#### Abstract

This paper investigates some parallel relations between the operators and 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 -operators, and their parallel interpretations as pseudocontractive properties of their counterpart -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.

#### 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 operator and the accretive properties of its counterpart operator 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 , the less restrictive asymptotic passivity conditions on being obtained if 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 by , , and the sets of real, positive real, and nonnegative real numbers, respectively, and by , , and the sets of real, positive real, and nonnegative integer numbers, respectively.

If the measurable function then is the truncation of on defined by for and for and any finite , where is the imaginary complex unit.

is the -space of the -integrable complex -vector functions of imaginary complex argument. In the same way, we can define spaces of truncated functions and from and , respectively.

It is well known that endowed with an inner product is a complete complex Hilbert space with norm (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 ,* leading to real nonnegative inner products*, as follows in the subsequent definitions.

*Definition 1. *An operator is said to be* passive *ifit is said to be* strictly passive* if there exists some real constant such thatand it is said to* be strongly strictly passive* if there exist some real constants and such thatA 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 operator is said to be positive ifand it is strictly positive if there exists some real constant such that It turns out that a strictly passive (resp., strictly positive) operator is also passive (resp., positive).

Proposition 3. *Consider operators such that is real , . Then, the following properties hold.**(i) is passive if and only if its adjoint operator is positive and, in particular, is passive if and only if its transpose operator is positive.**(ii) If 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) If is positive then it is self-adjoint and passive.**(iv) If 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) is positive and passive. If is normal then is positive and passive for any .*

*Proof. *Property (i) is a direct consequence of Definitions 1 and 2. On the other hand, is self-adjoint if and only if the inner product is real and . Then, the inner product is nonnegative real with . So, 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 , and is positive and passive. If, in addition, is normal then so that is positive and passive.

Proposition 4. *Proposition 3 also holds “mutatis-mutandis” for Properties (i)–(v) if is strictly passive/strictly positive.**Proposition 3 also holds “mutatis-mutandis” for Properties (i)–(v) if is passive/positive and if it is strictly passive/strictly positive.*

Roughly speaking, it is concluded from Propositions 3 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. *Let be an arbitrary real Banach space endowed with a scalar product from to . An operator with domain and range in is called as follows:

(a) -strictly accretive (or strictly accretive with constant ) if, for each , there is a , with being the normalized duality mapping, such that with .

The operator is strictly accretive if some such a positive constant exists [16].

(b) Accretive if, for each , there is such that, for each , [16].

(c) -asymptotically strictly accretive if there exist real constants and and a real sequence satisfying such that, for each ,The operator is asymptotically strictly accretive in the intermediate sense if some such a triple exists.

The operator is -asymptotically strongly strictly accretive if it is asymptotically strictly accretive with .

(d) -asymptotically strictly accretive in the intermediate sense if there exist real constants and and bounded real sequence satisfying such that, for each , The operator is asymptotically strictly accretive in the intermediate sense if some such a triple exists. The operator is -asymptotically strongly strictly accretive in the intermediate sense if it is -asymptotically strictly accretive in the intermediate sense with .

We give now incremental-type concepts of incremental passivity and incremental positivity to be then related to the accretive property as follows. First, Definition 5 is extended to operators on as follows.

*Definition 6. *An operator is as follows:

(a) Accretive if, provided that is real for each and all , one has that, for each and some, we have .

(b) is* K*-strictly accretive, if for some positive real constant and .

(c) is -asymptotically strictly accretive if, for each , there exist real constants and and a real sequence satisfying such that is asymptotically strongly strictly accretive if it is asymptotically strictly accretive with .

(d) -asymptotically strongly strictly accretive in the intermediate sense if, for each , there exist real constants and and a bounded real sequence satisfying such that is asymptotically strongly strictly accretive in the intermediate sense if it is asymptotically strictly accretive in the intermediate sense with .

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 operator is said to be incrementally passive ifand it is incrementally strictly passive if there exists some real constant such thatAn operator is said to be asymptotically incrementally strictly passive in the intermediate sense if there exists some real constant such thatThe counterpart definition to Definition 7 related to positivity follows.

