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

This paper investigates some parallel relations between the operators (I − 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.


Introduction
There is an important existing background literature available concerning passivity topics in dynamic systems.See, for instance, [1][2][3][4][5][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 closedloop 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 wellknown that the asymptotic hyperstability formalism covers particular cases, the so-called Lure's and Popov's absolute stability problems.See, for instance, [1][2][3][4][5][6][7][8][9][10][11][12][13][14][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][17][18][19][20][21][22] and the abundant included background literature in those background references.In particular, the socalled 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][17][18][19][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 squareintegrable 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.

Some Preliminaries
Denote by R, R + , and R 0+ the sets of real, positive real, and nonnegative real numbers, respectively, and by Z, Z + , and Z 0+ the sets of real, positive real, and nonnegative integer numbers, respectively.
it is said to be strictly passive if there exists some real constant  > 0 such that and it is said to be strongly strictly passive if there exist some real constants  > 0 and  > 0 such that 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.
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][17][18][19][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.
The operator  is strictly accretive if some such a positive constant  exists [16].
(c) ({  },  0 , )-asymptotically strictly accretive if there exist real constants  0 > 0 and  ≥ 0 and a real sequence The operator  is asymptotically strictly accretive in the intermediate sense if some such a triple ({  },  0 , ) exists.
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  2 (iR 0+ ; C  ) as follows.Definition 6.An operator  :  2 (iR 0+ ; C  ) →  2 (iR 0+ ; C  ) is as follows: (a) Accretive if, provided that ⟨ − ,  − ⟩  is real for each ,  ∈ () and all  ≥ 0, one has that, for each ,  ∈ () and some( − ) ∈ ( − ), we have 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 0 = 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. A rational function ĥ() of the complex variable  of real coefficients is positive real if (1) it is real for real , (2) it has no poles in the open right half plane, (3) its poles  = i at the imaginary axis, if any, are simple and their associate residues are simple, and (4) for all real  such that  = i is not a pole, Re ĥ(i) ≥ 0.
All these constraints together lead to Re ĥ() > 0 for Re  ≥ 0.
4. Assume that such a positive real rational function ĥ() is a transfer function of a realizable linear timeinvariant 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 ĥ() is both passive and positive since it is self-adjoint by nature with ĥ0 = 0 and we can also say that the associated dynamic system is positive and passive.As a result, its inputoutput time integral is nonnegative for all time.A simple example is, for instance, ĥ() = 1/( + ) for  ≥ 0 which is associated with the differential system ẏ () = −() + (), (0) =  0 .If  > 0 then the transfer function is strictly positive real (imaginary poles do not exist and the transfer function is stable satisfying also Re ĥ() > 0 for Re  ≥ 0 and all  ∈ (−∞, +∞)), and the associated dynamic system is strictly passive.If the transfer function is modified to ĥ () = ĥ() +  then Re ĥ () ≥  for Re  ≥ 0 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  > 0, 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 inputoutput energy for eventually negative controls.See, for instance, [31][32][33] and some references therein.
Some properties and relations for accretive operators on specific complex spaces are given and proved as follows.

Assume that
) is strictly accretive with constant  and bounded of norm   < 1/2 then it is strictly positive with  1 = (1 − 2  ) and, furthermore, it is strictly passive if the operator is self-adjoint.

