# On Bounded Strictly Positive Operators of Closed Range and Some Applications to Asymptotic Hyperstability of Dynamic Systems

Authors: M. De la Sen
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The problem discussed is the stability of two input-output feedforward and feedback relations, under an integral-type constraint defining an admissible class of feedback controllers. Sufficiency-type conditions are given for the positive, bounded and of closed range feed-forward operator to be strictly positive and then boundedly invertible, with its existing inverse being also a strictly positive operator. The general formalism is first established and the linked to properties of some typical contractive and pseudocontractive mappings while some real-world applications and links of the above formalism to asymptotic hyperstability of dynamic systems are discussed later on.


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## Body

## 1. Introduction

The properties of absolute stability and hyperstability and asymptotic hyperstability of dynamic systems are very important tools in dynamic systems since they are associated with the positivity and boundedness of the energy for all feedback controllers within a wide class characterized by a Popov-type integral inequality, then implying global Lyapunov’s stability [1–8]. The fact that such properties hold for a class of controllers defined by the Popov inequality, rather than for just some individual one, makes the related theory to be very useful against potential parametrical dispersion of components. The main objective of this paper is the investigation of the strict positivity and stability of bounded positive one-to-one operators with closed range on Hilbert spaces linked to contractive, pseudocontractive, asymptotically pseudocontractive, and asymptotically pseudocontractive in the intermediate sense mappings. See [9–21] and exhaustive list of references therein. Fixed point theory has also been proven to be useful to describe the asymptotic behaviour, stability and equilibrium points of differential, functional, and difference equations and systems of equations, and continuous-time, discrete-time, and hybrid dynamic systems. See, for instance, [22–27] and references therein. Further links with technical results and some real-world examples are established through the paper related to the relevant problems of absolute stability and asymptotic hyperstability of continuous-time and discrete-time dynamic systems [1–8]. Such dynamic systems possess the significant physical property that their associate input-output energy is non-negative and finite for all time. Thus, they are purely dissipative systems, for a wide class of feedback nonlinear time-varying controllers satisfying an integral input-output inequality what leads to the global Lyapunov’s stability for all controllers within such a class. Several operators are characterized but the most important one in the analysis is the one which maps the input space to the output space. Both such spaces are subspaces of a Hilbert space resulting to be, typically in real-world examples, either the space of square-integrable real or complex functions (or, in general, vector functions) or its corresponding square-summable counterparts. The relevant property needed for a positive operator to be strictly positive is seen to be that its minimum modulus be nonzero so as to ensure that it is invertible if it is of a closed range. Note, on the other hand, that the crucial property for the boundedness and stability of the operator restricted to the Hilbert space of interest is that it will be stable on its whole definition domain.


## 2. Problem Statement and Main Results

Through this paper, one considers the complex Hilbert spaceH on C and operators G:H→H and K:H→H which define the following associated relations:
(1)y=Gu+yh,u=Kr-φΓ(y),
where yh=yh(p) and p∈Cq is some given complex parameterizing vector, and r:Γ→H∣domK, φΓ:Γ×H→ranφΓ(⊂H), u:Γ→(H∣ranK)+ranφΓ(⊂H); andy=yh+yf, with yh:Γ×Cq→H and yf:Γ→H∣ranG. The set Γ is some appropriate domain to define the previous functions of interest. Examples which adjust to the previous structure are very common in the real world as, for instance, linear continuous-time dynamic systems (with Γ=R0+ being the nonnegative real set for picking up values t∈Γof the continuous-time argument) and linear discrete-time dynamic systems (with Γ=Z0+ being the nonnegative integer set for values t∈Γof the discrete-time argument) where r is an exogenous, or reference, signal, u is a feedback control, yh is the parameterized response to initial conditions, versus Gu which is the forced response, y is the measurable output to be controlled, and φΓ(y) is a nonlinear (and, eventually, time-varying) controller device.The inner products on the previous various Hilbert spaces are all denoted with the standard notation〈·,·〉 and mutually distinguished easily depending on context without explicit notational subscripts referred to each concrete space. Assume that Γ is an indicator set defining truncated elements of the Hilbert space as, for instance, a real interval or a subset of the nonnegative integers and Pt is a projection operator being a truncation operator so that xt=Ptx and for each t∈Γ, and we define the seminorm on H by ∥x∥t=∥xt∥=∥Ptx∥; ∀t∈Γ with Pt≠I and the family {∥∥t:t∈Γ} of seminorms defines the resolution topology on H since {Pt:t∈H} is a resolution of the identity [28]. Note that ∥x∥t1≥∥xt2∥ if t1≥t2. For instance, if Γ=R0+ then x(t) denotes a point value of x:R0+→R0+, for t∈Γ while xt denotes the strip x:[0,t]∩R0+→R0+. Through the paper the notation “*” stands for adjoint operators and also for complex conjugates of scalars or vectors depending on the context.The problem to be discussed in the paper is the stability of (1) under an integral-type constraint for the controller specified later on, which characterizes a whole admissible class of controllers rather than an individual controller. Conditions are given for the positive feed-forward operator G which is assumed to be bounded and of closed range is ensured to be also strictly positive, then boundedly invertible, with its existing inverse being also a strictly positive operator. If such an operator is bounded and strictly positive, then the inner products 〈Gu,u〉 and 〈G-1u,u〉 are both strictly positive and finitely upper-bounded for all nonzero input u. The general formalism is given in Section 2 together with some links to contractive and pseudocontractive mappings while some real-world applications to asymptotic hyperstability of dynamic systems are then given in Section 3. The following preliminary result holds.Proposition 1.
Assume thatG:H→H is a one-to-one linear operator with closed range. Then, the following properties hold(i)
G:H→His invertible with nonzero minimum modulus,(ii)
if, in addition,G:H→His positive (abbreviated notation beingG≽0), then〈Gu,u〉>0for any nonzerou:Γ→H,(iii)
there ist∈Γsuch that〈GPtu,Ptu〉>0for any nonzerou∈dom(G).Proof.
