# On Some Relations between Accretive, Positive, and Pseudocontractive Operators and Passivity Results in Hilbert Spaces and Nonlinear Dynamic Systems

Authors: M. De la Sen
Journal: Discrete Dynamics in Nature and Society
 (2017)

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## Abstract

This paper investigates some parallel relations between the operatorsI-G and G in Hilbert spaces in such a way that the pseudocontractivity, asymptotic pseudocontractivity, and asymptotic pseudocontractivity in the intermediate sense of one of them are equivalent to the accretivity, asymptotic accretivity, and asymptotic accretivity in the intermediate sense of the other operator. If the operators are self-adjoint then the obtained accretivity-type properties are also passivity-type properties. Such properties are very relevant in stability theory since they refer to global stability properties of passive feed-forward, in general, nonlinear, and time-varying controlled systems controlled via feedback by elements in a very general class of passive, in general, nonlinear, and time-varying controllers. These results allow the direct generalization of passivity results in controlled dynamic systems to wide classes of tandems of controlled systems and their controllers, described by G-operators, and their parallel interpretations as pseudocontractive properties of their counterpart I-G-operators. Some of the obtained results are also directly related to input-passivity, output-passivity, and hyperstability properties in controlled dynamic systems. Some illustrative examples are also given in the framework of dynamic systems described by extended square-integrable input and output signals.


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## Body

## 1. Introduction

There is an important existing background literature available concerning passivity topics in dynamic systems. See, for instance, [1–6]. The passivity property in dynamic systems is closely related to that of positivity of the operator which describes the input-output behaviour of the system and it is a very general issue of global stability. In particular, the so-called Popov’s hyperstability property of control systems has received a very important attention since it is basically related to the global closed-loop Lyapunov stability when (a) the feed-forward part of the control system (typically, the controlled system) is hyperstable and (b) the feedback part (typically, the controller) is any element belonging to a certain family of, in general, nonlinear and time-varying devices satisfying a hyperstability condition in terms of fulfilment of a Popov’s type inequality [7, 8]. Thus, the closed-loop system is hyperstable if the controlled system and its controller are both hyperstable in the above senses. In the case when the controlled system is linear and time-invariant, its hyperstability property can be mathematically characterized by its transfer matrix being positive real which is closely related to the dissipativity and passivity (or positivity) of such a system and this translates in parallel in the feature that its associated input-output energy is nonnegative for all time irrespective of the controller under operation. It is well-known that the asymptotic hyperstability formalism covers particular cases, the so-called Lure’s and Popov’s absolute stability problems. See, for instance, [1–15]. On the other hand, the so-called passivity property of dynamic systems can be described in the time-domain in terms of evolution of the so-called storage functions [6] and translates in the global Lypunov stability of all the feedback systems integrated by a feed-forward hyperstable controlled system and any controller belonging to the class of controllers satisfying a Popov’s type inequality. Passivity of dynamic systems is also important since it relies on both conservative and dissipative systems. It has become apparent that such a property admits a precise characterization through constraints on the operators describing the feed-forward and feedback parts of the controlled dynamic system. On the other hand, there is also a rich literature on fixed point theory which is very related to convergence of sequences to fixed points and to convergence of trajectory solutions and sequences to equilibrium points, in general, when applied to dynamic systems. See, for instance, [16–22] and the abundant included background literature in those background references. In particular, the so-called pseudocontractions, asymptotic pseudocontractions, and asymptotic pseudocontractions in the intermediate sense in the framework of Hilbert spaces have also received an important attention along the last three decades. See [16–20] and references therein. On the other hand, some research on stability of topological stability of time-varying maps has been given in [23] while some results on stability of certain positive linear operators have been provided in [24]. Also, weaker-type contractive assumptions have been addressed in [25] in the context of metric and geodesic spaces and related “ad hoc” results have been obtained. See also [26, 27] and references therein concerning Ulam’s type stability problems and stability conditions for switched dynamic systems.By taking advantage of certain formally obtained relations of the pseudocontractive properties of an operatorI-G and the accretive properties of its counterpart operator G in Hilbert spaces, the objective of this paper is to derive general conditions of the properties of accretivity, positivity, and passivity and their strict and asymptotic versions of an operator are asymptotically strictly pseudocontractive in the intermediate sense on a Hilbert space based on asymptotic pseudocontractive-type conditions on the operator I-G, the less restrictive asymptotic passivity conditions on G being obtained if I-G is asymptotically strictly, or strongly strictly pseudocontractive in the intermediate sense. The obtained results are applied for the Hilbert spaces of square-integrable vector-valued functions so as to formulate general conditions on stability, hyperstability and passivity, and their asymptotic versions, of controlled dynamic systems formulated in the framework of such spaces. The passivity of the whole controlled conditions is decomposed on passivity-type conditions on both the controlled system and its controller. Note that the properties of passivity and hyperstability are very relevant properties in the field of dynamic systems since they are formulated jointly for classes of systems and controllers rather than for individual ones. Some illustrative examples are also given and discussed.


## 2. Some Preliminaries

Denote byR, R+, and R0+ the sets of real, positive real, and nonnegative real numbers, respectively, and by Z, Z+, and Z0+ the sets of real, positive real, and nonnegative integer numbers, respectively.If the measurable functionf:iR0+→Cn then fT:R+→C is the truncation of f on 0,T defined by fTit=fit for t∈0,T and fTit=0 for t>T and any finite T∈R0+, where i=-1 is the imaginary complex unit.L
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∞ is the LpiR0+;Cn-space of the p≥1-integrable complex n-vector functions of imaginary complex argument. In the same way, we can define spaces of truncated functions LpeR0+;Cn and Lpen≡LpeR0+;Rn from LpR0+;Cn and Lpn≡LpR0+;Rn, respectively.It is well known thatLpC;Cn endowed with an inner product 〈·,·〉 is a complete complex Hilbert space with norm x=〈x,x〉p1/p (i.e., a Banach space with respect to the norm defined by such an inner product). We can extend from [1] the basic passivity concepts, of high relevance in stability, stabilization, and hyperstability problems of dynamic systems [1–6, 9], for real square-integrable operators to complex operators on L2eiR0+;Cn,leading to real nonnegative inner products, as follows in the subsequent definitions.Definition 1.
An operatorG:L2eiR0+;Cn→L2eiR0+;Cn is said to bepassiveif(1)x,GxT≥0,∀T≥0,∀x∈L2e,it is said to bestrictly passive if there exists some real constant ε>0 such that(2)x,GxT≥εxT22,∀T≥0,∀x∈L2e,and it is said tobe strongly strictly passive if there exist some real constants ε>0 and δ>0 such that(3)x,GxT≥εxT22+δGxT22,∀T≥0,∀x∈L2e.A strongly related concept to passivity is that of positivity. It is possible to extend the definition of positive operator [10] on Hilbert spaces to positive operators on the corresponding space of truncated functions.Definition 2.
An operatorG:L2eiR0+;Cn→L2eiR0+;Cn is said to be positive if(4)Gx,xT≥0,∀T≥0,∀x∈L2e,and it is strictly positive if there exists some real constant ε>0 such that (5)Gx,xT≥εxT22,∀T≥0,∀x∈L2e.It turns out that a strictly passive (resp., strictly positive) operator is also passive (resp., positive).Proposition 3.
Consider operatorsG:L2eiR0+;Cn→L2eiR0+;Cn such that 〈Gx,x〉T is real ∀T≥0, ∀x∈L2e. Then, the following properties hold.
(i)G:L2eiR0+;Cn→L2eiR0+;Cn is passive if and only if its adjoint operator G∗:L2eiR0+;Cn→L2eiR0+;Cn is positive and, in particular, G:L2eiR0+;Cn→L2eiR0+;Cn is passive if and only if its transpose operator GT:L2eiR0+;Cn→L2eiR0+;Cn is positive.
(ii) IfG:L2eiR0+;Cn→L2eiR0+;Cn is self-adjoint then it is positive if and only if it is passive (i.e., positivity and passivity are equivalent concepts for self-adjoint operators).
(iii) IfG:L2eiR0+;Cn→L2eiR0+;Cn is positive then it is self-adjoint and passive.
(iv) IfG:L2eiR0+;Cn→L2eiR0+;Cn is symmetric then it is positive if and only if it is passive (i.e., positivity and passivity are equivalent concepts for real symmetric operators).
(v)G∗G is positive and passive. If G is normal then GG∗ is positive and passive for any G:L2eiR0+;Cn→L2eiR0+;Cn.Proof.