*Definition 8. *An operator is said to be incrementally positive ifand it is incrementally strictly positive if there exists some real constant such that An operator is said to be incrementally strictly positive in the intermediate sense if there exists some real constant such that

*Remarks 9. *It turns out that one has the following:1.An incrementally strictly passive (resp., incrementally strictly positive) self-adjoint operator is also incrementally passive (resp., incrementally positive).2.In case , 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.If , incremental passivity and strict passivity are equivalent to passivity and strict passivity [1], respectively, and, furthermore, to positivity and strict positivity, respectively, if is self-adjoint.4.If 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, 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 operator on a Banach space is* strongly pseudocontractive* if there exists such that, for all and , the following inequality holds: is* pseudocontractive* if the above inequality holds with [4].2.The operator on a Banach space is* strongly accretive* if, for all and , there exists such that the following inequality holds for some and all : is* accretive* if the above inequality holds with , [4]. See [4, 28] and also [20] for the case of cyclic mappings.3.A rational function of the complex variable of real coefficients is* positive real* if () it is real for real , () it has no poles in the open right half plane, () its poles at the imaginary axis, if any, are simple and their associate residues are simple, and () for all real such that is not a pole, . All these constraints together lead to for .4.Assume that such a* positive real* rational function 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 is both* passive *and* positive *since it is self-adjoint by nature with 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, for which is associated with the differential system , . If then the transfer function is* strictly positive real* (imaginary poles do not exist and the transfer function is stable satisfying also for and all ), and the associated dynamic system is strictly passive. If the transfer function is modified to then for and all . The transfer function has a relative degree zero and it is said to be* strongly strictly positive real* (i.e., strictly positive real for any finite frequency and as frequency tends to infinity) if , with 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 is* incrementally passive* and* incrementally 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 the* positivity of the solution* does not imply* necessarily 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 that is accretive; then the following properties hold for any :*1.*, .*2.*If, in addition, is strictly accretive with constant k, odd superadditive, and bounded of norm then it is incrementally strictly positive with and also incrementally strictly passive if the operator is self-adjoint.*3.*Assume that . If is accretive then is positive and, furthermore, passive if is self-adjoint. If is strictly accretive with constant and bounded of norm then it is strictly positive with and, furthermore, it is strictly passive if the operator is self-adjoint.*4.*Properties (i)–(iii) hold “mutatis-mutandis” if .*

*Proof. *Since then by using Schwartz’s inequality and the linearity properties of the Hilbert space,and Property (i) follows by taking since is strictly accretive. Also,If is strictly accretive, odd superadditive and bounded of norm then and and Property (ii) is proved so that the identity mapping 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 . Property (iv) is direct from Properties (i)–(iii) by changing the operator domain from to

*Definition 12. *Let be a real Hilbert space . Then, an operator from (the Domain of ) to (the Image of ) is as follows:

(a) -pseudocontractive in the wide sense if there exist such thatThe operator is said to be pseudocontractive in the wide sense if such a pair exists.

The operator is nonexpansive if it is -pseudocontractive in the wide sense.

(b) Pseudocontractive ifand equivalently ifsee [16].

Note that if is pseudocontractive in the wide sense it is pseudocontractive as well and a pseudocontraction in the wide sense with is equivalent to a pseudocontraction.

(c) -strictly pseudocontractive if there exists a constant such that see [16].

The operator is said to be strictly pseudocontractive if such a constant exists.

(d) -asymptotically strictly pseudocontractive if there exists and a sequence with as such thatThe operator is said to be asymptotically strictly pseudocontractive if such a pair exists.

If then is (nonstrictly) asymptotically pseudocontractive, and one has, equivalently (see [18]):(e) -asymptotically strongly strictly pseudocontractive if there exist and a sequence with as such thatThe operator is said to be asymptotically strongly strictly pseudocontractive if such a triple exists. If then is (nonstrictly) asymptotically pseudocontractive.

(f) Asymptotically nonexpansive if it is -asymptotically strictly pseudocontractive.

(g) -asymptotically strongly strictly pseudocontractive in the intermediate sense if there exist and a real sequence satisfying such thatThe operator is said to be asymptotically strictly pseudocontractive in the intermediate sense if such a triple exists.

(h) -asymptotically strictly pseudocontractive in the intermediate sense if there exists and a real sequence satisfying such thatThe operator is said to be asymptotically strictly pseudocontractive in the intermediate sense if such pair exists.