SinceG on H is one-to-one with closed range, it is also invertible from the open mapping theorem and then bounded below, so that there is c∈R+ such that
(2)∥Gu∥=〈Gu,Gu〉1/2≥c∥u∥=c〈u,u〉1/2;∀u∈domG.
Then, the minimum modulus μ(G) of G satisfies
(3)μ(G)=inf{∥Gu∥∥u∥:u(∈domG)≠0}≥c>0
and Property (i) has been proven. Now, if G≽0, then there is a self-adjoint operator G~=G~*≽0 on X such that G=G~*G~=G~G~*=G~2 so that, since μ(G~)>0 from Property (i),
(4)〈Gu,u〉1/2=〈G~2u,u〉1/2=〈G~u,G~u〉1/2=∥G~u∥=∥G~u∥∥u∥∥u∥≥inf{∥G~u∥∥u∥:u(∈domG)≠0}∥u∥≥μ(G~)∥u∥>0;∀u(≠0):Γ⟶H
and Property (ii) is proven.
(iii)  Note that if,u≠0, then there is t∈Γ such that Ptu≠0 and 〈GPtu,Ptu〉>0 since one gets by Property (ii) that
(5)〈Gu,u〉1/2=supt∈Γ〈GPtu,Ptu〉1/2≥〈GPtu,Ptu〉1/2=〈G~2Ptu,Ptu〉1/2=∥G~(Ptu)∥=∥G~(Ptu)∥∥Ptu∥∥Ptu∥≥inf{∥G~(Ptu)∥∥Ptu∥:u(∈domG)≠0}∥Ptu∥≥μ(G~)∥Ptu∥>0;∀u(≠0):Γ⟶H.
Thus, 〈GPtu,Ptu〉>0 for some t∈Γ if u≠0. Hence, Property (iii) follows.Definition 2.
The operatorG:H→H is said to be strictly positive (denoted as G≻0) if it is positive (i.e., G≽0) and μ(G)>0.Note from Proposition1 that if G≽0 is a one-to-one operator on H with closed range, then it is invertible and G≻0.It is also direct to prove that Property (i) of Proposition1 is equivalent to its given assumption so that one has [28].Proposition 3.
G
:
H
→
H is a one-to-one linear bounded operator with closed range if and only if it is invertible with nonzero minimum modulus.Proposition 4.
IfG:H→H is a one-to-one linear bounded strictly positive operator with closed range, then it is invertible and G-1:H→H is also strictly positive with closed range and bounded and μ(G-1)>0 so that 〈G-1u,u〉>0 for any nonzero u:Γ→H.Proof.
Note that1/∥G-1∥=μ(G)>0 from Proposition 1. Thus, ∥G-1∥<∞ so that G-1 is bounded. Since G is bounded, then ∥G∥<∞ and 1/∥G∥=μ(G-1)>0. Thus, G-1:H→H is also one-to-one with closed range from Proposition 3. Then, G-1=G~-1G~*-1=G~-2 is self-adjoint, and since μ(G~-1)>0, one has from Property (i) of Proposition 1 and Definition 2 that G-1≻0 since
(6)〈G-1u,u〉1/2=〈G~-2u,u〉1/2=〈G~-1u,G~-1u〉1/2=∥G~-1u∥=μ(G-1)∥u∥>0,∀u(≠0):Γ⟶H.
Note that, if G≽0, then 〈Gu,u〉 can be zero for some nonzero u:Γ→H. The following result refers to the fulfilment of relationships (1) for all t∈Γ, that is, on the space He={ft=Ptf:(f:Γ→PC(Γ));∀t∈Γ} provided that G≻0 and bounded. Under some additional weak boundedness conditions, it is proven the stability of (1) with u and y belonging to H. Note that He is not a Hilbert space (even though H is a Hilbert space) since it is not ensured that, for any f:Γ→PC(Γ), ft:Γ→H as (Γ∋)t→∞.An important result follows.Theorem 5.
Assume that (1) holds for all t∈Γ, that is, G:He→He and K:He→He, where He={ft=Ptf:(f:Γ→PC(Γ));∀t∈Γ}, yh=yh(p) and p∈Cq is some given complex parameterizing vector, r:Γ→He∣domK, φΓ:Γ×He→ranφΓ, u:Γ→(He∣ranK)+ranφΓ; andy=yh+yf, with yh:Γ×Cq→H and yf:Γ→He∣ranG. Assume also that(1)
yh=yh(p) is bounded and Ptyh→0 as (Γ∋)t→∞,(2)
G:He→He is stable (or, equivalently, G:He∣H→H is bounded and causal), one-to-one, and with closed range,(3)
G≻0,(4)
K:H→H is bounded,(5)
r:Γ→H is bounded,(6)
〈Pty,Pt(φΓ(y))〉≥-γt≥-γ>-∞; ∀t∈Γ.Then, u, yf, y:Γ→H, and they are bounded. Also, if r≡0, then u(t)→0, yf(t)→0, y(t)→0 as (Γ∋)t→∞.Proof.
Direct calculations yield(7)〈y,u〉=〈Gu+yh,Kr-φΓ(y)〉=〈Gu,Kr〉+〈yh,Kr〉-(〈Gu,φ(y)〉+〈yh,φΓ(y)〉)=〈Gu,Kr〉+〈yh,Kr〉+〈Gu,u〉-〈yh,φΓ(y)〉-〈Gu,Kr〉=〈Gu,u〉+〈yh,Kr-φΓ(y)〉≥μ2(G~)∥u∥2+〈yh,Kr-φΓ(y)〉=μ2(G~)supt∈Γ∥Ptu∥2+〈yh,Kr-φΓ(y)〉≥μ2(G~)∥Ptu∥2+〈yh,Kr-φΓ(y)〉;∀t∈Γ
since u=Kr-φΓ(t,y) and 〈Gu,u〉≥μ2(G~)∥u∥2≥μ2(G~)∥Ptu∥2>0 for any nonzero control from Proposition 1. One gets in the same way that
(8)〈Pty,Ptu〉≥μ2(G~)∥Ptu∥2+〈Pt(yh),Pt(Kr-φΓ(y))〉,∀t∈Γ.