Property (i) is a direct consequence of Definitions1 and 2. On the other hand, G:L2eiR0+;Cn→L2eiR0+;Cn is self-adjoint if and only if the inner product 〈Gx,x〉T is real and 〈Gx,x〉T≥0,∀T≥0,∀x∈L2e. Then, the inner product 〈Gx,x〉T is nonnegative real with 〈Gx,x〉T=〈x,G∗x〉T=〈x,Gx〉T≥0,∀T≥0,∀x∈L2e. So, G:L2eiR0+;Cn→L2eiR0+;Cn is passive. From the same nonnegative real equalities, one concludes that if the operator is self-adjoint and passive then it is positive. Property (ii) is proved. Similarly, one concludes that if the operator is positive then it has to be self-adjoint and then passive, hence Property (iii). Property (iv) is a particular conclusion of the above ones for symmetric real operators. To prove Property (v), note that 〈G∗Gx,x〉T=〈Gx,Gx〉T=〈GxT,GxT〉=〈x,G∗Gx〉=GxT22≥0;∀x∈L2eiR0+;Cn,∀T≥0, and G∗G is positive and passive. If, in addition, G is normal then GG∗=G∗G so that GG∗ is positive and passive.Proposition 4.
Proposition3 also holds “mutatis-mutandis” for Properties (i)–(v) if G:L2eiR0+;Cn→L2eiR0+;Cn is strictly passive/strictly positive.
Proposition3 also holds “mutatis-mutandis” for Properties (i)–(v) if G:L2eiR0+;Cn→L2eiR0+;Cn is passive/positive and if it is strictly passive/strictly positive.Roughly speaking, it is concluded from Propositions3 and 4 that passivity (resp., strict passivity) and positivity (resp., strict positivity) are equivalent properties for complex self-adjoint and real symmetric operators. It is well-known that fixed point theory is a very useful tool to analyze stability and convergence problems in different applications, like, for instance, stability of continuous-time and discrete-time differential difference and hybrid equations, dynamic systems, and iterative computational processes. A main objective of this research is to discuss links between passivity properties versus pseudocontractive properties of operators in Hilbert spaces as well as generalize passivity bearing in mind the weaker pseudocontraction concept of that of pseudocontraction in the intermediate sense. See, for instance, [16–20]. Now, the passivity concepts for operators are related to those of pseudocontractions and pseudocontractions in the intermediate sense for alternative operators which are directly related to passive ones. The definition of accretive operators [16] can be applied to the space of truncated functions as follows.Definition 5.
LetE be an arbitrary real Banach space endowed with a scalar product 〈·,·〉 from E×E to R. An operator U with domain DU and range RU in E is called as follows:
(a)K-strictly accretive (or strictly accretive with constant K) if, for each x,y∈DU, there is a jx-y∈Jx-y, with J being the normalized duality mapping, such that 〈Ux-Uy,jx-y〉≥Kx-y2 with K>0.
The operatorU is strictly accretive if some such a positive constant K exists [16].
(b) Accretive if, for eachx,y∈DU, there is jx-y∈Jx-y such that, for each x,y∈DU, 〈Ux-Uy,jx-y〉≥0 [16].
(c)Kn,K0,K-asymptotically strictly accretive if there exist real constants K0>0 and K≥0 and a real sequence Kn⊂0,∞ satisfying liminfn→∞Kn≥K0 such that, for each x,y∈DU,(6)Uny,Unx,y-x≥Knx-y+KI-Uny-I-Unx;∀n∈Z+.
The operatorU is asymptotically strictly accretive in the intermediate sense if some such a triple Kn,K0,K exists.
The operatorU is Kn,K0,K-asymptotically strongly strictly accretive if it is Kn,K0,K asymptotically strictly accretive with K>0.
(d)Kn,K0,K-asymptotically strictly accretive in the intermediate sense if there exist real constants K0>0 and K≥0 and bounded real sequence Kn satisfying liminfn→∞Kn≥K0 such that, for each x,y∈DU,(7)liminfn→∞Uny-Unx,y-x-Knx-y-KI-Uny-I-Unx≥0.
The operatorU is asymptotically strictly accretive in the intermediate sense if some such a triple Kn,K0,K exists. The operator U is Kn,K0,K-asymptotically strongly strictly accretive in the intermediate sense if it is Kn,K0,K-asymptotically strictly accretive in the intermediate sense with K>0.
We give now incremental-type concepts of incremental passivity and incremental positivity to be then related to the accretive property as follows. First, Definition5 is extended to operators on L2eiR0+;Cn as follows.Definition 6.
An operatorU:L2eiR0+;Cn→L2eiR0+;Cn is as follows:
(a) Accretive if, provided that〈Ux-Uy,x-y〉T is real for each x,y∈DU and all T≥0, one has that, for each x,y∈DU and somejx-y∈Jx-y, we have 〈Ux-Uy,jx-y〉T≥0.
(b)U:L2eiR0+;Cn→L2eiR0+;Cn isK-strictly accretive, if 〈Ux-Uy,jx-y〉T≥Kx-yT22 for some positive real constant K and ∀T>0.
(c)U:L2eiR0+;Cn→L2eiR0+;Cn is Kn,K0,K-asymptotically strictly accretive if, for each x,y∈DU, there exist real constants K0>0 and K≥0 and a real sequence Kn⊂0,∞ satisfying liminfn→∞Kn≥K0 such that (8)Uny,Unx,y-x≥Knx-yT22+KI-Uny-I-UnxT22;∀n∈Z+,∀T>0.U:L2eiR0+;Cn→L2eiR0+;Cn is asymptotically strongly strictly accretive if it is asymptotically strictly accretive with K>0.
(d)Kn,K0,K-asymptotically strongly strictly accretive in the intermediate sense if, for each x,y∈DU, there exist real constants K0>0 and K≥0 and a bounded real sequence Kn satisfying liminfn→∞Kn≥K0 such that(9)liminfn→∞Uny,Unx,y-x-Knx-yT22-KI-Unx-I-UnyT22≥0,∀T>0.U:L2eiR0+;Cn→L2eiR0+;Cn is asymptotically strongly strictly accretive in the intermediate sense if it is asymptotically strictly accretive in the intermediate sense with K>0.The following definition is given extending the concepts of passivity to incremental passivity and to asymptotic incremental passivity in the intermediate sense.Definition 7.
An operatorG:L2eiR0+;Cn→L2eiR0+;Cn is said to be incrementally passive if(10)x-y,Gx-GyT≥0,∀T≥0,∀x,y∈L2e,and it is incrementally strictly passive if there exists some real constant ε>0 such that(11)x-y,Gx-GyT≥εx-yT22,∀T≥0,∀x,y∈L2e.An operator G:L2eiR0+;Cn→L2eiR0+;Cn is said to be asymptotically incrementally strictly passive in the intermediate sense if there exists some real constant ε>0 such that(12)liminfn→∞x-y,Gx-GyT-εx-yT22≥0,∀T≥0,∀x,y∈L2e.The counterpart definition to Definition 7 related to positivity follows.Definition 8.
An operatorG:L2eiR0+;Cn→L2eiR0+;Cn is said to be incrementally positive if(13)Gx-Gy,x-yT≥0,∀T≥0,∀x,y∈L2e,and it is incrementally strictly positive if there exists some real constant ε>0 such that (14)Gx-Gy,x-yT≥εx-yT22,∀T≥0,∀x,y∈L2e.An operator G:L2eiR0+;Cn→L2eiR0+;Cn is said to be incrementally strictly positive in the intermediate sense if there exists some real constant ε>0 such that(15)liminfn→∞Gx-Gy,x-yT-εx-yT22≥0,∀T≥0,∀x,y∈L2e.Remarks 9.
It turns out that one has the following:1.
An incrementally strictly passive (resp., incrementally strictly positive) self-adjoint operatorG is also incrementally passive (resp., incrementally positive).2.
In caseG0=0, the accretive property (resp., the strict accretive property) is equivalent to incremental positivity (resp., strict positivity) and to the respective incremental passivity concepts for self-adjoint operators.3.
IfG0=0, incremental passivity and strict passivity are equivalent to passivity and strict passivity [1], respectively, and, furthermore, to positivity and strict positivity, respectively, if G is self-adjoint.4.
IfG is self-adjoint with the same domain and codomain on a Hilbert space of finite dimension then the operator is Hermitian and in particular symmetric if the Hilbert space is real. In this case positivity, strict positivity, accretivity, and strict accretivity are equivalent to passivity, strict passivity, incremental passivity, and strict incremental passivity, respectively. If, in addition, G0=0 then the respective incremental properties are equivalent to the standard ones.Examples 10.
Simple examples of some of the relevant previously introduced operators are now described.1.