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  = 0 since  :  2 (iR 0+ ; If  is strictly accretive, odd superadditive and bounded of norm and 2 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  =  = 0. Property (iv) is direct from Properties (i)-(iii) by changing the operator domain from iR 0+ to R 0+ .Definition 12. Let  be a real Hilbert space .Then, an operator  from () (the Domain of ) to () (the Image of ) is as follows: (a) ( 1 , )-pseudocontractive in the wide sense if there exist The operator  : () → () is said to be pseudocontractive in the wide sense if such a pair and equivalently if see [16].Note that if  : () → () is pseudocontractive in the wide sense it is pseudocontractive as well and a pseudocontraction in the wide sense with  1 =  = 1 is equivalent to a pseudocontraction.
The operator  : () → () is said to be strictly pseudocontractive if such a constant  exists.
Then, ( − ) is asymptotically strongly strictly accretive in the intermediate sense satisfying for some  ∈ [0, 1) and a convergent real sequence ( Proof.Firstly, assume that, ∀,  ∈ () is ({  }, )-asymptotically strictly pseudocontractive, then one has for some sequence {  } ⊂ R + , with   =   (, ), that where {  } is a real nonnegative sequence Relation ( 38) is equivalent to  Recall that positivity is equivalent to passivity for selfadjoint 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.The above result still holds with  0 = 1 for some  ∈ [0, 1), that is, {  } → (1 − )/(3 − ), if  : () → () is ({  }, )-asymptotically strictly pseudocontractive in the intermediate sense.
Proof.Properties (i) and (ii) follow from their counterparts of Theorem 14 and Property (iii) follows from Theorem 14(v) (see also Remarks 9).
If, furthermore,  is self-adjoint then it is also incrementally strictly passive.If, in addition, 0 = 0 then  is also strictly positive and strictly passive.
If, in addition,  is odd superadditive and bounded of norm   < 1/2 then it is incrementally strictly positive.
If, furthermore,  is self-adjoint then it is also incrementally strictly passive.If, in addition, 0 = 0 then  is also strictly positive and strictly passive.
If, in addition,  is self-adjoint then it is also incrementally passive.If, furthermore, 0 = 0 then  is also positive and passive.
(vi) Assume that ( − ) is ({  }, )-asymptotically strictly pseudocontractive in the intermediate sense.Then,  is ({  },  0 , )-asymptotically strongly accretive in the intermediate sense with  = (1 − )/(3 − ) and If, in addition,  is self-adjoint then it is also incrementally asymptotically strictly passive.Remark 20.It turns out that Corollary 19 is applicable to operators on  2 (R 0+ ; R  ) endowed with the same scalar product and of easy generalization to operators on . . .
∀ > 0, with   ≥ 0 since  is an accretive operator on  2 (R 0+ ; R  ), such that the real constant   > 0 if the operator  is strongly accretive, where () > 0 is the minimum modulus of  since it is one-to-one and of closed range, [10].Now, (51) implies that  :  2 (R 0+ ; R  ) →  2 (R 0+ ; R  ) 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 lim →∞ inf (  ) ≥ 0. On the other hand, for any finite  ∈ Z + , the composite operator   on  2 (R 0+ ; R  ), resulting from composition of  of  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 (  ) > 0 since   on  2 (R 0+ ; R  ) is one-to-one and of closed range for any finite  ∈ Z + since  is one-to-one and of closed range.