Since 〈Pty,Pt(φΓ(y))〉≥-γt>-∞; ∀t∈Γ, one gets also that
(9)〈y,u〉=〈y,Kr-φΓ(y)〉=〈y,Kr〉-〈y,φΓ(y)〉≤〈y,Kr〉+supt∈Γγt;∀t∈Γ,(10)〈Pty,Ptu〉≤〈Pty,Pt(Kr)〉+γt;∀t∈Γ.
One gets from (7), (9), (8), and (10) that
(11)〈y,Kr〉+γ≥μ2(G~)∥u∥2+〈yh,Kr-φΓ(y)〉≥μ2(G~)∥Ptu∥2+〈yh,Kr-φΓ(y)〉,∀t∈Γ,(12)〈Pty,Pt(Kr)〉γt≥μ2(G~)∥Ptu∥2〈Pt(yh),Pt(Kr-φΓ(y))〉;∀t∈Γ,
where 0<γ=supt∈Γγt<∞. Now, since K is a bounded operator, r is a bounded function, Ptyh→0 as t→∞, and G≻0 is bounded and one-to-one with closed range so that G~≻0 is also bounded and one-to-one with closed range implying from Proposition 1 that μ(G)=μ2(G~)>0, and one gets from (12) that
(13)liminfΓ∋t→∞[-μ2(G~)∥Ptu∥2]γt+|〈Pt(GPtu)〉,Pt(Kr)>|-μ2(G~)∥Ptu∥2]≥liminfΓ∋t→∞[-μ2(G~)∥Ptu∥2]γt+〈Pt(GPtu)〉,Pt(Kr)>-μ2(G~)∥Ptu∥2]≥0.
Assume that there is some unbounded u:Γ→H. Then, the subsequent contradiction
(14)0≤liminfΓ∋t→∞[γt+μ2(G~)(λ-∥Ptu∥)∥Ptu∥]=-∞
follows from (13) for some λ∈R0+ since μ(G~)>0. Then any u:Γ→H is bounded. Since the operator G on He is bounded, it is stable, and then G:He∣H→H is also bounded and causal, and, since the function u:Γ→H is bounded, then yf:Γ→H is also bounded with ∥yf∥≤∥G∥∥u∥ and ∥Ptyf∥≤∥PtGPt∥∥u∥≤∥G∥∥u∥; ∀t∈Γ, and y:Γ→H is also bounded since y=yh+yf. On the other hand, if r:Γ→H is identically zero, then one gets from (13) 0≤μ(G~)limsupΓ∋t→∞∥Ptu∥≤γ<∞, and, since μ(G~)>0, then ∃limΓ∋t→∞u(t)=0.
Also, it is clear that, sinceu:Γ→He, and since u:Γ→H is bounded and converges asymptotically to zero and ∥yf∥≤∥G∥∥u∥, then u:Γ→H, yf is bounded, yf:Γ→H, and then ∃limΓ∋t→∞y(t)=limΓ∋t→∞yf(t)=0 since yh:Γ→H is bounded and asymptotically vanishing.The assumption 6 of Theorem5 can be relaxed leading to the following stronger result.Corollary 6.
Theorem5 holds if its assumption 6 is relaxed to liminft→∞〈Pty,Pt(φΓ(y))〉≥-γ>-∞.Proof.
Note that (7) still holds since it is independent of assumption 6. The constraint (12) is modified as follows:
(15)liminfΓ∋t→∞〈Pty,Pt(Kr)〉+γt-μ2(G~)∥Ptu∥2-〈Pt(yh),Pt(Kr-φΓ(y))〉≥0
which makes (13) to remain valid, and Theorem 5 still holds.In a physical context,E=〈y,u〉 is the whole input-output energy of (1), E(t)=〈Pty,Ptu〉 is the input-output energy dissipated on [0,t]∩Γ, and (u*y)(t) is the instantaneous input-output power at t∈Γ while 〈Gu+yh,Kr〉 is the energy supplied by the external source. Particular cases of interest in control engineering are (a) if the reference input r≡0, then the feedback control system is a regulator evolving only from its initial conditions, (b) if such reference is a constant real level, then the control system is a position servomechanism, (c) if the reference r(t)=Kt for t∈Γ, then the control system is a velocity servomechanism and so forth.On the other hand, the extended Popov-type control inequality of the controller〈Pty,Pt(φΓ(y))〉≥-γ>-∞ and G≻0 implies that 0≤E(t)≤γ<∞, (0<E≤γ<∞ for any nonzero control input with compact support); ∀t∈Γ and all φΓ(y) satisfying the assumption 6 of Theorem 5; that is the input-output energy is nonnegative and bounded; ∀t∈Γ. The use of such a constraint allows the simultaneous investigation of the maintenance of the positivity and stability properties of (1) under a class of nonlinear time-varying controllers (defined by such a Popov constraint itself) rather than for a particular controller device belonging to such a class.Note thatG on He is stable since ∥Gu∥t≤M∥u∥t for some finite ∈R+; ∀t∈Γ and, equivalently, G:He∣H→H is bounded. Now, one concludes from Proposition 4 for the system defined by the inverse operator G-1 that 0≤〈Pt(G-1u),Ptu〉<∞ for any admissible control input u since G-1≻0, bounded and causal.The following result basically reformulates Theorem5 if G∣H is a strictly positive pseudocontraction. Since the contribution of initial conditions and a bounded exogenous reference do not modify the stability properties, as seen from Theorem 5, they are assumed to be null in the sequel.Theorem 7.
Assume that the relationships of (1) hold for all t∈Γ with r≡0, yh≡0, G:He→He, φΓ:Γ×He→ranφΓ and, furthermore,(1)
G:He∣H→H is bounded and causal, one-to-one, and with closed range.(2)
G≻0.(3)
〈Pty,Pt(φΓ(y))〉≥-γt≥-γ>-∞; ∀t∈Γ.Then, u,yf,y:Γ→H and are bounded, and u(t)→0, y(t)→0 as (Γ∋)t→∞. Furthermore, one gets for any, u1,u2:Γ→ranφΓ that
(16)〈u1,u2〉+〈u1,u2〉*≤2γμ2(G~).