The operatorT:X→2X on a Banach space X isstrongly pseudocontractive if there exists t>1 such that, for all x,y∈DT and r>0, the following inequality holds:(16)1+rx+y-rtTx-Ty≥x-y.T ispseudocontractive if the above inequality holds with t=1 [4].2.
The operatorT:X→2X on a Banach space X isstrongly accretive if, for all x,y∈DT and r>0, there exists k>0 such that the following inequality holds for some k>0 and all x,y∈DT:(17)1-rkx-y+rTx-Ty≥x-y.T isaccretive if the above inequality holds with k=0, [4]. See [4, 28] and also [20] for the case of cyclic mappings.3.
A rational functionh^s of the complex variable s of real coefficients ispositive real if (1) it is real for real s, (2) it has no poles in the open right half plane, (3) its poles s=iω at the imaginary axis, if any, are simple and their associate residues are simple, and (4) for all real ω such that s=iω is not a pole, Reh^iω≥0. All these constraints together lead to Reh^s>0 for Res≥0.4.
Assume that such apositive real rational function h^s is a transfer function of a realizable linear time-invariant system of one single input and one single output. That is, it has nonnegative relative degree (i.e., nonmore zeros than poles) so that it describes in Laplace transforms the input-output relation (i.e., the zero initial state response) of such a dynamic system. Then, the operator h^s is bothpassiveandpositivesince it is self-adjoint by nature with h^0=0 and we can also say that the associated dynamic system is positive and passive. As a result, its input-output time integral is nonnegative for all time. A simple example is, for instance, h^s=1/s+a for a≥0 which is associated with the differential system y˙t=-ayt+ut, y0=y0. If a>0 then the transfer function isstrictly positive real (imaginary poles do not exist and the transfer function is stable satisfying also Reh^s>0 for Res≥0 and all ω∈-∞,+∞), and the associated dynamic system is strictly passive. If the transfer function is modified to h^′s=h^s+d then Reh^′s≥d for Res≥0 and all ω∈-∞,+∞. The transfer function has a relative degree zero and it is said to bestrongly strictly positive real (i.e., strictly positive real for any finite frequency and as frequency tends to ± infinity) if d>0, with d being a positive direct input-output interconnection gain in the dynamic system. Since the dynamic system is linear, the above properties imply also that it isincrementally passive andincrementally positive. See, for instance, [3, 4, 7, 29, 30]. The above examples are easily extendable to the discrete case, to the continuous-time and discrete-time multivariable cases (i.e., the cases when the output and/or the input can be vectors of dimensions greater than one), and also to dynamic systems of state dimensions being greater than one.
(a)
It can be pointed out that the external positivity of a dynamic system in the sense that the solution trajectory solution (roughly speaking, the system output) is nonnegative for all time under arbitrary nonnegative initial conditions and nonnegative controls for all time is a different problem to the positivity and related passivity discussed here. Note that thepositivity of the solution does not implynecessarily stability. Also, such an external positivity concept does not imply positivity for all time of the input-output energy for eventually negative controls. See, for instance, [31–33] and some references therein.Some properties and relations for accretive operators on specific complex spaces are given and proved as follows.Theorem 11.
Assume thatU:L2eiR0+;Cn→L2eiR0+;Cn is accretive; then the following properties hold for any x,y∈DU:1.
〈Ux-Uy,x-y〉T+2Ux-UyT2≥0, ∀T≥0.2.
If, in addition,U:L2eiR0+;Cn→L2eiR0+;Cn is strictly accretive with constant k, odd superadditive, and bounded of norm kU<1/2 then it is incrementally strictly positive with k1=k1-2kU and also incrementally strictly passive if the operator is self-adjoint.3.
Assume thatU0=0. If U:L2eiR0+;Cn→L2eiR0+;Cn is accretive then U is positive and, furthermore, passive if U is self-adjoint. If U:L2eiR0+;Cn→L2eiR0+;Cn is strictly accretive with constant k and bounded of norm kU<1/2 then it is strictly positive with k1=k1-2kU and, furthermore, it is strictly passive if the operator is self-adjoint.4.
Properties (i)–(iii) hold “mutatis-mutandis” ifU:L2eiR0+;Cn→L2eiR0+;Cn.Proof.
Sincejx-yT=x-yT then by using Schwartz’s inequality and the linearity properties of the Hilbert space,(18)Ux-Uy,x-yT+2Ux-UyT2x-yT2≥Ux-Uy,x-yT+Ux-UyT2jx-y+x-yT2≥Ux-Uy,x-yT+Ux-Uy,jx-y-x-yT=Ux-Uy,jx-y≥kx-yT22,and Property (i) follows by taking k=0 since U:L2eiR0+;Cn→L2eiR0+;Cn is strictly accretive. Also,(19)Ux-Uy,x-yT≥kx-yT2-2Ux-UyTx-yT2If U is strictly accretive, odd superadditive and bounded of norm kU<1/2 then (20)Ux-Uy,x-yT+2Ux-yTx-yT≥Ux-Uy,x-yT+2Ux+U-yT2x-yT2≥kx-yT22.and 〈Ux-Uy,x-y〉T≥k1-2kUx-yT22=k1x-yT22 and Property (ii) is proved so that the identity mapping jx=x fulfills the accretive property. Strict passivity/incremental strict passivity for a self-adjoint operator follows from Proposition 4. The first part of Property (iii) follows from Property (i), and the second part follows from Property (ii) without requiring odd superadditivity and boundedness, if Uy=y=0. Property (iv) is direct from Properties (i)–(iii) by changing the operator domain from iR0+ to R0+.Definition 12.
LetH be a real Hilbert space H. Then, an operator T from DT (the Domain of T) to RT (the Image of T) is as follows:
(a)K1,K-pseudocontractive in the wide sense if there exist K1,K∈0,1 such that(21)Tx-Ty2≤K1x-y2+KI-Tx-I-Ty2;∀x,y∈DT.The operator T:D(T)→R(T) is said to be pseudocontractive in the wide sense if such a pair K1,K exists.
The operatorT is nonexpansive if it is 1,0-pseudocontractive in the wide sense.
(b) Pseudocontractive if(22)Tx-Ty2≤x-y2+I-Tx-I-Ty2;∀x,y∈DT,and equivalently if(23)Tx-Ty,x-y≤x-y2⟺I-Tx-I-Ty,x-y≥0;∀x,y∈DT;see [16].
Note that ifT:DT→RT is pseudocontractive in the wide sense it is pseudocontractive as well and a pseudocontraction in the wide sense with K1=K=1 is equivalent to a pseudocontraction.
(c)K-strictly pseudocontractive if there exists a constant K∈0,1 such that (24)Tx-Ty2≤x-y2+KI-Tx-I-Ty2;∀x,y∈DT;see [16].
The operatorT:D(T)→R(T) is said to be strictly pseudocontractive if such a constant K exists.
(d)Kn,K-asymptotically strictly pseudocontractive if there exists K∈0,1 and a sequence Kn⊂0,∞ with Kn→1 as n→∞ such that(25)Tnx-Tny2≤Knx-y2+KI-Tnx-I-Tny2;∀x,y∈DT,∀n≥1∈Z.The operator T:DT→RT is said to be asymptotically strictly pseudocontractive if such a pair Kn,K exists.
IfK=1 then T:DT→RT is (nonstrictly) asymptotically pseudocontractive, and one has, equivalently (see [18]):(26)Tnx-Tny,x-y≤1+Kn2x-y2;∀x,y∈DT,∀n≥1∈Z.
(e)Kn,K0,K-asymptotically strongly strictly pseudocontractive if there exist K0,K∈0,1 and a sequence Kn⊂0,∞ with Kn→K0 as n→∞ such that(27)Tnx-Tny2≤Knx-y2+KI-Tnx-I-Tny2;∀x,y∈DT,∀n∈Z+.The operator T:D(T)→R(T) is said to be asymptotically strongly strictly pseudocontractive if such a triple Kn,K0,K exists. If K=1 then T:DT→RT is (nonstrictly) asymptotically pseudocontractive.
(f) Asymptotically nonexpansive if it isKn,0-asymptotically strictly pseudocontractive.
(g)Kn,K0,K-asymptotically strongly strictly pseudocontractive in the intermediate sense if there exist K0,K∈0,1 and a real sequence Kn⊂0,∞ satisfying Kn→K0 such that(28)limsupn→∞supx,y∈DTTnx-Tny2-Knx-y2-KI-Tnx-I-Tny2≤0.The operator T:DT→RT is said to be asymptotically strictly pseudocontractive in the intermediate sense if such a triple Kn,K0,K exists.