Asymptotic Passivity in Dynamic Systems
We first give some elementary concepts of usefulness to set the passivity framework.The notation for the spaces of real -square-integrable and truncated -square-integrable functions of nonnegative real domain is simplified due to subsequent extensive use as 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.
(iii) The binary relation stable wb if one of the operators ( −   ) for some  ∈ {1, 2} is asymptotically strongly (or strict or strict strongly) pseudocontractive in the intermediate sense and bounded while ( −   ) ( ̸ = ) ∈ {1,2} is (at least) asymptotically pseudocontractive in the intermediate sense.
Remark 23.While the operators   on  2 (R 0+ ; R  ),  0  = , for  = 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   =  −   for  = 1, 2 which satisfy the recursive relations: for  = 1, 2, ∀ ∈ Z + .Note that Theorem 22(iii) includes the case when both operators ( −   ) on  2 (R 0+ ; R  ) for  = 1, 2 are asymptotically strictly pseudocontractive.Note also that since the binary relation The following result is of interest relating the convergence properties of the operators (−) and  while also relating the potential fixed points of ( − ) to the convergence properties of sequences generated through the operator .Remark 26.Theorem 22 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]: where  = () ∈ R  ,  = () ∈ R  , and  = () ∈ R ℓ are the state, input, and output vectors and, respectively,  and  are smooth vector-valued functions on , and  is a smooth matrix-valued function on .
From Definitions 1 and 27, we have immediately the following simple direct result.
Proposition 28.Assume that  = ℓ and that  :  2 (R 0+ ; R  ) →  2 (R 0+ ; R  ) in Then one has the following: strictly passive) then the system ( 60) is   -passive (resp., strictly   -passive) and conversely.Remark 29.Inequality (64) refers to the operator  being strictly passive if the sum of the two right-hand-side constants is positive.Conversely, control system (60) is   -strictly passive.Borrowing the terminology of [13], if   > 0, then system (60) is said to be strictly input passive while if   > 0, then system (52) is said to be strictly output passive.If both right-hand side constants of (64) are positive, then the operator  is, furthermore, strongly strictly passive.
The last right-hand-side inequality of (63), that is, ⟨, ⟩  ≥ −, 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.) and () → 0 and () → 0 as  → ∞ for any given bounded initial conditions of the state.
(ii) Assume that  =  and  = −, with  and  being the input and output signals of dimension  ≥ 1 in   2 .Then, the closed-loop system is globally asymptotically Lyapunov's stable if  is strictly input passive and bounded and  is passive with zero-independent constant or if  is strictly input passive and bounded and  is passive with zero-independent constant.Proof.Since () 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, Re(()) ≥  1 > 0 for all  ∈ C with Re  ≥ 0. Thus, one has, for any real constant  ∈ ( 1 , ∞) and some real constant  1 > 0 and for any given control   2 with support of nonzero measure, where  min (⋅) denotes the minimum eigenvalue of the symmetric (⋅)-matrix, so that the controlled system  =  is input-strictly passive, and the operator  is strictly positive, strictly passive, and strictly accretive.) and   () → 0 and   () → 0 as  → ∞ so that () → 0 and () → 0 as  → ∞ for any bounded initial conditions.Property (i) has been proved.
To prove Property (ii), note that, in this case, if  2 () and  2 () are the  2 -gains wb of  and , then one has for any control being nonzero except possibly on a set of zero measure that which holds provided that  1 ∈ (0,   / 2 ()],  2 () < +∞,   =   = 0, and   ≥ 0,   > 0; that is,  is bounded and strictly input passive and  is passive, or which holds provided that  2 ∈ (0,   / 2 ()],  2 () < +∞,   =   = 0,   ≥ 0, and   > 0; that is,  is bounded and strictly input passive and  is passive.On the other hand if  is bounded and strictly input passive and  is passive with  0 = 0, then it is also passive with any finite positive constant   so that one gets from (66) that   () → 0 and   () → 0 as  → ∞ for any  > 0 and then () → 0 and () → 0 as  → ∞, for any given initial conditions.A similar conclusion arises from (67) if  is bounded and strictly input passive and  is passive with  0 = 0. Property (ii) has been proved.
Theorem 30 can be directly reformulated in the discretetime framework related to the space ℓ  2 .

Further Examples
Example 1.Consider the iterated linear continuous-time dynamic feedback system: for given initial conditions  0 =   for  = 1, 2, which is described by the sequences of operators  , [1,13].Note also that the particular case that the inputs  1 and  2 are independent of  is included.
On the other hand, if   is asymptotically pseudocontractive in the intermediate sense for some  ∈ {1, 2} and   is asymptotically strictly (or strongly strictly) pseudocontractive in the intermediate sense for ( ̸ = ) ∈ {1, 2} then ( −   ) are asymptotically strictly passive and incrementally asymptotically passive for  = 1, 2 and the binary relation  (−  1 )(−  2 ) is asymptotically   2 -stable wb (Theorem 22).As a result, if  1,2 are square-integrable, then  1,2 and  1,2 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.If ( −   ) is either asymptotically strictly or strongly strictly pseudocontractive in the intermediate sense for some  ∈ {1, 2} then   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., both  1 ,  2 are not jointly linear) then (70a) and (70b) do not necessarily hold.However, the following equations hold also if the operators are nonlinear [1]: (73a) that is, if both inputs are zero in finite discrete-time  0 then {  } ≥ 0 ≡ 0,  ,+1 =  1  1 +  2  2 ; ,  = 1, 2, ∀ ∈ Z + , (73b) for some no-necessarily unique, gain sequences {  }, ,  = 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 {  } and {  } of dimension  = 2( 1 +  2 ): Thus, if (  − ), respectively , is either asymptotically strictly, or strongly strictly, pseudocontractive, in the intermediate sense, then , respectively (  − ), 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][35][36][37][38] are under study.
set , a mapping  :  → , and a binary relation   ⊂  2 on  defined by  as   = {(, ) :  ∈ }.A binary relation  on    is said to be    -stable if [ ∈    ∧ (, ) ∈ ] ⇒  ∈    . is said to be   -stable with finite gain (wfg) if there exist finite nonnegative real constants   (gain) and   (bias) such that -stable with finite gain and zero bias (wb) if   = 0. Note that, if  is    -stable wfg, then it is trivially    -stable wb since  ∈    .