ) k ) dt ≤ u r F ( r ) . (cid:3) Proof. (of Theorem 14). First note that if lim sup k T ( t ) k > u r = ∞ for all r so there can be nostability. In this case, none of the three conditions involving integrals can hold. So it is enough to considerthe case when k T ( t ) k is eventually bounded by 1. Since Z t H ( k T ( t ) k ) dt < ∞ for any bounded function H (recall that t < ∞ since we have eliminated the other case), we may assumethat k T ( t ) k ≤ t : the integral conditions are unaffected by finite translations in time as is thestability of the semigroup.Recall that condition (i) is the Datko-Pazy Theorem ([Dat70], [Paz72], [Paz83]). Consider the function F ( x ) = e − px for some fixed 0 < p < ∞ . Then F is decreasing and F ( ∞ ) = 0.By Lemma 15, ∞ X r =0 u r e − p ( r +1) ≤ Z ∞ F ( − log k T ( t ) k ) dt = Z ∞ k T ( t ) k p dt < ∞ . For any given r ∗ observe that since the u r are nonincreasing, r ∗ − X r =0 u r e − p ( r +1) ≥ u r ∗ r ∗ X r =1 e − pr , and since P r e − p ( r +1) = C < ∞ , lim r ∗ r ∗ − X r =0 u r e − p ( r +1) ≥ lim r ∗ u r ∗ C, and therefore lim r →∞ u r ≤ C − ∞ X r =0 u r e − p ( r +1) < ∞ , hence the semigroup is stable.Condition (ii) is a weakening of the Pazy condition: set F ( x ) = x − p for the appropriate 1 < p < ∞ .By Lemma 15, ∞ X r =0 u r ( r + 1) − p ≤ Z ∞ ( − log k T ( t ) k ) − p dt < ∞ . Then, as above, since P r ( r + 1) − p = C < ∞ ,lim r →∞ u r ≤ C − ∞ X r =0 u r ( r + 1) − p < ∞ , so the semigroup is stable.Now condition (iii): set F ( x ) = x . By Lemma 15, ∞ X r =0 u r r + 1 ≤ Z ∞ dt − log k T ( t ) k < ∞ . Proceeding as above, r ∗ − X r =0 u r r + 1 ≥ u r ∗ r ∗ X r =1 r , and since P ∞ r =1 1 r = ∞ this means that lim r →∞ u r = 0 (and in fact converges to 0 faster than the inverseof the harmonic sum (1 + · · · + r ) − ). The semigroup is therefore superstable.Finally condition (iv): set F p ( x ) = x − p for 0 < p . Then, by Lemma 15,sup p> ∞ X r =0 u r ( r + 1) − p ≤ sup p> Z ∞ ( − log k T ( t ) k ) p dt < ∞ , here we use that F p ≤ F p ′ for p ≥ p ′ so the hypothesis for p ↓ p > P r u r = ∞ . Then for any K there exists r K such that P r K − r =0 u r ≥ K . Then ∞ X r =0 u r ( r + 1) − p ≥ r K − X r =0 u r r − pK ≥ Kr − pK , for any p > p> ∞ X r =0 u r ( r + 1) − p ≥ K. But K is arbitrary so ∞ > sup p> ∞ X r =0 u r ( r + 1) − p = ∞ is a contradiction. The semigroup therefore has finite time extinction. In fact the semigroup goes extinctat time k = ∞ X r =0 u r = sup p> Z ∞ ( − log k T ( t ) k ) − p dt (details here are left to the reader). (cid:3) Counterexamples
We construct examples of semigroups demonstrating that finite time extinction is strictly strongerthan superstability and that superstability is strictly stronger than stability. In particular, we answera question of Balakrishnan [Bal05] on the existence of superstable semigroups not vanishing in finitetime with generator being a differential operator (what he terms a “physical system”). We also remarkon a (previously known) example showing that the spectrum of the generator does not fully determinesuperstability.9.1.
Superstable Without Finite Time Extinction.
Consider the Gaussian (probability) measure µ on R + = [0 , ∞ ) given by dµ ( x ) = q π exp( − x ) dx . Let X = L ( R + , µ ). Define the semigroup T ( t ) f ( s ) = f ( s − t ) for s ≥ t and T ( t ) f ( s ) = 0 otherwiseon X . The reader may verify that this in fact a semigroup with generator A = − dds and domain theappropriate Sobolev space.Now for f ∈ L ( R + , µ ) with k f k = 1 we have k T ( t ) f k = Z ∞ | f ( s − t ) | dµ ( s ) = Z ∞ | f ( v ) | r π e − ( t + v )22 dv ≤ e − t Z ∞ | f ( v ) | dµ ( v ) = e − t , UPERSTABILITY AND FINITE TIME EXTINCTION FOR C -SEMIGROUPS 11 since e − ( t + v ) ≤ e − t for t, v ≥
0. So k T ( t ) k ≤ exp( − t ) → T ( t ) = 0 for any t .Taking f to be a norm one (with respect to µ ) function concentrated near 0 we see that k T ( t ) k =exp( − t ) and so t r = 2 √ r and u r = 2( √ r + 1 − √ r ) → P u r = ∞ .Hence superstability can occur without finite time extinction (even when the generator is merely aderivative). The space ( R + , µ ) is a variant of the classical Gaussian measure space which arises naturallyin the context of stochastic systems and quantum systems, among other areas. Our example can easilybe extended to any system with a Gaussian measure (details are left to the interested reader).9.2. Finite Time Extinction.