(h)Kn,K-asymptotically strictly pseudocontractive in the intermediate sense if there exists K∈0,1 and a real sequence Kn⊂0,∞ satisfying Kn→1 such that(29)limsupn→∞supx,y∈DTTnx-Tny2-Knx-y2-KI-Tnx-I-Tny2≤0.The operator T:DT→RT is said to be asymptotically strictly pseudocontractive in the intermediate sense if such pair Kn,K exists.
(i)Kn-asymptotically pseudocontractive in the intermediate sense if there exists a real sequence Kn⊂0,∞ satisfying Kn→1 such that(30)limsupn→∞supx,y∈DTTnx-Tny2-Knx-y2-I-Tnx-I-Tny2≤0;equivalently, (31)limsupn→∞supx,y∈DTTnx-Tny,x-y-1+Kn2x-y2≤0.The operator T:DT→RT is said to be asymptotically strictly pseudocontractive in the intermediate sense if such a sequence Kn exists.
(j)Kn,K0-asymptotically strongly pseudocontractive in the intermediate sense if there exists K0∈0,1 and a real sequence Kn⊂0,∞ satisfying Kn→K0 such that(32)limsupn→∞supx,y∈DTTnx-Tny2-Knx-y2-I-Tnx-I-Tny2≤0.The operator T:DT→RT is said to be asymptotically strongly pseudocontractive in the intermediate sense if such a sequence Kn exists.
The following result is obvious from Definition12(g)–(i).Proposition 13.
IfT:DT→RT is asymptotically strongly strictly pseudocontractive in the intermediate sense then it is asymptotically strictly pseudocontractive in the intermediate sense.
IfT:DT→RT is asymptotically strictly pseudocontractive in the intermediate sense then it is asymptotically pseudocontractive in the intermediate sense.
IfT:DT→RT is asymptotically strongly strictly pseudocontractive in the intermediate sense then it is asymptotically pseudocontractive in the intermediate sense.
IfT:DT→RT is asymptotically strongly pseudocontractive in the intermediate sense then it is asymptotically pseudocontractive in the intermediate sense.Theorem 14.
LetH be a real Hilbert space. Then, the following properties hold:
(i) Assume thatT:DT→RT is K1,K-pseudocontractive in the wide sense. Then I-T is strongly accretive and it satisfies(33)I-Ty-I-Tx,y-x≥3-2K-K13-Kx-y2;∀x,y∈DT.
(ii) Assume thatT:DT→RT is K-strictly pseudocontractive. Then I-T is strongly accretive and it satisfies(34)I-Ty-I-Tx,y-x≥1-K3-Kx-y2;∀x,y∈DT.
(iii) Assume thatT:DT→RT is pseudocontractive. Then, I-T is accretive.
(iv) Assume thatT:DT→RT is asymptotically strictly pseudocontractive. Then, I-T is asymptotically strictly accretive.
(v) Assume thatT:DT→RT is Kn,K0,K-asymptotically strongly strictly pseudocontractive in the intermediate sense. Then, I-T is asymptotically strongly strictly accretive in the intermediate sense satisfying for some K0,K∈0,1 and a convergent real sequence Kn⊂0,∞ with Kn→K0 that(35)liminfn→∞I-Tny-I-Tnx,y-x-2-K-K03-Kx-y2-1-K3-KTny-Tnx2≥0;∀x,y∈DT.
(vi) Assume thatT:DT→RT is Kn,K-asymptotically strictly pseudocontractive in the intermediate sense. Then, I-T is asymptotically strongly strictly accretive in the intermediate sense satisfying for some K∈0,1 and a convergent real sequence Kn⊂0,∞ with Kn→1 that(36)liminfn→∞I-Tny-I-Tnx,y-x-1-K3-Kx-y2-1-K3-KTny-Tnx2≥0;∀x,y∈DT.
(vii) Assume thatT:DT→RT is Kn,K-asymptotically strongly pseudocontractive in the intermediate sense. Then, I-T is asymptotically strictly accretive in the intermediate sense satisfying for some K0∈0,1 and a convergent real sequence Kn⊂0,∞ with Kn→K0 that(37)liminfn→∞I-Tny-I-Tnx,y-x-1-K02x-y2≥0;∀x,y∈DT.Proof.
Firstly, assume that,∀x,y∈DT is Kn,K-asymptotically strictly pseudocontractive, then one has for some sequence βn⊂R+, with βn=βnx,y, that(38)Tnx-Tny2=Knx-y2+KI-Tnx-I-Tny2-βn;∀n∈Z+=x-y2+I-Tnx-I-Tny2-αn-βn;∀x,y∈DT,∀n∈Z+,where αn is a real nonnegative sequence (39)αn=αnx,y=1-Knx-y2+1-KI-Tnx-I-Tny2;∀n≥1∈Z+.Relation (38) is equivalent to(40)Tny-Tnx,Tny-Tnx=x-y,x-y+Tnx-Tny-x+y,Tnx-Tny-x+y-αn-βn=y-x,y-x+Tnx-Tny,Tnx-Tny+Tnx-Tny,y-x+y-x,Tnx-Tny+y-xy-x-αn-βn=0;∀x,y∈DT,∀n∈Z+,so that 2〈x-y,x-y〉-〈Tny-Tnx,y-x〉-αn-βn=0 and then(41)Tny-Tnx,y-x=x-y2-αn+βn2≤x-y2-αn2=y-x,y-x-αn2;∀x,y∈DT,∀n∈Z+,orI-Tny,I-Tnx,y-x≥αn+βn2;∀x,y∈DT,∀n∈Z+.Then, one gets from (39)(42)αn=1-Knx-y,x-y+1-KI-Tnx-I-Tny,I-Tnx-I-Tny=1-Knx-y,x-y+1-Kx-y+Tny-Tnx,x-y+Tny-Tnx=1-Kny-x,y-x+1-Ky-x,y-x+1-Kx-y,Tny-Tnx+1-KTny-Tnx,y-x+1-KTny-Tnx,Tny-Tnx=2-K-Kny-x,y-x-1-KI-Tny-I-Tnx,y-x+1-KTny-Tnx,Tny-Tnx;∀x,y∈DT,∀n∈Z+.Now, one gets from (42) into (41) that(43)3-K2I-Tny-I-Tnx,y-x≥2-K-Kn2x-y,x-y+1-K2Tny-Tnx,Tny-Tnx;∀x,y∈DT,∀n∈Z+so that(44a)I-Tny-I-Tnx,y-x≥2-K-Kn3-Kx-y2+1-K3-KTny-Tnx2;∀x,y∈DT,∀n∈Z+,(44b)≥2-K-Kn3-Kx-y2;∀x,y∈DT,∀n∈Z+.Property (i) follows from (44a) and (44b) with the replacement Tn→T. Property (ii) follows from (44a) and (44b) with the replacement Tn→T in (44a) and (44b), the definition of strict pseudocontraction, and Kn=1, ∀n≥∈Z+ leading to 〈I-Tny,I-Tnx,y-x〉≥1-K/3-Kx-y2, ∀x,y∈DT, hence Property (ii). Properties (iii)-(iv) follow from Property (ii) with K=1 and K∈0,1. Property (v) follows with the replacement Tn→T in (44a) and (44b) and Kn→K0 with K0<1. Property (vi) and Property (vii), respectively, follow with the replacement Tn→T in (44a) and (44b) and, respectively, Kn→1 and K<1, and Kn→K0 and K=1.Recall that positivity is equivalent to passivity for self-adjoint operators and that accretivity can be interpreted as incremental positivity for inner products of pairs of elements in the operator domains and their respective images. The above result on pseudocontractions is now linked with some previous parallel positivity and passivity results from Proposition3 and Theorems 11 and 14 on the extended space of truncated square-integrable vector functions.Remark 15.
Note from Theorems14(v)-(vi) the important fact that if T:DT→RT is either Kn,K-asymptotically strictly pseudocontractive in the intermediate sense or Kn,K0,K asymptotically strongly strictly pseudocontractive in the intermediate sense then I-T is in both cases asymptotically strongly strictly accretive in the intermediate sense.The following two results follow from Theorem14.Corollary 16.
LetH be a real Hilbert space. Then, the following properties hold:
(i) Assume thatT:DT→RT is K1,K-pseudocontractive and self-adjoint in the wide sense. Then, I-T is K--strongly accretive, K--incrementally strictly passive, K--incrementally strictly positive with K-=3-2K-K1/3-K, K--strictly passive, and K--strictly positive. If, in addition, T0=0, then the incremental properties are equivalent to the nonincremental counterparts.
(ii) Assume thatT:DT→RT is K-strictly pseudocontractive. Then I-T is K-′-strictly accretive with K-′=1-K/3-K. If, furthermore, T:DT→RT is self-adjoint then it is equivalently K-′-incrementally strictly passive and K-′-incrementally strictly positive. If, in addition, T0=0, then the incremental properties are equivalent to the nonincremental counterparts.