If, in addition, G∣H is a pseudocontraction, then
(17)0≤〈Gu1-Gu2,u1-u2〉≤γmin(1,4μ2(G~))
with the lower-bound equating zero if and only if u1=u2.Proof.
Take the relation proved in Theorem5  〈y,u〉=〈Gu+yh,Kr-φΓ(y)〉under zero exogenous reference and initial conditions in (1) to yield
(18)0<μ2(G~)〈u,u〉≤〈Gu,u〉=〈G~u,G~u〉≤γ<∞.
Since G≻0 then G=G*, and one gets for u=u1-u2(19)0≤〈Gu1-Gu2,u1-u2〉=〈G~u1-G~u2,G~u1-G~u2〉=〈G~u1,G~u1〉+〈G~u2,G~u2〉-〈G~u1,G~u2〉-〈G~u2,G~u1〉=〈Gu1,u1〉+〈Gu2,u2〉-〈Gu1,u2〉-〈Gu2,u1〉≤2γ-〈Gu1,u2〉-〈u2,Gu1〉=2γ-〈Gu1,u2〉-〈Gu1,u2〉*⟹0≤max(0,〈Gu1,u2〉+〈Gu1,u2〉*)≤〈Gu1-Gu2,u1-u2〉-〈Gu1,u2〉-〈Gu1,u2〉*≤2γ<∞0≤∥u1-u2∥2=〈u1-u2,u1-u2〉=∥u1∥2+∥u2∥2-〈u1,u2〉-〈u1,u2〉*≤2γμ2(G~)-〈u1,u2〉-〈u1,u2〉*
and  〈u1,u2〉+〈u1,u2〉*≤2γ/μ2(G~). Assume that G≻0 is, furthermore, a pseucontraction on H. Then,
(20)0≤〈Gu1-Gu2,u1-u2〉=〈G~u1-G~u2,G~u1-G~u2〉≤∥u1-u2∥2
and, equivalently,
(21)∥Gu1-Gu2∥2≤∥u1-u2∥2+∥(I-G)u1-(I-G)u2∥20≤〈Gu1-Gu2,u1-u2〉≤〈G~u1-G~u2,G~u1-G~u2〉≤∥u1-u2∥2≤2γμ2(G~)-〈u1,u2〉-〈u1,u2〉*
implies that
(22)〈u1,u2〉+〈u1,u2〉*<〈Gu1-Gu2,u1-u2〉+〈u1,u2〉+〈u1,u2〉*≤2γμ2(G~),
and the following cases can occur.(a)
0≤〈Gu1-Gu2,u1-u2〉≤min(γ,2γ/μ2(G~)-〈u1,u2〉-〈u1,u2〉*) if the controls u1 and u2 fulfil 0<〈u1,u2〉+〈u1,u2〉*<2γ/μ2(G~).(b)
0≤〈Gu1-Gu2,u1-u2〉≤min(γ,2γ/μ2(G~)-|〈u1,u2〉+〈u1,u2〉*|)if the controls u1 and u2 fulfil 〈u1,u2〉+〈u1,u2〉*<0.(c)
0<2γ/μ2(G~)≤〈Gu1-Gu2,u1-u2〉≤γif the controls u1 and u2 fulfil 〈u1,u2〉+〈u1,u2〉*≥2γ/μ2(G~). This case is only feasible with equality.Combining the three cases one gets that
(23)0≤〈Gu1-Gu2,u1-u2〉≤γmin(1,4μ2(G~))
with the lower-bound equating zero if and only if u1=u2; that is u=u1-u2=0.Basically, Theorem7 states that a strictly positive operator, which is also a pseudocontraction, subject to a feedback control law satisfying a Popov-type inequality keeps the boundedness of the input-output energy with a modified upper-bound which improves that associated to the Popov inequality if the minimum modulus of G satisfies μ(G)>4. The following result guarantees the fulfilment of Theorem 5 if G:H→H is strictly positive and asymptotically pseudocontractive in the intermediate sense under a modified Popov-type inequality.Theorem 8.
Assume that(1)
G≻0 is one-to-one, bounded, causal, and of closed range with minimum modulus μ(G)>α,(2)
G:H→H is asymptotically pseudocontractive in the intermediate sense satisfying the constraint,
(24)0≤〈Pty-1-Pty-2,Ptu-1-Ptu-2〉≤αt∥Ptu-1-Ptu-2∥2;∀t∈Γ
for some real convergent sequence {αt}t∈Γ in [α,∞) such that αt→α∈(0,1] as t(∈Γ)→∞ and zero initial conditions and exogenous reference in (1), where
(25)y-t=yt+T-yt=Pt+Ty-Pty,u-t=ut+T-ut=Pt+Tu-Ptu
are incremental values of y and u with t,t+T(≻t)∈Γ being adjacent elements in the strict ordering on Γ if such an indexing set is discrete and [t,t+T] being a closed interval of nonzero constant Lebesgue measure T in Γ if such an indexing set is real,(3)
the following inequality holds:(26)liminfΓ∋t→∞(〈Pty,Pt(φΓ(y-))〉+αt∥u-∥t2)≥0.
Then, u,y:Γ→H, and they are bounded, and, furthermore, u(t)→0, y(t)→0 as (Γ∋)t→∞ under a zero exogenous input and initial conditions.Proof.
SinceG:H→H is asymptotically pseudocontractive in the intermediate sense
(27)0≤〈Pty-1-Pty-2,Ptu-1-Ptu-2〉=〈PtGPtu-1-PtGPtu-2,Ptu-1-Ptu-2〉≤αt∥Ptu-1-Ptu-2∥2;∀t∈Γ
for zero initial conditions and exogenous reference in (1). Note that these particular conditions do not modify the boundedness-type stability properties related to the injection of any bounded exogenous reference under bounded initial conditions and some real convergent sequence {αt}t∈Γ in [α,∞) such that αt→α∈(0,1) as t(∈Γ)→∞. Since
(28)liminfΓ∋t→∞(〈Pty-,Pt(φΓ(y-))〉+αt∥u-∥t2)=liminfΓ∋t→∞(αt∥Ptu-∥2-〈Pty-,Ptu-〉)≥0,
then one has
(29)0≤μ2(G~)∥u-t∥2=μ2(G~)∥u-∥t2≤αt∥u-∥t2⟹(-αt∥u-∥t2≤(μ2(G~)-αt)∥u-∥t2≤0),∀t∈Γ
so that, if α=limΓ∋t→∞αt<μ2(G~), then limsupΓ∋t→∞(μ2(G~)-αt)∥u-∥t2≤0 and u(t)→0 as Γ∈t→∞. u:Γ→H is bounded since it is piecewise continuous with eventual bounded discontinuities, and T>0 and finite. Since G on He and G restricted to H are stable, y:Γ→H is also bounded and converges to zero.A particular case of Theorem8 of interest is as follows.Corollary 9.
