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Abstract and Applied Analysis

Volume 2013 (2013), Article ID 639576, 12 pages

http://dx.doi.org/10.1155/2013/639576

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

Institute of Research and Development of Processes, University of the Basque Country, Campus of Leioa (Bizkaia), Apartado 644, 48080 Bilbao, Spain

Received 7 May 2013; Accepted 27 May 2013

Academic Editor: Luca Guerrini

Copyright © 2013 M. De la Sen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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

#### 1. Introduction

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

#### 2. Problem Statement and Main Results

Through this paper, one considers the complex Hilbert space on and operators and which define the following associated relations: where and is some given complex parameterizing vector, and , , ; and, with and . The set is some appropriate domain to define the previous functions of interest. Examples which adjust to the previous structure are very common in the real world as, for instance, linear continuous-time dynamic systems (with being the nonnegative real set for picking up values of the continuous-time argument) and linear discrete-time dynamic systems (with being the nonnegative integer set for values of the discrete-time argument) where is an exogenous, or reference, signal, is a feedback control, is the parameterized response to initial conditions, versus which is the forced response, is the measurable output to be controlled, and is a nonlinear (and, eventually, time-varying) controller device.

The inner products on the previous various Hilbert spaces are all denoted with the standard notation and mutually distinguished easily depending on context without explicit notational subscripts referred to each concrete space. Assume that is an indicator set defining truncated elements of the Hilbert space as, for instance, a real interval or a subset of the nonnegative integers and is a projection operator being a truncation operator so that and for each , and we define the seminorm on by ; with and the family of seminorms defines the resolution topology on since is a resolution of the identity [28]. Note that if . For instance, if then denotes a point value of , for while denotes the strip . Through the paper the notation “*” stands for adjoint operators and also for complex conjugates of scalars or vectors depending on the context.

The problem to be discussed in the paper is the stability of (1) under an integral-type constraint for the controller specified later on, which characterizes a whole admissible class of controllers rather than an individual controller. Conditions are given for the positive feed-forward operator which is assumed to be bounded and of closed range is ensured to be also strictly positive, then boundedly invertible, with its existing inverse being also a strictly positive operator. If such an operator is bounded and strictly positive, then the inner products and are both strictly positive and finitely upper-bounded for all nonzero input . The general formalism is given in Section 2 together with some links to contractive and pseudocontractive mappings while some real-world applications to asymptotic hyperstability of dynamic systems are then given in Section 3. The following preliminary result holds.

Proposition 1. *Assume that is a one-to-one linear operator with closed range. Then, the following properties hold*(i)* is invertible with nonzero minimum modulus,*(ii)*if, in addition, ** is positive (abbreviated notation being **), then** for any nonzero **,*(iii)*there is ** such that ** for any nonzero *.

*Proof. *Since on is one-to-one with closed range, it is also invertible from the open mapping theorem and then bounded below, so that there is such that
Then, the minimum modulus of satisfies
and Property (i) has been proven. Now, if , then there is a self-adjoint operator on such that so that, since from Property (i),
and Property (ii) is proven.

(iii) Note that if, , then there is such that and since one gets by Property (ii) that
Thus, for some if . Hence, Property (iii) follows.

*Definition 2. *The operator is said to be strictly positive (denoted as ) if it is positive (i.e., ) and .

Note from Proposition 1 that if is a one-to-one operator on with closed range, then it is invertible and .

It is also direct to prove that Property (i) of Proposition 1 is equivalent to its given assumption so that one has [28].

Proposition 3. * is a one-to-one linear bounded operator with closed range if and only if it is invertible with nonzero minimum modulus.*

Proposition 4. *If is a one-to-one linear bounded strictly positive operator with closed range, then it is invertible and is also strictly positive with closed range and bounded and so that for any nonzero .*

*Proof. *Note that from Proposition 1. Thus, so that is bounded. Since is bounded, then and . Thus, is also one-to-one with closed range from Proposition 3. Then, is self-adjoint, and since , one has from Property (i) of Proposition 1 and Definition 2 that since
Note that, if , then can be zero for some nonzero . The following result refers to the fulfilment of relationships (1) for all , that is, on the space provided that and bounded. Under some additional weak boundedness conditions, it is proven the stability of (1) with and belonging to . Note that is not a Hilbert space (even though is a Hilbert space) since it is not ensured that, for any , as .

An important result follows.