(iii) Assume thatT:DT→RT is Kn,K0,K-asymptotically strongly strictly pseudocontractive in the intermediate sense and self-adjoint with I-T0=0 (equivalently with T0=0 if I-T is linear). Then, I-T is K-n,K- asymptotically strictly passive (and, equivalently, asymptotically strictly positive) in the intermediate sense with K-n=2-K-Kn/3-K, ∀n∈Z+, such that K-n→2-K-K0/3-K as n→∞ and K-=1-K/3-K for some K0,K∈0,1 and a convergent real sequence Kn⊂0,∞ with Kn→K0.
The above result still holds withK0=1 for some K∈0,1, that is, K-n→1-K/3-K, if T:DT→RT is Kn,K-asymptotically strictly pseudocontractive in the intermediate sense.Proof.
Properties (i) and (ii) follow from their counterparts of Theorem14 and Property (iii) follows from Theorem 14(v) (see also Remarks 9).Corollary 17.
LetH be a real Hilbert space. Then, the following properties hold:
(i) Assume thatT:DT→RT is K1,K-pseudocontractive in the wide sense. Then, the 3-2K-K1/3-K-strongly accretive operator I-T satisfies(45)I-TTx-I-Tx,Tx-x≥3-2K-K13-Kx-Tx2.
(ii) Assume thatT:DT→RT is K-strictly pseudocontractive. Then,(46)I-TTx-I-Tx,Tx-x≥1-K3-Kx-Tx2;∀x∈DT.
(iii) Assume thatT:DT→RT is Kn,K0,K-asymptotically strongly strictly pseudocontractive in the intermediate sense for some K,K0∈0,1 and 0,∞⊃Kn→K0. Then,(47)liminfn→∞I-TnTx-I-Tnx,Tx-x-2-K-Kn3-Kx-Tx2-1-K3-KTn+1x-Tnx2≥0;∀x∈DT.The result still holds with K0=1 if T:DT→RT is Kn,K-asymptotically strictly pseudocontractive in the intermediate sense.
(iv) IfT:DT→RT is linear then the commuting assumptions of Properties (i)–(iii) are removed.
(v) IfT:DT→RT is linear and asymptotically strongly strictly pseudocontractive in the intermediate sense for some K,K0∈0,1 and 0,∞⊃Kn→K0 and Tn is pointwise convergent to T0:DT→RT everywhere in its definition domain with z=zx=limn→∞Tnx=T0x for any x∈DT then (48)liminfn→∞Ty-x+T0x-y,y-x-2-K-K03-Ky-x2-1-K3-KT0y-x2≥0;∀x,y∈DT(49)liminfn→∞T2-I-T0T-Ix,T-Ix-2-K-K03-KT-Ix2-1-K3-KT0T-Ix2≥0;∀x∈DT.The result still holds with K0=1 if T:DT→RT is Kn,K-asymptotically strictly pseudocontractive in the intermediate sense. Also, the result still holds with K=1 if T:DT→RT is asymptotically strongly pseudocontractive in the intermediate sense.Proof.
Properties (i) and (ii) follow from their counterparts of Theorem14 while Property (iii) follows from Property (v) of Theorem 14, by fixing y=Tx for any given x∈DT. Property (iv) is direct if T:DT→RT is linear. Relation (45) of Property (v) follows from Properties (iii) and (iv), without equalizing y to Tx for x,y∈DT, zx=limn→∞Tnx=limn→∞Tn+1x=T0x leading to(50)liminfn→∞Ty-x+T0y-T0x,y-x-2-K-Kn3-Ky-x2-1-K3-KT0y-T0x2≥0;∀x,y∈DT,for any x∈DT and the linearity of the limit operator T0. Relation (46) follows from linearity and associated commuting property of T0 and T and (45) for y=Tx and x∈DT.Remark 18.
Note that Theorem14 and Corollaries 16 and 17 are directly applicable to operators defined on the extended real space L2eiR0+;Cn of truncated functions on any real interval 0,T by defining on this space the scalar product 〈·,·〉T=〈·T,·T〉 for any real T∈R+. In particular, we have the subsequent result.Corollary 19.
LetI-G:L2eiR0+;Cn→L2eiR0+;Cn be endowed with the scalar product on L2eiR0+;Cn being everywhere real. Then, the following properties hold.
(i) Assume thatI-G is K1,K-pseudocontractive on L2eiR0+;Cn in the wide sense. Then G is α-strongly accretive with α=3-2K-K1/3-K.
If, in addition,G is odd superadditive and bounded of norm kU<1/2 then it is strictly positive.
If, furthermore,G is self-adjoint then it is also incrementally strictly passive. If, in addition, G0=0 then G is also strictly positive and strictly passive.
(ii) Assume thatI-G is K-strictly pseudocontractive. Then G is α-strongly accretive with α=1-K/3-K and incrementally strictly positive.
If, in addition,G is odd superadditive and bounded of norm kU<1/2 then it is incrementally strictly positive.
If, furthermore,G is self-adjoint then it is also incrementally strictly passive. If, in addition, G0=0 then G is also strictly positive and strictly passive.
(iii) Assume thatI-G is pseudocontractive. Then, G is accretive and incrementally positive.
If, in addition,G is self-adjoint then it is also incrementally passive. If, furthermore, G0=0 then G is also positive and passive.
(iv) Assume thatI-G is Kn,K-asymptotically strictly pseudocontractive. Then, G is K-n,K-1-asymptotically strongly accretive with 0,∞∋K-n=2-K-Kn/3-K→K-1=1-K/3-K as n→∞, such that 0,∞⊃Kn→1 and K∈0,1, and incrementally asymptotically strictly positive. If, in addition, G is self-adjoint then it is also incrementally asymptotically strictly passive. If, furthermore, G0=0 then G is also asymptotically strictly positive and asymptotically strictly passive.
(v) Assume thatI-G is Kn,K0,K-asymptotically strongly strictly pseudocontractive in the intermediate sense. Then, G is K-n,K-0,K--asymptotically strongly accretive in the intermediate sense with K-=1-K/3-K and K-n→K-0=2-K-K0/3-K such that 0,∞⊃Kn→K0∈0,1 and K∈0,1. If, in addition, G is self-adjoint then it is also incrementally asymptotically strictly passive. If, furthermore, G0=0 then G is also asymptotically strictly positive and asymptotically strictly passive.
(vi) Assume thatI-G is Kn,K-asymptotically strictly pseudocontractive in the intermediate sense. Then, G is K-n,K-0,K--asymptotically strongly accretive in the intermediate sense with K-=1-K/3-K and K-n→K-0=1-K/3-K such that 0,∞⊃Kn→1 and K∈0,1. If, in addition, G is self-adjoint then it is also incrementally asymptotically strictly passive.Remark 20.
It turns out that Corollary19 is applicable to operators on L2eR0+;Rp endowed with the same scalar product and of easy generalization to operators on LpeR0+;Rp for p∈1,∞.Proposition 21.
Assume thatG:L2eR0+;Rp→L2eR0+;Rp is self-adjoint, one-to-one, of closed range, and strongly accretive. Then1.
G:L2eR0+;Rp→L2eR0+;Rp is asymptotically accretive, incrementally asymptotically positive, incrementally asymptotically passive, asymptotically positive, and asymptotically passive;2.
the operatorGn on L2eR0+;Rp is strictly accretive, incrementally strictly positive, incrementally strictly passive, asymptotically strictly positive, and strictly passive for any finite n∈Z+.Proof.
Note that ifG is an accretive operator on L2eR+;Rp then 〈Gx-Gy,x-y〉T≥0 for each x,y∈DG. Thus, Gnx∈L2eR0+;Rp for any Gnx∈L2eR0+;Rp; ∀n∈Z+, one has(51)G2x-G2y,Gx-GyT≥μG2Gx-Gy,x-yT=μG2Gx-Gy,x-yT≥μG2εgx-yT22⋮Gnx-Gny,Gn-1x-Gn-1yT≥μG2n-1Gx-Gy,x-yT≥μG2n-1εgx-yT22≥0∀T>0, with εg≥0 since G is an accretive operator on L2eR0+;Rp, such that the real constant εg>0 if the operator G is strongly accretive, where μG>0 is the minimum modulus of G since it is one-to-one and of closed range, [10]. Now, (51) implies that G:L2eR0+;Rp→L2eR0+;Rp is asymptotically accretive, incrementally asymptotically positive, incrementally asymptotically passive (since the operator is self-adjoint), asymptotically positive, and asymptotically passive, since, in addition, the operator maps “0” into “0” and limn→∞inf⁡μGn≥0. On the other hand, for any finite n∈Z+, the composite operator Gn on L2eR0+;Rp, resulting from composition of G of n times on itself, is strictly accretive, incrementally strictly positive, incrementally strictly passive, asymptotically strictly positive, and strictly passive, since, in addition, the operator maps “0” into “0” and μGn>0 since Gn on L2eR0+;Rp is one-to-one and of closed range for any finite n∈Z+ since G is one-to-one and of closed range.