Theorem8 holds if the assumption 2 is replaced by G:H→H being a pseudocontraction.Proof.
It follows since Theorem8 holds, in particular, under the condition αt=α=1; ∀t∈Γ.IfG:H→H is strictly positive and contractive, we obtain the subsequent result.Theorem 10.
Assume that(1)
G≻0 is one-to-one, bounded, causal, and of closed range.(2)
G:H→H satisfies the following positive-bounded and contractive constraints for some given β∈Γ and u=u1-u2:
(30)0≤〈PβGPβu1-PβGPβu2,Pβu1-Pβu2〉-〈P0GP0u1-P0GP0u2,P0u1-P0u2〉≤M<∞(31)0≤〈Pt+2βGPt+2βu1-Pt+2βGPt+2βu2,Pt+2βu1-Pt+2βu2〉-〈Pt+βGPt+Tu1-Pt+βGPt+βu2,Pt+αu1-Pt+βu2〉≤χt〈Pt+βGPt+ϕu1-Pt+βGPt+βu2,Pt+βu1-Pt+βu2〉-χt〈PtGPtu1-PtGPtu2,Ptu1-Ptu2〉;∀t∈Γ
with {χt}t∈Γ being a real sequence subject to 0≤χt≤σ<∞ and ∏Γ∋tt+β[χt]=χ<1 with β, (t+nβ)∈Γ;  ∀n∈Z+,  ∀t∈Γ provided that 0∈Γ is the first element of Γ, and(3)
〈Pty,Pt(φΓ(y))〉≥-γt≥-γ=M/(χ-1)>-∞; ∀t∈Γ.Then, u,y:Γ→H, and they are bounded, and, furthermore, u(t)→0, y(t)→0 as (Γ∋)t→∞ under a zero exogenous input and initial conditions.Proof.
Note from (30) that
(32)0≤〈PnβGPnβu1-PnβGPnβu2,P(n-1)βu1-P(n-1)βu2〉≤χn-1(〈PβGPβu1-PβGPβu2,Pβu1-Pβu2〉-〈P0GP0u1-P0GP0u2,P0u1-P0u2〉(〈PβGPβu1-PβGPβu2,Pβu1-Pβu2〉)=Mχn-1≤M<∞
since nβ∈Γ; ∀n∈Z+, and 0∈Γ is the first element of Γ. Thus, since 0≤χ<1, one gets
(33)0≤∑n∈Z+〈PnβGPnβu1-PnβGPnβu2,P(n-1)βu1-P(n-1)βu2〉≤〈Gu1-Gu2,u1-u2〉≤γ=M1-χ<∞
and also
(34)0≤∑n=1N‍〈PnβGPnβu1-PnβGPnβu2,P(n-1)βu1-P(n-1)βu2〉≤〈Gu1-Gu2,u1-u2〉≤γ=M1-χ<∞,∀N∈Z+.
Thus, u(t)=u1(t)-u2(t)→0 as Γ∈t→∞. Since u≡0 is an admissible control, one concludes that any admissible control is bounded and it converges asymptotically to zero. The output y(t) has a similar property.


## 3. Application Examples

Example 1.
Define the truncated function within the time interval[0,t]; ∀t∈R0+ of f:R→R as follows:
(35)ft(τ)=Ptf={f(τ)forτ∈[0,t]0otherwise.
Thus, the output of a single-input single-output linear time-invariant continuous-time dynamic system of nth order and initial state x(0)=x0∈Rn under a piecewise continuous control with eventual isolated bounded discontinuities u:R→R∩H, where H=L2(R0+)≡L2[0,∞) the Hilbert space of the square-integrable functions on R0+ is
(36)y(t)=∫0tg(t,τ)u(τ)dτ+c(t,x0)=∫-∞∞g(t,τ)ut(τ)dτ+c(t,x0)=(g*u)(t)+c(t,x0),∀t∈R0+,
where Γ=R0+={z∈R:z≥0}, g:R×R→R is the impulse response, c(t,x0) is the zero-input response (i.e., the response contribution due to initial conditions) for initial sate x0, and “*” stands for the convolution integral operator. Since the dynamic system is realizable, g(t,τ)=0 for τ>t. The complex function G:C→C defined as G(s)=L(g(t)) is the transfer function, where L stands for the Laplace transform of the impulse response where it exists. After defining y(t)=0 for t<0, the input-output energy obeys the following relations by using twice Parseval theorem:
(37)E(t)=∫0ty(τ)u(τ)dτ=∫-∞∞y(τ)ut(τ)dτ=12π∫-∞∞Y(iω)Ut(-iω)dω=〈y,u〉t=〈y,ut〉=〈yt,ut〉=〈yt,u〉=12π〈Y,Ut〉=12π∫-∞∞G(iω)Ut(iω)Ut(-iω)dω+∫0tc(τ,x0)u(τ)dτ=12π∫-∞∞G(iω)|Ut(iω)|2dω+∫0tc(τ,x0)u(τ)dτ=12π〈GUt,Ut〉+∫0tc(τ,x0)u(τ)dτ=12π〈G,|Ut|2〉+∫0tc(τ,x0)u(τ)dτ≥12π(minω∈R0+ReG(iω))∫-∞∞|Ut(iω)|2dω=(minω∈R0+ReG(iω))∫-∞∞|ut(τ)|2dτ+∫0tc(τ,x0)u(τ)dτ=(minω∈R0+ReG(iω))∫0t|u(τ)|2dτ+∫0tc(τ,x0)u(τ)dτ=(minω∈R0+ReG(iω))〈ut,ut〉+∫0tc(τ,x0)u(τ)dτ=(minω∈R0+ReG(iω))〈u,u〉t+∫0tc(τ,x0)u(τ)dτ;∀t∈R0+,
where F(iω) is the pointwise value at frequency ω of F:C→C, the Fourier transform of f:R→R provided that it exists with i=-1 being the complex unit. Note that, in the previous expressions, the integral expressions have been also denoted by inner products 〈·,·〉t≡〈·t,·〉≡〈·,·t〉 on the time interval [0,t] for the given t∈R0+, all of them being equivalent to inner products of truncated functions for the given t∈R0+on the Hilbert space L2[0,∞). Equivalently, integrals of complex Fourier transforms on the whole imaginary axis are got through Parseval’s theorem and denoted by 〈GUt,Ut〉 involving the impulse response (i.e., the transfer function evaluated on the imaginary complex axis) of the system and the Fourier transform of the truncated input. Now, assume that the controller is
(38)u(t)=k(t)r(t)-φ(t,y(t));∀t∈R0+.r:R→R is an exogenous reference signal which is piecewise continuous on R0+, and φ:[0,t]×R→R is any piecewise continuous nonlinear time-varying function which satisfies the following integral-type constraint:
(39)∫0tφ(τ,y(τ))y(τ)dτ≥-γ(t,x0),∀t∈R0+,
then
(40)E(t)=∫0ty(τ)u(τ)dτ=∫0ty(τ)k(τ)r(τ)dτ-∫0ty(τ)φ(τ,y(τ))dτ≤γ(t,x0)+∫0ty(τ)k(τ)r(τ)dτ.