Theorem 5. *Assume that (1) holds for all , that is, and , where , and is some given complex parameterizing vector, , , ; and, with and . Assume also that*(1)* is bounded and as ,*(2)* is stable (or, equivalently, is bounded and causal), one-to-one, and with closed range,*(3)*,*(4)* is bounded,*(5)* is bounded,*(6)*; .**Then, , , , and they are bounded. Also, if , then , , as .*

*Proof. *Direct calculations yield
since and for any nonzero control from Proposition 1. One gets in the same way that
Since ; , one gets also that
One gets from (7), (9), (8), and (10) that
where . Now, since is a bounded operator, is a bounded function, as , and is bounded and one-to-one with closed range so that is also bounded and one-to-one with closed range implying from Proposition 1 that , and one gets from (12) that
Assume that there is some unbounded . Then, the subsequent contradiction
follows from (13) for some since . Then any is bounded. Since the operator on is bounded, it is stable, and then is also bounded and causal, and, since the function is bounded, then is also bounded with and ; , and is also bounded since . On the other hand, if is identically zero, then one gets from (13) , and, since , then .

Also, it is clear that, since , and since is bounded and converges asymptotically to zero and , then , is bounded, , and then since is bounded and asymptotically vanishing.

The assumption 6 of Theorem 5 can be relaxed leading to the following stronger result.

Corollary 6. *Theorem 5 holds if its assumption 6 is relaxed to .*

*Proof. *Note that (7) still holds since it is independent of assumption 6. The constraint (12) is modified as follows:
which makes (13) to remain valid, and Theorem 5 still holds.

In a physical context, is the whole input-output energy of (1), is the input-output energy dissipated on , and is the instantaneous input-output power at while is the energy supplied by the external source. Particular cases of interest in control engineering are (a) if the reference input , then the feedback control system is a regulator evolving only from its initial conditions, (b) if such reference is a constant real level, then the control system is a position servomechanism, (c) if the reference for , then the control system is a velocity servomechanism and so forth.

On the other hand, the extended Popov-type control inequality of the controller and implies that , ( for any nonzero control input with compact support); and all satisfying the assumption 6 of Theorem 5; that is the input-output energy is nonnegative and bounded; . The use of such a constraint allows the simultaneous investigation of the maintenance of the positivity and stability properties of (1) under a class of nonlinear time-varying controllers (defined by such a Popov constraint itself) rather than for a particular controller device belonging to such a class.

Note that on is stable since for some finite ; and, equivalently, is bounded. Now, one concludes from Proposition 4 for the system defined by the inverse operator that for any admissible control input since , bounded and causal.

The following result basically reformulates Theorem 5 if is a strictly positive pseudocontraction. Since the contribution of initial conditions and a bounded exogenous reference do not modify the stability properties, as seen from Theorem 5, they are assumed to be null in the sequel.

Theorem 7. *Assume that the relationships of (1) hold for all with , , , and, furthermore,*(1)* is bounded and causal, one-to-one, and with closed range.*(2)*.*(3)*; .**Then, and are bounded, and , as . Furthermore, one gets for any, that
**
If, in addition, is a pseudocontraction, then
**
with the lower-bound equating zero if and only if .*

*Proof. *Take the relation proved in Theorem 5 under zero exogenous reference and initial conditions in (1) to yield
Since then , and one gets for
and . Assume that is, furthermore, a pseucontraction on . Then,
and, equivalently,
implies that
and the following cases can occur.(a) if the controls and fulfil .(b)if the controls and fulfil . (c)if the controls and fulfil . This case is only feasible with equality.Combining the three cases one gets that
with the lower-bound equating zero if and only if ; that is .

Basically, Theorem 7 states that a strictly positive operator, which is also a pseudocontraction, subject to a feedback control law satisfying a Popov-type inequality keeps the boundedness of the input-output energy with a modified upper-bound which improves that associated to the Popov inequality if the minimum modulus of satisfies . The following result guarantees the fulfilment of Theorem 5 if is strictly positive and asymptotically pseudocontractive in the intermediate sense under a modified Popov-type inequality.