Define the semigroup T ( t ) f ( s ) = f ( s + t ) for s + t ≤ T ( t ) f ( s ) = 0 otherwiseon X = { f ∈ L [0 ,
1] : f (0) = f (1) = f ′ (0) = f ′ (1) = 0 } . The reader may verify that this is a semigroupwith generator A = dds and domain Sobolev space. It is clear that T ( t ) = 0 for all t ≥ t r = 1 for all r > u r = 0 for r > P u r = 1 < ∞ .9.3. Stable but Not Superstable.
For completeness, we mention the fairly trivial example T ( t ) f ( s ) = e − νt f ( s ) (so the generator is A = νI and the domain is D = L ) is clearly stable with index ν . Here t r isdefined by e − νt r = e − r so t r = rν − meaning u r = − ν − .9.4. Empty Spectrum (for the Generator) but not Superstable.
For completeness, we mentionan example due to Hille and Phillips [HP57] (chapter 23, section 16).We first present a superstable semigroup which will be used to develop the actual example of interest.Define T ( t ) f ( s ) := t ) R s ( s − u ) t − f ( u ) du . That this is a semigroup follows from Euler integral identities.The generator A is the derivative of the convolution with log minus a constant: Af ( s ) = dds R s log( s − u ) f ( u ) du − γf ( s ) ( γ is Euler’s constant). Then spec ( A ) = ∅ and k T ( t ) k ≈ t Γ( t ) . So ω = −∞ and T ( t ) → T ( t ) = 0 for any t . This semigroup is in fact superstable but does not have finite timeextinction.With this construction in hand, we construct the desired example: for ξ ∈ C with Re ξ >
0, define J ξ f ( s ) = ξ ) R s ( s − u ) ξ − f ( u ) du . When ξ is taken to be a positive real this yields the semigroup above.There is an analytic extension of J ξ to ξ purely imaginary. Let T ( t ) = J it . The generator of this semigroupis iA where A is the generator from above. Then spec ( iA ) = ∅ but 0 / ∈ spec ( T ( t )), e.g. the operators arenot quasinilpotent hence not superstable.The reader is referred to [HP57] for details on these semigroups. References [Bal81] A.V. Balakrishnan,
Applied functional analysis , Springer-Verlag, New York, NY, 1981.[Bal05] ,
Superstability of systems , Applied Mathematics and Computation (2005), 321–326.[Dat70] R. Datko,
Extending a theorem of A. M. Liapunov to Hilbert space , J. Math. Anal. Appl. (1970), 610–616.[GK70] I.C. Gohberg and M.G. Krein, Theory and applications of Volterra operators in Hilbert space , American Mathematical Society,Providence, Rhode Island, 1970.[HP57] E. Hille and R. Phillips,
Functional analysis and semi-groups , American Mathematical Society, Providence, Rhode Island,1957.[Lum01] G. Lumer,
On the growth orders of the resolvents for an explicit class of superstable semigroups , Ulmer Seminare ¨uberFunktionalanalysis und Differentialgleichungen (2001).[LX12] D. Liu and G. Xu, Superstability of wave equations on a tree-shape network , Control Conference (CCC), 2012 31st Chinese,IEEE (2012), 1257–1262.[NR92] R. Nagel and F. R¨abiger,
Superstable operators on Banach spaces , Israel J. Math (1992), no. 1-2, 213–226.[Paz72] A. Pazy, On the applicability of Lyapunovs theorem in Hilbert space , SIAM J. Math. Anal. (1972), 291–294.[Paz83] , Semigroups of operators and applications to partial differential equations , Springer-Verlag, Berlin, 1983.[RW95] F. R¨abiger and M. Wolff,
Superstable semigroups of operators , Indagationes Mathematicae (1995), 481–494.[SLX13] Y. Shang, D. Liu, and G. Xu, Super-stability and the spectrum of one-dimensional wave equations on general feedback controllednetworks , IMA Journal of Mathematical Control and Information (2013).[Udw05] F.E. Udwadia,
Boundary control. quiet boundaries, super-stability and super-instability , Applied Mathematics and Computa-tion (2005), no. 2, 327–349.[Udw12] ,
On the longitudinal vibrations of a bar with viscous boundaries: Super-stability, super-instability and loss of damping ,International Journal of Engineering Science (2012), no. 1, 79–100. Department of Mathematics, Vanderbilt University, Nashville, USA
E-mail address : [email protected] Delft Center for Systems and Control, Delft University of Technology, The Netherlands
E-mail address : [email protected] Department of Economics and Business Modeling, University of West Timisoara, Romania
E-mail address ::