Note that any hodograph G(iω) has the symmetry rules ReG(iω)=ReG(-iω) and ImG(iω)=-ImG(-iω). Also, μ(G)≥minω∈R0+ReG(iω)>0. Thus, one gets by combining (37) and (40)
(41)(minω∈R0+ReG(iω))∫0t|u(τ)|2dτ+∫0tc(τ,x0)u(τ)dτ≤γ(t,x0)+∫0ty(τ)k(τ)r(τ)dτ;∀t∈R0+.
Decompose [0,t]=I1ε(t)∪I2ε(t) for each t∈R0+, where
(42)I1ε(t)={τ∈[0,t]:|u(τ)|≤ε},I2ε(t)={τ∈[0,t]:|u(τ)|>ε},
for some given prefixed ε∈R+. Note that one (but not both) of the disjoint sets Iiε(t) for i=1,2 can be empty. Then, by direct calculations one gets the following:
(43)(minω∈R0+ReG(iω))∫0t|u(τ)|2dτ+∫0tc(τ,x0)u(τ)dτ≥(minω∈R0+ReG(iω))∫0t|u(τ)|2dτ-ε∫I1(t)|c(τ,x0)|dτ+∫I2(t)c(τ,x0)u(τ)|u(τ)|2dτ≥(minω∈R0+ReG(iω))∫0t|u(τ)|2dτ-ε∫I1(t)|c(τ,x0)|dτ-∫I2(t)|c(τ,x0)u(τ)||u(τ)|2dτ≥(minω∈R0+ReG(iω)-ε-1maxt∈R0+|c(t,x0)|)×∫0t|u(τ)|2dτ-ε∫I1(t)|c(τ,x0)|dτ;∀t∈R0+.
Assume that supt∈R0+|k(t)|≤k- and supt∈R0+|r(t)|≤r-. Then, one gets from (41) and (43) that
(44)∫0t(|u(τ)|-k-r-maxω∈R0+|G(iω)|)|u(τ)|dτ≤1minω∈R0+ReG(iω)-ε-1maxt∈R0+|c(t,x0)|×(γ(t,x0)+ε∫I1(t)|c(τ,x0)|dτ),∀t∈R0+.
This relation leads to the following result.Proposition 11.
Assume that(1)
k,r∈L∞(R0+)so thatsupt∈R0+|k(t)|≤k-<∞andsupt∈R0+|r(t)|≤r-<∞,(2)
the transfer functionG(s)is strongly strictly positive real; that is,ReG(s)>0for all complexswithRes≥0,(3)
+∞>limsupt→∞(γ(t,x0))≥liminft→∞(γ(t,x0))>0;∀x0∈Rn.Then, one gets the following properties for any given initial state x0∈Rn.(i)
u,y∈L∞(R0+).(ii)
liminft→∞∫0ty(τ)k(τ)r(τ)dτ>-∞;liminft→∞(u2(t)-y(t)k(t)r(t))≤0.(iii)
If|u(t)|2≥y(t)k(t)r(t); ∀t∈R0+, then∃limt→∞(|u(t)|2-y(t)k(t)r(t))=0. If, in addition,k(t)=kandr(t)=rare nonzero constants;∀t∈R0+,then∃limt→∞y(t)=y∞and∃limt→∞u(t)=u∞=kr∫0∞g(τ)dτand∃limt→∞y(t)=y∞=kr(∫0∞g(τ)dτ)2.(iv)
ifr≡0, thenu(t)→0andy(t)→0ast→∞and are both square-integrable onR0+;∀x0∈Rn. Thus, the closed-loop dynamic system (36), (38) is asymptotically hyperstable (i.e., globally asymptotically Lyapunov’s stable, [1–3]) since the state of any minimal state-space realization is also square-integrable onR0+, and it converges asymptotically to zero as time tends to infinity for any controller deviceφ:[0,t]×R→Rsatisfying (39).Proof.
Since the transfer functionG(s) is strictly positive real then it is strictly stable (i.e. all its poles are in Res≤-ρ<0 for some ρ∈R+) and ReG(s)>0 for all complex s with Res≥0. Since it is, furthermore, strongly positive real (i.e., a strictly positive operator on L2(R0+)), and it is associated to a dynamic system, so that it is realizable, then it is rational with pole-zero excess is zero (otherwise, if the pole-zero excess was +1, then it could not be strongly strictly positive real since limω→±∞ReG(iω)=0, and if the pole-zero excess was −1 then it would not be realizable.) Since it has the same number of zeros and poles, and it is strongly strictly positive real, then its modulus is everywhere bounded in its definition domain, invertible, and of bounded inverse, so that one has
(45)0<minω∈R0+ReG(iω)≤maxω∈R0+|G(iω)|<+∞.