Theorem 8. *Assume that*(1)* is one-to-one, bounded, causal, and of closed range with minimum modulus ,*(2)* is asymptotically pseudocontractive in the intermediate sense satisfying the constraint,
for some real convergent sequence in such that as and zero initial conditions and exogenous reference in (1), where
are incremental values of and with being adjacent elements in the strict ordering on if such an indexing set is discrete and being a closed interval of nonzero constant Lebesgue measure in if such an indexing set is real,*(3)*the following inequality holds:
Then, , and they are bounded, and, furthermore, , as under a zero exogenous input and initial conditions.*

*Proof. *Since is asymptotically pseudocontractive in the intermediate sense
for zero initial conditions and exogenous reference in (1). Note that these particular conditions do not modify the boundedness-type stability properties related to the injection of any bounded exogenous reference under bounded initial conditions and some real convergent sequence in such that as . Since
then one has
so that, if , then and as . is bounded since it is piecewise continuous with eventual bounded discontinuities, and and finite. Since on and restricted to are stable, is also bounded and converges to zero.

A particular case of Theorem 8 of interest is as follows.

Corollary 9. *Theorem 8 holds if the assumption 2 is replaced by being a pseudocontraction.*

*Proof. *It follows since Theorem 8 holds, in particular, under the condition ; .

If is strictly positive and contractive, we obtain the subsequent result.

Theorem 10. *Assume that *(1)* is one-to-one, bounded, causal, and of closed range.*(2)* satisfies the following positive-bounded and contractive constraints for some given and :
with being a real sequence subject to and with , ; , provided that is the first element of , and*(3)*; .**Then, , and they are bounded, and, furthermore, , as under a zero exogenous input and initial conditions.*

*Proof. *Note from (30) that
since ; , and is the first element of . Thus, since , one gets
and also
Thus, as . Since is an admissible control, one concludes that any admissible control is bounded and it converges asymptotically to zero. The output has a similar property.

#### 3. Application Examples

*Example 1. *Define the truncated function within the time interval ; of as follows:
Thus, the output of a single-input single-output linear time-invariant continuous-time dynamic system of th order and initial state under a piecewise continuous control with eventual isolated bounded discontinuities , where the Hilbert space of the square-integrable functions on is
where , is the impulse response, is the zero-input response (i.e., the response contribution due to initial conditions) for initial sate , and “” stands for the convolution integral operator. Since the dynamic system is realizable, for . The complex function defined as is the transfer function, where stands for the Laplace transform of the impulse response where it exists. After defining for , the input-output energy obeys the following relations by using twice Parseval theorem:
where is the pointwise value at frequency of , the Fourier transform of provided that it exists with being the complex unit. Note that, in the previous expressions, the integral expressions have been also denoted by inner products on the time interval for the given , all of them being equivalent to inner products of truncated functions for the given on the Hilbert space . Equivalently, integrals of complex Fourier transforms on the whole imaginary axis are got through Parseval’s theorem and denoted by involving the impulse response (i.e., the transfer function evaluated on the imaginary complex axis) of the system and the Fourier transform of the truncated input. Now, assume that the controller is
is an exogenous reference signal which is piecewise continuous on , and is any piecewise continuous nonlinear time-varying function which satisfies the following integral-type constraint:
then
Note that any hodograph has the symmetry rules and . Also, . Thus, one gets by combining (37) and (40)
Decompose for each , where
for some given prefixed . Note that one (but not both) of the disjoint sets for can be empty. Then, by direct calculations one gets the following:
Assume that and . Then, one gets from (41) and (43) that
This relation leads to the following result.

Proposition 11. *Assume that*(1)*, ** so that ** and **,*(2)*the transfer function ** is strongly strictly positive real; that is, ** for all complex ** with **,*(3)*; *.*Then, one gets the following properties for any given initial state .*(i)*, **.*(ii)*; **.*(iii)*If *; *, then **. If, in addition, ** and ** are nonzero constants; *,* then ** and ** and **. *(iv)*if **, then ** and ** as ** and are both square-integrable on **; **. Thus, the closed-loop dynamic system (36), (38) is asymptotically hyperstable (i.e., globally asymptotically Lyapunov’s stable, [1–3]) since the state of any minimal state-space realization is also square-integrable on **, and it converges asymptotically to zero as time tends to infinity for any controller device ** satisfying (39).*

*Proof. *Since the transfer function is strictly positive real then it is strictly stable (i.e. all its poles are in for some ) and for all complex with . Since it is, furthermore, strongly positive real (i.e., a strictly positive operator on ), and it is associated to a dynamic system, so that it is realizable, then it is rational with pole-zero excess is zero (otherwise, if the pole-zero excess was +1, then it could not be strongly strictly positive real since , and if the pole-zero excess was −1 then it would not be realizable.) Since it has the same number of zeros and poles, and it is strongly strictly positive real, then its modulus is everywhere bounded in its definition domain, invertible, and of bounded inverse, so that one has
Note that since at exponential rate since the dynamic system is strictly stable. Since can be chosen arbitrarily to build the disjoint union equalizing ; , then choose . Now, assume that is unbounded. Since, it is piecewise continuous with eventual bounded discontinuities, then which implies that is strictly increasing so that the subsequent contradiction follows
Thus, . Since is strictly stable and , then . Property (i) has been proved. On the other hand, if , then , and the above contradiction holds. Then, . Note also that if , then the subsequent contradiction follows
Then, . Property (ii) has been proven.