## 3. Asymptotic Passivity in Dynamic Systems

We first give some elementary concepts of usefulness to set the passivity framework. The notation for the spaces of realn-square-integrable and truncated n-square-integrable functions of nonnegative real domain is simplified due to subsequent extensive use as Lpn≡LpR0+;Rn and Lpen=LpeR0+;Rn, respectively, for p∈1,∞.Consider a setX, a mapping T:X→X, and a binary relation RT⊂X2 on X defined by T as RT=x,Tx:x∈X. A binary relation R on Lpen is said to be Lpn-stable if x∈Lpn∧x,y∈R⇒y∈Lpn. R is said to be Lp-stable with finite gain (wfg) if there exist finite nonnegative real constants γp (gain) and bp (bias) such that(52)x∈Lpn∧x,y∈R⟹yp≤γpxp+bpand Lpn-stable with finite gain and zero bias (wb) if bp=0. Note that, if R is Lpn-stable wfg, then it is trivially Lpn-stable wb since y∈Lpn.The following result generalizes a well-known passivity result of [11], also included in [1] in the context of a general framework setting on passivity, which is addressed based on some of the results given in the above section for asymptotic pseudocontractions in the intermediate sense. The result relies on the strict passivity of a tandem of dynamic systems consisting of a controlled system and its controller.Theorem 22.
Consider the real Hilbert spaceL2R0+;Rp endowed with the inner product 〈·,·〉. Assume that the self-adjoint operators I-Gin:L2eR0+;Rp→L2eR0+;Rp for i=1,2 are invertible (i.e., one-to-one and of closed range) for n≥n0, some n0∈Z+, and that I-Gi:L2eR0+;Rp→L2eR0+;Rp for i=1,2 satisfies I-Gin0=0 for n≥n0 (if both operators are linear, it suffices that Gin0=0; i=1,2) while input/output/feedback relations for outputs, inputs, and errors, respectively, yin, uin, and ein, i=1,2, are subject to(53)1yin=Gineinfori=1,22e1n=u1n-y2n,e2n=u2n+y1n;uin∈L2ep≡L2eR0+;Rpfori=1,2,∀n∈Z+. Then, the following properties hold for i=1,2:
(i) (a)Gi:L2eR0+;Rp→L2eR0+;Rp is K-in,K0i,K-i-asymptotically strongly strictly accretive with K-i=1-Ki/3-Ki and K-in→K-0i=2-Ki-K0i/3-Ki if I-Gin:L2eR0+;Rp→L2eR0+;Rp is Kin,K0i,Ki-asymptotically strongly strictly pseudocontractive in the intermediate sense such that 0,∞⊃Kin→K0i∈0,1 and Ki∈0,1.
(b)Gi:L2eR0+;Rp→L2eR0+;Rp is K-in,K-i-asymptotically strongly strictly accretive in the intermediate sense with K-i=1-Ki/3-Ki and K-in→K-0i=1-Ki/3-Ki for i=1,2 if I-Gin:L2eR0+;Rp→L2eR0+;Rp is Kin,K0i,Ki-asymptotically strictly pseudocontractive in the intermediate sense such that 0,∞⊃Kin→1 and Ki∈0,1.
(c)Gi:L2eR0+;Rp→L2eR0+;Rp is K-in,K-0i-asymptotically strictly accretive in the intermediate sense with K-in→K-0i=1-K0i/2 for i=1,2 if I-Gin:L2eR0+;Rp→L2eR0+;Rp is Kin,K0i-asymptotically strongly pseudocontractive in the intermediate sense such that 0,∞⊃Kin→K0i∈0,1.
(ii)Gi:L2eR0+;Rp→L2eR0+;Rp, i=1,2, is asymptotically incrementally strictly passive (and asymptotically strictly passive, asymptotically incrementally strictly positive, and asymptotically strictly positive) in the intermediate sense.
(iii) The binary relationRG1nG2nu,y=u1n,u2n,y1n,y2n:u1,2∈L2ep defined by (47)-(48) is L2p-stable wb.
(iv)RG1nG2nu,y=u1n,u2n,y1n,y2n:u1,2∈L2ep is L2p-stable wb if one of the operators I-Gi for some i∈1,2 is asymptotically strongly (or strict or strict strongly) pseudocontractive in the intermediate sense and bounded while I-Gjj≠i∈1,2 is (at least) asymptotically pseudocontractive in the intermediate sense.Proof.
From Theorem14(v), one has via the replacements I-Tn→Gin for i=1,2(54)liminfn→∞Ginuin-Ginu-in,uin-u-inT-2-Ki-Kin3-Kiuin-u-niT22-1-Ki3-KiI-Ginuin-I-Ginu-inT22≥0;∀T>0,∀yin,y-in∈DGi for i=1,2. Since K-in→K-i0=2-Ki-Ki0/3-Ki with Ki0=1 in the strict case and Ki0∈0,1;i=1,2 in the strong strict case and 0<Ki<1, i=1,2, leading to 0<Ki+Ki0<2 for i=1,2 one has from (54) for all integer n≥n0 and some n0∈Z+, by taking u-i=0 for i=1,2, that(55)uin,GinuinT=Ginui,uiT≥εinuinT22+εiI-GinuinT22≥εinuinT22+εiλinGinuinT22;∀T>0,for n≥n0, where εi=1-Ki/3-Ki>0, εinn≥n0>0, and λinn≥n0>0 with εin=2-Ki-Kin/3-Ki and λin=μI-Gin/μGin>0,n≥n0, so that δin=εinλin,n≥n0 for i=1,2, since I-Gin:L2eR+;Rp→L2eR+;Rp are one-to-one and of closed range, equivalently, invertible so that μI-Gin>0, [10], for i=1,2 and all n≥n0. Property (i) follows from (54)-(55). Direct calculations with (50) subject to the input/output/feedback constraints (a) and (b) yield(56)y1n,e1nT+y2n,e2nT=y1n,u1nT+y2n,u2nT=G1nu1n,e1nT+G2nu2n,e2nT=G1nu1n,u1nT+G2nu2n,u2nT≥∑i=12εineinT22+δinGinuinT22+εnfor some sequence εn⊂R+, with εn=εnT→0 as n→∞, ∀T>0, so that (57)liminfn→∞y1n,e1nT+y2n,e2nT-∑i=12εineinT22+δinGinuinT22≥0.It follows from (57) after some routine calculations as those given in [1] that RG1nG2nu,y is L2p-stable wb for n≥n0 and asymptotically L2p-stable wb since y1,y2∈L2p for any given inputs u1,u2∈L2ep; thus Property (ii) follows since, from Property (i), liminfn→∞εin+δjn>0 for j,i≠j∈1,2. Property (ii) is direct from the relations of accretivity and passivity (Theorem 14). Property (iii) follows since the condition liminfn→∞εin+δjn>0 for j,i≠j∈1,2 still holds if1.
at least one of the operatorsI-Gi on L2ep≡L2eR+;Rp for some i=1,2 is bounded (then L2p-stable wb [1]) and (at least) asymptotically strongly pseudocontractive in the intermediate sense, leading to (at least) asymptotic strict accretivity (then implying the strict passivity) in the intermediate sense of the corresponding Gi,2.
the other operatorI-Gj is (at least) asymptotically pseudocontractive in the intermediate sense (see Definition 12(d)–(h) and Theorems 14(v)–(vii) leading to accretivity (then implying the passivity) in the intermediate sense of the corresponding Gj.
Configuration (53) of strict passive bounded operator with a passive one is sufficient to guarantee the L2p-closed-loop stability [1] so the parallel conditions in the intermediate sense also guarantee the stability property.Remark 23.
While the operatorsGi on L2eR0+;Rp, Gi0=I, for i=1,2, are asymptotically strictly passive (resp., passive) according to Theorem 22, the corresponding ones being asymptotically pseudocontractive in the intermediate sense or, simply, asymptotically pseudocontractive (resp., passive) are Ti=I-Gi for i=1,2 which satisfy the recursive relations:(58)Tin=I-Gin,T0=0Tin+1=I-Gin+1=I-GinI-T=I-I-TinI-Tfor i=1,2, ∀n∈Z+. Note that Theorem 22(iii) includes the case when both operators I-Gi on L2eR0+;Rp for i=1,2 are asymptotically strictly pseudocontractive. Note also that since the binary relation RG1nG2nu,e=u1n,u2n,e1n,e2n:u1,2∈L2ep is asymptotically L2p-stable wb under the conditions of Theorem 22(i) and under the weaker conditions of Theorem 22(iii) then the input-to-error binary Rue=u1n,u2n,e1n,u2n:u1,2∈L2ep.