Note that ∫I1(t)|c(τ,x0)|dτ≤∫0∞|c(τ,x0)|dτ=v(x0)<+∞ since limt→∞|c(t,x0)|=0 at exponential rate since the dynamic system is strictly stable. Since ε∈R+ can be chosen arbitrarily to build the disjoint union I1ε(t)∪I2ε(t) equalizing [0,t]; ∀t∈R0+, then choose ε>|c(t,x0)|/(minω∈R0+ReG(iω)). Now, assume that u:R→R is unbounded. Since, it is piecewise continuous with eventual bounded discontinuities, then limt→∞∫0t|u(τ)|dτ=∞ which implies that ∫0t(|u(τ)|-k-r-maxω∈R0+|G(iω)|)|u(τ)|dτ is strictly increasing so that the subsequent contradiction follows
(46)+∞=limt→∞∫0t(|u(τ)|-k-r-maxω∈R0+|G(iω)|)|u(τ)|dτ≤limt→∞∫0t(|u(τ)|2-k(τ)r(t)y(τ))dτ≤1minω∈R0+ReG(iω)-ε-1maxt∈R0+|c(t,x0)|×(γ(t,x0)+ε∫I1(t)|c(τ,x0)|dτ)<+∞,∀t∈R0+.
Thus, u∈L∞(R0+). Since G(s) is strictly stable and u∈L∞(R0+), then y∈L∞(R0+). Property (i) has been proved. On the other hand, if  liminft→∞(|u(t)|2-y(t)k(t)r(t))>0, then limt→∞∫0t(|u(τ)|2-k(τ)r(t)y(τ))dτ=+∞, and the above contradiction holds. Then, liminft→∞(|u(t)|2-y(t)k(t)r(t))≤0. Note also that if limt→∞∫0ty(τ)k(τ)r(τ)dτ=-∞, then the subsequent contradiction follows
(47)0≤limt→∞∫0t|u(τ)|2dτ≤1minω∈R0+ReG(iω)-ε-1maxt∈R0+|c(t,x0)|×(γ(t,x0)+∫0t[ε|c(τ,x0)|+k(τ)r(t)y(τ)]dτ)=-∞,∀t∈R0+.
Then, liminft→∞∫0ty(τ)k(τ)r(τ)dτ>-∞. Property (ii) has been proven.
Note that∃limt→∞(|u(t)|2-y(t)k(t)r(t))=0 if |u(t)|2≥y(t)k(t)r(t); ∀t∈R0+ is a direct consequence of liminft→∞(|u(t)|2-y(t)k(t)r(t))≤0 from Property (ii). This proves the first part of Property (iii). Also, if k(t)=k and r(t)=r are nonzero constants; ∀t∈R0+, then limt→∞(|u(t)|2-y(t)kr)=0.
Now, ifr(t) is identically zero in R0+, then
(48)limt→∞∫0t|u(τ)|2dτ≤1minω∈R0+ReG(iω)-ε-1maxt∈R0+|c(t,x0)|×(γ(t,x0)+ε∫I1(t)|c(τ,x0)|dτ)<+∞,×(γ(t,x0)+ε∫I1(t)|c(τ,x0)|dτ)<∞,∀t∈R0+
leads to limt→∞u(t)=0 exponentially and the limt→∞y(t)=0; ∀x0∈Rn since G(s) is strongly strictly positive real so that the internal state of any minimal state-space realization is uniformly bounded, and it converges asymptotically to zero as time tends to infinity. Thus, asymptotic hyperstability follows for any φ:[0,t]×R→R satisfying (38). As a result, Property (iv) has been proven.Note that the property of asymptotic hyperstability is independent of each particular controller provided that it belongs to a class that satisfies the integral relation (39) for some positive finite real γ. The particular case when the nonlinear controller is nonlinear, but time-invariant, while satisfying the corresponding integral constraint (39), is said to be the Popov-type absolute stability problem implying closed-loop global asymptotic Lyapunov’s stability. If the input-output euclidean inner product (associated with instantaneous power) under the integral symbol, rather than the inner product on the Hilbert space (associated with the energy), satisfies a parallel inequality, then the problem is said to be that of the Lure’s absolute stability problem [4–8]. It is, therefore, useful to describe the global asymptotic stability of classes of closed-loop systems of the given form under certain tolerated components dispersions. Proposition 11 also implies directly that any nonminimal state-space realization associated with strictly stable zero-pole cancellations of the transfer function is globally asymptotically Lyapunov stable. This follows since the transfer function remains invariant under zero-pole cancellations, so it is identical to that of the minimum state space realization, so that the operator is kept strictly positive and invertible although either controllability or observability (or both) becomes lost [29–31]. A generalization of the previous result to the study of hyperstability of composite connections [32] as well to Ulman-type extended stability [33, 34] of continuous-time dynamic systems can be performed based on the study given in [32].The subsequent example is a discrete version of the previous one.Example 2.