(i) -asymptotically pseudocontractive in the intermediate sense if there exists a real sequence satisfying such thatequivalently, The operator is said to be asymptotically strictly pseudocontractive in the intermediate sense if such a sequence exists.

(j) -asymptotically strongly pseudocontractive in the intermediate sense if there exists and a real sequence satisfying such thatThe operator is said to be asymptotically strongly pseudocontractive in the intermediate sense if such a sequence exists.

The following result is obvious from Definition 12(g)–(i).

Proposition 13. *If is asymptotically strongly strictly pseudocontractive in the intermediate sense then it is asymptotically strictly pseudocontractive in the intermediate sense.**If is asymptotically strictly pseudocontractive in the intermediate sense then it is asymptotically pseudocontractive in the intermediate sense.**If is asymptotically strongly strictly pseudocontractive in the intermediate sense then it is asymptotically pseudocontractive in the intermediate sense.**If is asymptotically strongly pseudocontractive in the intermediate sense then it is asymptotically pseudocontractive in the intermediate sense.*

Theorem 14. *Let be a real Hilbert space. Then, the following properties hold:**(i) Assume that is -pseudocontractive in the wide sense. Then is strongly accretive and it satisfies**(ii) Assume that is -strictly pseudocontractive. Then is strongly accretive and it satisfies**(iii) Assume that is pseudocontractive. Then, is accretive.**(iv) Assume that is asymptotically strictly pseudocontractive. Then, is asymptotically strictly accretive.**(v) Assume that is -asymptotically strongly strictly pseudocontractive in the intermediate sense. Then, is asymptotically strongly strictly accretive in the intermediate sense satisfying for some and a convergent real sequence with that**(vi) Assume that is -asymptotically strictly pseudocontractive in the intermediate sense. Then, is asymptotically strongly strictly accretive in the intermediate sense satisfying for some and a convergent real sequence with that**(vii) Assume that is -asymptotically strongly pseudocontractive in the intermediate sense. Then, is asymptotically strictly accretive in the intermediate sense satisfying for some and a convergent real sequence with that*

*Proof. *Firstly, assume that, is -asymptotically strictly pseudocontractive, then one has for some sequence , with , thatwhere is a real nonnegative sequence Relation (38) is equivalent toso that and thenThen, one gets from (39)Now, one gets from (42) into (41) thatso thatProperty (i) follows from (44a) and (44b) with the replacement . Property (ii) follows from (44a) and (44b) with the replacement in (44a) and (44b), the definition of strict pseudocontraction, and , leading to , , hence Property (ii). Properties (iii)-(iv) follow from Property (ii) with and . Property (v) follows with the replacement in (44a) and (44b) and with . Property (vi) and Property (vii), respectively, follow with the replacement in (44a) and (44b) and, respectively, and , and and .

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 Proposition 3 and Theorems 11 and 14 on the extended space of truncated square-integrable vector functions.

*Remark 15. *Note from Theorems 14(v)-(vi) the important fact that if is either -asymptotically strictly pseudocontractive in the intermediate sense or asymptotically strongly strictly pseudocontractive in the intermediate sense then is in both cases asymptotically strongly strictly accretive in the intermediate sense.

The following two results follow from Theorem 14.

Corollary 16. *Let be a real Hilbert space. Then, the following properties hold:**(i) Assume that is -pseudocontractive and self-adjoint in the wide sense. Then, is -strongly accretive, -incrementally strictly passive, -incrementally strictly positive with , -strictly passive, and -strictly positive. If, in addition, , then the incremental properties are equivalent to the nonincremental counterparts.**(ii) Assume that is -strictly pseudocontractive. Then is -strictly accretive with . If, furthermore, is self-adjoint then it is equivalently -incrementally strictly passive and -incrementally strictly positive. If, in addition, , then the incremental properties are equivalent to the nonincremental counterparts.**(iii) Assume that is -asymptotically strongly strictly pseudocontractive in the intermediate sense and self-adjoint with (equivalently with if is linear). Then, is asymptotically strictly passive (and, equivalently, asymptotically strictly positive) in the intermediate sense with , , such that as and for some and a convergent real sequence *