Note that if ; is a direct consequence of from Property (ii). This proves the first part of Property (iii). Also, if and are nonzero constants; , then .

Now, if is identically zero in , then
leads to exponentially and the ; since is strongly strictly positive real so that the internal state of any minimal state-space realization is uniformly bounded, and it converges asymptotically to zero as time tends to infinity. Thus, asymptotic hyperstability follows for any satisfying (38). As a result, Property (iv) has been proven.

Note that the property of asymptotic hyperstability is independent of each particular controller provided that it belongs to a class that satisfies the integral relation (39) for some positive finite real . The particular case when the nonlinear controller is nonlinear, but time-invariant, while satisfying the corresponding integral constraint (39), is said to be the Popov-type absolute stability problem implying closed-loop global asymptotic Lyapunov’s stability. If the input-output euclidean inner product (associated with instantaneous power) under the integral symbol, rather than the inner product on the Hilbert space (associated with the energy), satisfies a parallel inequality, then the problem is said to be that of the Lure’s absolute stability problem [4–8]. It is, therefore, useful to describe the global asymptotic stability of classes of closed-loop systems of the given form under certain tolerated components dispersions. Proposition 11 also implies directly that any nonminimal state-space realization associated with strictly stable zero-pole cancellations of the transfer function is globally asymptotically Lyapunov stable. This follows since the transfer function remains invariant under zero-pole cancellations, so it is identical to that of the minimum state space realization, so that the operator is kept strictly positive and invertible although either controllability or observability (or both) becomes lost [29–31]. A generalization of the previous result to the study of hyperstability of composite connections [32] as well to Ulman-type extended stability [33, 34] of continuous-time dynamic systems can be performed based on the study given in [32].

The subsequent example is a discrete version of the previous one.

*Example 2. *Example 1 has a direct parallel discrete-time counterpart as discussed in the sequel. Define the truncated sequence on ; of the real sequence as follows:
where is the sampling period. Thus, the output of a single-input single-output linear continuous-time dynamic system of th order and initial state under a piecewise continuous control with eventual isolated bounded discontinuities , now the Hilbert space being , is
where , “” stands for the discrete convolution operator, and is the impulse response sequence since the dynamic system is realizable for . If this dynamic system is the same system as in the previous example subject to a piecewise control sequence , with ; , then ; where is the one-step delay operator such that . In this case, the discrete controller is
is an exogenous reference sequence, and are the elements of any nonlinear time-varying real sequence which satisfies the following summation-type constraint:
A close discussion to that of the former example by using the discrete Parseval theorem and inner products on the Hilbert space of square-summable sequences yields
By using (53), one gets a discrete counterpart of (44) as follows:
which leads to the subsequent result which is the discrete-time counterpart of Proposition 11, whose proof is close to that of Proposition 11, and it is then omitted.

Proposition 12. *Assume that*(1)*, ** so that ** and **,*(2)*the discrete function ** is strongly strictly positive real; that is, ** for all complex ** with **.*(3)*; **.**Then, one gets the following properties for any given initial state .*(i)*, **.*(ii)*; **.*(iii)*If *; *, then **. If, in addition, ** and ** are nonzero constants; **, then ** and ** and *.(iv)*If **, then ** and ** as **, and they are both square-summable on **; **. Thus, the closed-loop discrete dynamic system (50)-(51) is asymptotically hyperstable for any controller device of output sequence ** satisfying the discrete summation inequality **; **.*

The following example links asymptotic hyperstability of a discrete dynamic system with a unique equilibrium point which is also a fixed point.

*Example 3. *Assume that, in Example 2, a feedback stabilizing discrete control law satisfying the constraint ; , is injected to the system (1), neglecting initial conditions, and equivalently if the initial conditions are zero (this assumption does not affect the stability study,) we get