Example1 has a direct parallel discrete-time counterpart as discussed in the sequel. Define the truncated sequence on [0,kT]; ∀k∈Γ=Z0+ of the real sequence {fk≡f(kT)}k∈Z0+ as follows:
(49)fj(k)=Pjf(kT)={fj≡f(jT)forj∈[0,k]∩Z0otherwise;∀k∈Z0+,
where T>0 is the sampling period. Thus, the output of a single-input single-output linear continuous-time dynamic system of nth order and initial state x(0)=x0∈Rn under a piecewise continuous control with eventual isolated bounded discontinuities u:R→R∩H, now the Hilbert space being H=ℓ2(Z0+), is
(50)yk=∑j=0kgd(k,j)uj+c(k,x0)=∑j=-∞∞gd(k,j)uk(j)+ck(x0)=(gd∘u)(k)+ck(x0),∀k∈Z0+,
where Z0+={z∈Z:z≥0}, “∘” stands for the discrete convolution operator, ck=ck(x0)≡ck(kT,x0) and {gd(k,j)}k,j∈Z0+ is the impulse response sequence since the dynamic system is realizable gd(k,j)=0 for j>k. If this dynamic system is the same system as in the previous example subject to a piecewise control sequence {uk}k∈Z0+, with uk=u(kT); ∀k∈Z0+, then gd(k,0)=(1-q-1)L-1(G(s)/s)t=kT; ∀k∈Z0+ where q-1 is the one-step delay operator such that fk=q-1fk+1. In this case, the discrete controller is
(51)uk=kkrk-φk(k,yk);∀k∈Z0+.r:R→R is an exogenous reference sequence, and φ:[0,k]×R→R are the elements of any nonlinear time-varying real sequence {φk=φk(k,yk)}k∈Z0+ which satisfies the following summation-type constraint:
(52)∑j=0kφj(j,yj)yj≥-γk(x0);∀k∈Z0+.
A close discussion to that of the former example by using the discrete Parseval theorem and inner products on the Hilbert space of square-summable sequences ℓ2(Z0+) yields
(53)Ek=∑j=0kyjuj=∑j=-∞∞yjuj(k)=12π∮|z|=1z-1Y(z)U(k,z-1)dz=12π∫02πY(eiφ)U(k,e-iφ)e-iφdφ=〈y,u(k)〉=12π〈Y,U(k)〉=12π〈GdU(k),U(k)〉+∑j=0kcj(x0)uj=12π〈Gd,|U(k)|2〉+∑j=0kcj(x0)uj≥12π(minφ∈[0,2π]ReGd(eiφ))∫02π|U(k,eiφ)|2dφ=(minφ∈[0,2π]ReGd(eiφ))(∑j=-∞∞uj2(k))+∑j=0kcj(x0)uj=(minφ∈[0,2π]ReGd(eiφ))(∑j=0kuj2(k))+∑j=0kcj(x0)uj,∀k∈Z0+.
By using (53), one gets a discrete counterpart of (44) as follows:
(54)∑j=0k(|uj|-k-r-maxφ∈[0,2π]|Gd(eiφ)|)|uj|≤1minφ∈[0,2π]ReGd(eiφ)-ε-1maxk∈Z0+|ck(x0)|×(γk(x0)+ε∑j=0k|ck(x0)|),∀k∈Z0+
which leads to the subsequent result which is the discrete-time counterpart of Proposition 11, whose proof is close to that of Proposition 11, and it is then omitted.Proposition 12.
Assume that(1)
k,r∈ℓ∞(Z0+)so thatsupj∈Z0+|kj|≤k-<∞andsupk∈Z0+|rk|≤r-<∞,(2)
the discrete functionGd(z)is strongly strictly positive real; that is,ReG(z)>0for all complexzwith|z|≥1.(3)
+∞>limsupk→∞(γk(x0))≥liminft→∞(γk(x0))>0;∀x0∈Rn.Then, one gets the following properties for any given initial state x0∈Rn.(i)
u,y∈ℓ∞(Z0+).(ii)
liminfk→∞(∑j=0kyjkjrj)>-∞;liminfj→∞(uj2-yjkjrj)≤0.(iii)
Ifuj2≥yjkjrj; ∀j∈Z0+, then∃limj→∞(uj2-yjkjrj)=0. If, in addition,kj=kandrj=rare nonzero constants;j∈Z0+, then∃limj→∞yj=y∞and∃limj→∞uj=u∞=kr(∑j=0∞gj)and∃limj→∞yj=y∞=kr(∑j=0∞gj)2.(iv)
Ifr≡0, thenuj→0andyj→0asj→∞, and they are both square-summable onZ0+;∀x0∈Rn. Thus, the closed-loop discrete dynamic system (50)-(51) is asymptotically hyperstable for any controller device of output sequence{φk=φk(k,yk)}k∈Z0+satisfying the discrete summation inequality∑j=0kφj(j,yj)yj≥-γk(x0);∀k∈Z0+.The following example links asymptotic hyperstability of a discrete dynamic system with a unique equilibrium point which is also a fixed point.Example 3.
Assume that, in Example2, a feedback stabilizing discrete control law ut=-φt(t,yt-1)yt-1 satisfying the constraint ∑j=0tφj(j,yj)yj≥-γt(x0)≥-γ>∞; ∀j, t∈Z0+ is injected to the system (1), neglecting initial conditions, and equivalently if the initial conditions are zero (this assumption does not affect the stability study,) we get
(55)Pty=yt=-(PtG)u=-(PtGφt(t,yt-1)Pt-1)y=-(PtGφt(t,Pt-1y)Pt-1)y;∀t∈Z0+
so that the closed-loop system can be described by the operator Q:He|ℓ2(Z0+)→He|ℓ2(Z0+) represented as
(56)yt=PtQPt-1y=-PtGφt(t,yt-1)yt-1;∀t∈Z0+
or, equivalently, as
(57)Pt(I+Gφt(t,Pt-1y)Pt-1)y=0;∀t∈Z0+.
Assume that Q:ℓ2[0,z)→ℓ2[0,z) for any z∈Z+ is stable, positive, one-to-one, and of closed range. Then, Q:He|ℓ2(Z0+)→He|ℓ2(Z0+), where He=⋃z∈Z+ℓ2[0,z) is positive, bounded and of closed range, invertible and of nonzero minimum modulus; and
(58)0≤E(t)=〈PtQPt-1y,Pty〉≤γ<∞;∀t∈Z0+.
Since E(t) is nonnegative, bounded, and nondecreasing, yt→0 as t(∈Z0+)→∞, and then ut=-φt(t,yt-1)yt-1→0 as t(∈Z0+)→∞. One gets for any given finite integer T>0 that limΓ∋t→∞(Pt+Tu-Pru)=limΓ∋t→∞(Pt+Ty-Pty)=0. Thus, y-(T)≡0(∈ℓ2[t,t+T]) is the unique fixed point of Q:He|ℓ2[t,t+T]→He|ℓ2[t+T,t+2T]; ∀t∈Γ.



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