The following result is of interest relating the convergence properties of the operatorsI-G and G while also relating the potential fixed points of I-G to the convergence properties of sequences generated through the operator G.Theorem 24.
The following properties hold:
(i) IfC is a nonempty closed convex subset of a real Hilbert space H and T:C→C is uniformly L-Lipschitz and asymptotically pseudocontractive in the intermediate sense then the set FixT of fixed points of T is a closed convex subset of H. If, furthermore, FixT≠⌀ then I-T is demiclosed at zero (Lemmas  1.4 and 1.5 of Kim et al. [18]).
(ii) Assume thatI-G:C→C is uniformly L-Lipschitz and asymptotically pseudocontractive in the intermediate sense, where C is a nonempty closed convex subset of L2p≡L2R0+;Rp and that L2p is endowed with the usual inner product. Then one has the following:1.
FixI-G is a closed convex subset of L2p.2.
IfFixI-G≠⌀ then G is demiclosed at zero.3.
Assume thatFixI-G≠⌀, that 0∈C, and that Gn→G∗ (i.e., Gn converges pointwise to G∗). Thus, if xn⊂C→wx∗∈C and Gnxn⊂C→0 (i.e., if xn converges weakly to x∗, namely, 〈xn,y〉→〈x∗,y〉, ∀y∈C, and Gnxn converges strongly to 0∈C) then G∗x∗→0. If, in addition, G∗0=0 and G∗ is injective at zero then x∗=0 so that 〈xn,y〉→0, ∀y∈C as a result.
(iii) Property (ii) also holds ifI-G is replaced with a sequence of L-Lipschitz and asymptotically pseudocontractive in the intermediate sense operators I-Gn:C→C such that C is a nonempty closed convex subset of L2p, with 0∈C and Gn→G∗.Proof.
SinceI-G:C→C is uniformly L-Lipschitz and asymptotically pseudocontractive in the intermediate sense, C is a nonempty closed convex subset of L2p,FixI-G≠⌀ [then G is demiclosed at 0 [21] from Property (i)], 0∈C,Gn→G∗, xn→wx∗, and Gnxn→0, then one has that(59)Gnxn-G∗xn⟶0⟹G∗xn⟶0⟹G∗x∗=0.If, furthermore, G∗0=0 and G∗ is injective at zero then x∗=0 since G∗x∗=0∧G∗0=0⇒x∗=0. Hence, Property (ii) follows.One gets directly from Theorem24 in view of Proposition 13 the following result.Corollary 25.
Theorem24 holds for T:C→C being uniformly L-Lipschitz and asymptotically strictly, or strongly strictly, pseudocontractive in the intermediate sense.Remark 26.
Theorem22 is applicable to asymptotic passivity and incremental asymptotic passivity of, in general, a nonlinear dynamic system described by two operators connected in feedback form, one of them describing the controlled object while the other one describes the feedback controller. The passivity conditions are guaranteed if two associated related operators are, respectively, asymptotically strictly pseudocontractive and/or asymptotically strictly pseudocontractive in the intermediate sense. The related discussion follows below.
Consider the nonlinear control system [12]:(60)x˙t=fx+Lxu,y=Gx,where x=xt∈Rn, u=ut∈Rm, and y=yt∈Rl are the state, input, and output vectors and, respectively, f and G are smooth vector-valued functions on x, and L is a smooth matrix-valued function on x.Definition 27 (see [12]).
System (60) is called strictly S-passive (resp., S-passive) if there exists a nonnegative scalar function (storage function) Vx and a scalar function μx, where μx>0 for x≠0, such that (61)Vx≤Vx0+∫0tuTσSyσ-μxσdσ,respectively,(62)Vx≤Vx0+∫0tuTσSyσdσfor any t≥0 and for any solution of system (60) satisfying x0=x0 and xt≡x, where S is a prespecified m×l real matrix.From Definitions1 and 27, we have immediately the following simple direct result.Proposition 28.
Assume thatm=l and that G:L2eR0+;Rm→L2eR0+;Rm in (60). Then one has the following:1.
IfG:L2eR0+;Rm→L2eR0+;Rm is passive (resp., strictly passive) then the system (60) is Im-passive (resp., strictly Im-passive) and conversely.2.
IfG:L2eR0+;Rm→L2eR0+;Rm is strictly passive or strongly strictly passive then the system (60) is Im-strictly passive and conversely.Proof.
One has from (61) that(63)u,yT=u,GuT≥εμu,uT+Vx-Vx0≥εμuT22-β≥-βfor any T≥0 since Vx is everywhere nonnegative where εμ=infT≥0inft≥0μt/uT2≥0, the inequality being strict if (60) is strictly Im-passive, and εμ=0 if (60) is Im-passive, with β=Vx0≥0. Conversely, the existence of the storage function being constant, that is, Vx=Vx0, proves the converse assertion; that is, if (60) is Im-passive then the operator G is passive (resp., strictly passive). Property (i) has been proved. To prove Property (ii), define the storage function V=Vy=VGx=infT≥0δ0G/yT〈y,y〉T for any T≥0 and some real constant δ0G>0 is taken as a nonnegative function of the output, then, from (54) for δG=infT≥0δ0G/yT,(64)u,GuT≥εμuT22+δGyT22-βfor any real T≥0 with εμ+δG>0 if G:L2eR+;Rm→L2eR+;Rm is strongly strictly passive, hence Property (ii).Remark 29.
Inequality (64) refers to the operator G being strictly passive if the sum of the two right-hand-side constants is positive. Conversely, control system (60) is Im-strictly passive. Borrowing the terminology of [13], if εμ>0, then system (60) is said to be strictly input passive while if δG>0, then system (52) is said to be strictly output passive.
If both right-hand side constants of (64) are positive, then the operator G is, furthermore, strongly strictly passive.
The last right-hand-side inequality of (63), that is, 〈u,y〉T≥-β, is commonly referred to as Popov’s inequality [3–5, 12, 22] which is a basic tool to characterize the hyperstability and asymptotic hyperstability of feedback systems where the feed-forward loop is a passive linear dynamic system while the nonlinear feedback controller belongs to a general class satisfying such a passivity-type constraint. A well-known related result is as follows.Theorem 30.
The following properties hold.
(i) Assume that1.
y=Gu with u and y being the input and output signals of dimension p≥1 in L2ep and G being described in the frequency domain by a strictly positive real transfer matrix G^s,2.
u=-Ky with K∈KΩ being a feedback nonlinear and perhaps time-varying controller causal operator for the controlled system described by G satisfying the inequality 〈y,Ky〉T≥-β;∀K∈KΩ, for any arbitrary real constant β>0.
Then, the closed-loop controlled system is globally asymptotically Lyapunov’s stable for any controllerK belonging to the set KΩ; that is, it is asymptotically hyperstable for the class KΩ; thus u and y are bounded square-integrable functions on 0,∞ (i.e., in L2ep) and ut→0 and yt→0 as t→∞ for any given bounded initial conditions of the state.
(ii) Assume thaty=Gu and u=-Ky, with u and y being the input and output signals of dimension p≥1 in L2ep. Then, the closed-loop system is globally asymptotically Lyapunov’s stable if G is strictly input passive and bounded and K is passive with zero-independent constant or if K is strictly input passive and bounded and G is passive with zero-independent constant.Proof.
SinceGs is a positive real transfer matrix then it is a bounded self-adjoint causal operator which is strictly stable with strictly positive real, that is, ReGs≥β1>0 for all s∈C with Res≥0. Thus, one has, for any real constant β∈β1,∞ and some real constant β1>0 and for any given control L2ep with support of nonzero measure,(65)∞>β≥-Ky,yT=-KyT,yT=-KyT,yTT=-KyT,yTT=-Ky,yTT=y,uT=Gu,uT=GuT,uT=GuT,uTT=GuT,uT=Gu,uTT=12π∫-∞∞u^TT-iωG^iωu^Tiωdω≥14πminω∈R0+λminG^iω+G^T-iω∫-∞∞u^TT-iωu^Tiωdω≥β14πuT22>0;∀T>0,where λmin· denotes the minimum eigenvalue of the symmetric ·-matrix, so that the controlled system y=Gu is input-strictly passive, and the operator G is strictly positive, strictly passive, and strictly accretive. Thus, the Fourier transform u^T of uT exists since u∈L2e and the Laplace transform G^s and the associated Fourier transform G^iω=G^T-iω of G exist as well for any complex number s=σ+iω with σ∈R+ and ω∈R0+ since G^ is strictly positive real, then analytic for σ=Res≥0 and subject to λminG^iω+G^T-iω≥β1>0, ∀ω∈R0+, where λmin· denotes the minimum eigenvalue and the superscript “T” denotes transpose. Thus, Parseval’s theorem can be applied in (65). Furthermore, the controller operator K is passive since 〈y,Ky〉T≥-β; ∀T>0, since β≥0 is arbitrary, u andy are bounded square-integrable functions on 0,∞ (i.e., in L2ep) and uTt→0 and yTt→0 as t→∞ so that ut→0 and yt→0 as t→∞ for any bounded initial conditions. Property (i) has been proved.
To prove Property (ii), note that, in this case, ifγ2G and γ2K are the L2-gains wb of G and K, then one has for any control being nonzero except possibly on a set of zero measure that(66)+∞>minβy-εyyT22,γ2GuT22≥-y,KyT=y,uT=Gu,uT≥-βu+εuuT22≥γ1γ2GuT22>0;∀T>0,which holds provided that γ1∈0,εu/γ2G, γ2G<+∞, βu=εy=0, and βy≥0, εu>0; that is, G is bounded and strictly input passive and K is passive, or(67)0<γ2γ2KyT22≤-βy+εyyT22≤y,KyT=-y,uT=-Gu,uT≤minβu-εuuT22,γ2KyT22<+∞;∀T>0,which holds provided that γ2∈0,εy/γ2K, γ2K<+∞, βy=εu=0, βu≥0, and εy>0; that is, K is bounded and strictly input passive and G is passive. On the other hand if G is bounded and strictly input passive and K is passive with βy0=0, then it is also passive with any finite positive constant βy so that one gets from (66) that uTt→0 and yTt→0 as t→∞ for any T>0 and then ut→0 and yt→0 as t→∞, for any given initial conditions. A similar conclusion arises from (67) if K is bounded and strictly input passive and G is passive with βu0=0. Property (ii) has been proved.Theorem30 can be directly reformulated in the discrete-time framework related to the space l2ep.


## 4. Further Examples

Example 1.
Consider the iterated linear continuous-time dynamic feedback system:(68)ae1n=u1n-y2n,e2n=u2n+y1n,uin∈L2ep≡L2eR0+;Rpfori=1,2;∀n∈Z+(69a)by1n=I-G1ne1n=I-G1nu1n-y2n;∀n∈Z+(69b)y2n=I-G2ne2n=I-G2nu2n+y1n;∀n∈Z+,for given initial conditions yi0=ai for i=1,2, which is described by the sequences of operators I-Gin on L2ep. Then, if Gin are linear for i=1,2, (70a)y1n=I-G1nu1n-I-G1nI-G2nu2n+y1n=I+I-G1nI-G2n-1I-G1nu1n-I-G1nI-G2nu2n;∀n∈Z+,(70b)y2n=I-G2nu2n+y1n=I-G2nu2n-I-G2nI-G1nu1n+y2n=I+I-G2nI-G1n-1I-G2nu2n-I-G2nI-G1nu1n;∀n∈Z+provided that the given inverses exists. Then (68) and (69a)-(69b) are well-posed in the sense that for each control input un=u1nT,u2nT∈L2e2p the output yn=y1nT,y2nT∈L2e2p and so the input-output error en=e1nT,e2nT∈L2e2p exist and are unique and, equivalently, the operators I+I-GinI-Gjn-1 on L2ep for i,j≠i∈1,2 are causal and then well-posed and then (67) holds. Note that a mapping A:Lqep→Lqep for q∈1,∞ is causal if AfT=AfTT, ∀T≥0, ∀f∈Lqep, [1, 13]. Note also that the particular case that the inputs u1 and u2 are independent of n is included.
On the other hand, ifGi is asymptotically pseudocontractive in the intermediate sense for some i∈1,2 and Gj is asymptotically strictly (or strongly strictly) pseudocontractive in the intermediate sense for j≠i∈1,2 then I-Gi are asymptotically strictly passive and incrementally asymptotically passive for i=1,2 and the binary relation RI-G1nI-G2n is asymptotically L2p-stable wb (Theorem 22). As a result, if u1,2 are square-integrable, then e1,2 and y1,2 are also square-integrable and converge asymptotically to zero except possibly on a set of zero measures.
A dual problem to the above one is as follows. IfI-Gi is either asymptotically strictly or strongly strictly pseudocontractive in the intermediate sense for some i∈1,2 then Gi is asymptotically strictly positive and passive (if self-adjoint), asymptotically strongly strictly accretive, and incrementally asymptotically passive (see Remark 15).Example 2.
If the above system is not linear (i.e., bothG1,G2 are not jointly linear) then (70a) and (70b) do not necessarily hold. However, the following equations hold also if the operators are nonlinear [1]:(71)e=u-Fyy=I-Ge=I-Gu-FI-Ge⟺I+FI-Ge=uy=I-Gu-Fy⟺I+I-GFy=I-Gufor given initial conditions yi0=ai for i=1,2, where e=e1T,e2TT, y=y1T,y2TT, u=u1T,u2TT, G=BlockDiagG1,G2,F=0Ip-Ip0. Then, the linearity assumption of Example 1 is not necessary since the error can be described via e=I+FG-1u provided that the above inverse exists. Thus, the given results of well-posedness, passivity, and wb-stability still hold.Remark 31.
Examples1 and 2 can be directly extended to the case that ui, ei, and yi have distinct dimensions pi for i=1,2 with p1≠p2. In particular, F=0Ip1-Ip20 in Example 2. The formalism can be also extended directly to the case ui,ei,yi∈Lqepi for i=1,2 and q∈1,∞.Example 3.
Reconsider Example1 in the discrete-time framework with the operators being nonlinear and p1≠p2, in general. Then, uin∈l2epi≡l2eZ0+;Rpi (i.e., the set of square summable real sequences) for i,j=1,2 so that (69a) and (69b) become modified as(72a)y1,n+1=I-G1ne1n=I-G1nu1n-y2n;∀n∈Z+(72b)y2,n+1=I-G2ne2n=I-G2nu2n+y1n;∀n∈Z+.The problem is assumed to be well-posed then uin∈l2epi≡l2eZ0+;Rpi; i,j=1,2 implies that yin,ein∈l2epi≡l2eZ0+;Rpi; i,j=1,2. Assume that(73a)u1n0=0∧u2n0=0,n0∈Z0+⟹u1n0=0∧u2n0=0,n≥n0∈Z0+;that is, if both inputs are zero in finite discrete-time n0 then uinn≥n0≡0,(73b)ui,n+1=Ki1nu1n+Ki2nu2n;i,j=1,2,∀n∈Z+,for some no-necessarily unique, gain sequences Kijn, i,j=1,2, Note that, since, from the previous assumption, if both input sequences are simultaneously zero in finite time, then they become identically zero afterwards, any input sequences satisfying such a constraint can be described in this way. Consider the discrete extended dynamic system of input and output sequences u-n and y-n of dimension p=2p1+p2:(74)y-n+1=y1,n+1y2,n+1u1,n+1u2,n+1=0G1n-Ip1Ip1-G1n0Ip2+G2n00Ip2-G2n00K11nK12n00K21nK22nu-n;u-n=y1ny2nu1nu2n;∀n∈Z+.Under a compact description, one has equivalently y-n+1=Ip-G-nu-n, ∀n∈Z+, where(75)G-n=Ip1Ip1-G1nG1n-Ip10-Ip2+G2nIp20G2n-Ip200Ip1-K11n-K12n00-K21nIp2-K22n;∀n∈Z+,or, dually, y-n+1=Gnu-n, ∀n∈Z+, where(76)Gn=Ip-G-n=0G1n-Ip1Ip1-G1n0Ip2+G2n00Ip2-G2n00K11nK12n00K21nK22n;∀n∈Z+.Thus, if Ip-G-, respectively G, is either asymptotically strictly, or strongly strictly, pseudocontractive, in the intermediate sense, then G-, respectively Ip-G, is asymptotically strictly passive, asymptotically strongly strictly accretive, and incrementally asymptotically passive. The eventual possible extensions of the pseudocontractive conditions related to positive realness/passivity in both continuous-time and discrete-time formalisms in dynamic systems including the eventual presence of known or unknown internal and external delays and parametrical disturbances based on previous background results [34–38] are under study.



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