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

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 𝐻 on 𝐂 and operators 𝐺∶𝐻→𝐻 and 𝐾∶𝐻→𝐻 which define the following associated relations: 𝑦=𝐺𝑢+𝑦_ℎ,𝑢=𝐾𝑟−𝜑_Γ𝑦, (1) where 𝑦_ℎ=𝑦_ℎ(𝑝) and 𝑝∈𝐂^𝑞 is some given complex parameterizing vector, and 𝑟∶Γ→𝐻∣dom𝐾, 𝜑_Γ∶Γ×𝐻→ran𝜑_Γ(⊂𝐻), 𝑢∶Γ→(𝐻∣ran𝐾)+ran𝜑_Γ(⊂𝐻); and𝑦=𝑦_ℎ+𝑦_𝑓, with 𝑦_ℎ∶Γ×𝐂^𝑞→𝐻 and 𝑦_𝑓∶Γ→𝐻∣ran𝐺. The set Γ is some appropriate domain to define the previous functions of interest. Examples which adjust to the previous structure are very common in the real world as, for instance, linear continuous-time dynamic systems (with Γ=𝐑₀₊ being the nonnegative real set for picking up values 𝑡∈Γof the continuous-time argument) and linear discrete-time dynamic systems (with Γ=𝐙₀₊ being the nonnegative integer set for values 𝑡∈Γof the discrete-time argument) where 𝑟 is an exogenous, or reference, signal, 𝑢 is a feedback control, 𝑦_ℎ is the parameterized response to initial conditions, versus 𝐺𝑢 which is the forced response, 𝑦 is the measurable output to be controlled, and 𝜑_Γ(𝑦) is a nonlinear (and, eventually, time-varying) controller device.

The inner products on the previous various Hilbert spaces are all denoted with the standard notation ⟨⋅,⋅⟩ and mutually distinguished easily depending on context without explicit notational subscripts referred to each concrete space. Assume that Γ is an indicator set defining truncated elements of the Hilbert space as, for instance, a real interval or a subset of the nonnegative integers and 𝑃_𝑡 is a projection operator being a truncation operator so that 𝑥_𝑡=𝑃_𝑡𝑥 and for each 𝑡∈Γ, and we define the seminorm on 𝐻 by ‖𝑥‖_𝑡=‖𝑥_𝑡‖=‖𝑃_𝑡𝑥‖; ∀𝑡∈Γ with 𝑃_𝑡≠𝐼 and the family {‖‖_𝑡∶𝑡∈Γ} of seminorms defines the resolution topology on 𝐻 since {𝑃_𝑡∶𝑡∈𝐻} is a resolution of the identity [28]. Note that ‖𝑥‖_𝑡1≥‖𝑥_𝑡2‖ if 𝑡₁≥𝑡₂. For instance, if Γ=𝐑₀₊ then 𝑥(𝑡) denotes a point value of 𝑥∶𝐑₀₊→𝐑₀₊, for 𝑡∈Γ while 𝑥_𝑡 denotes the strip 𝑥∶[0,t]∩𝐑₀₊→𝐑₀₊. Through the paper the notation “*” stands for adjoint operators and also for complex conjugates of scalars or vectors depending on the context.

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

Proposition 1. Assume that 𝐺∶𝐻→𝐻 is a one-to-one linear operator with closed range. Then, the following properties hold(i)𝐺∶𝐻→𝐻 is invertible with nonzero minimum modulus,(ii)if, in addition, 𝐺∶𝐻→𝐻 is positive (abbreviated notation being 𝐺≽0), then⟨𝐺𝑢,𝑢⟩>0 for any nonzero 𝑢∶Γ→𝐻,(iii)there is 𝑡∈Γ such that ⟨𝐺𝑃_𝑡𝑢,𝑃_𝑡𝑢⟩>0 for any nonzero 𝑢∈dom(𝐺).

Proof. Since 𝐺 on 𝐻 is one-to-one with closed range, it is also invertible from the open mapping theorem and then bounded below, so that there is 𝑐∈𝐑₊ such that ‖𝐺𝑢‖=⟨𝐺𝑢,𝐺𝑢⟩^1/2≥𝑐‖𝑢‖=𝑐⟨𝑢,𝑢⟩^1/2;∀𝑢∈dom𝐺. (2) Then, the minimum modulus 𝜇(𝐺) of 𝐺 satisfies 𝜇(𝐺)‖‖=inf𝐺𝑢‖𝑢‖∶𝑢(∈dom𝐺)≠0≥𝑐>0 (3) and Property (i) has been proven. Now, if 𝐺≽0, then there is a self-adjoint operator 𝐺=𝐺^∗≽0 on 𝑋 such that 𝐺=𝐺^∗𝐺=𝐺𝐺^∗=𝐺² so that, since 𝜇(𝐺)>0 from Property (i), ⟨𝐺𝑢,𝑢⟩^1/2=𝐺²𝑢,𝑢^1/2=𝐺𝑢,𝐺𝑢^1/2=𝐺𝑢=𝐺𝑢‖𝑢‖‖𝑢‖≥inf𝐺𝑢‖𝑢‖()𝑢‖𝐺‖𝑢‖()∶𝑢∈dom𝐺≠0‖≥𝜇>0;∀𝑢≠0∶Γ⟶𝐻 (4) and Property (ii) is proven.
(iii) Note that if, 𝑢≠0, then there is 𝑡∈Γ such that 𝑃_𝑡𝑢≠0 and ⟨𝐺𝑃_𝑡𝑢,𝑃_𝑡𝑢⟩>0 since one gets by Property (ii) that ⟨𝐺𝑢,𝑢⟩^1/2=sup_𝑡∈Γ⟨𝐺𝑃_𝑡𝑢,𝑃_𝑡𝑢⟩^1/2≥⟨𝐺𝑃_𝑡𝑢,𝑃_𝑡𝑢⟩^1/2=⟨𝐺²𝑃_𝑡𝑢,𝑃_𝑡𝑢⟩^1/2=𝐺𝑃_𝑡𝑢=𝐺𝑃_𝑡𝑢𝑃_𝑡𝑢𝑃_𝑡𝑢𝐺𝑃≥inf^𝑡𝑢𝑃_𝑡𝑢∶𝑢(∈dom𝐺)≠0𝑃_𝑡𝑢≥𝜇𝐺𝑃_𝑡𝑢>0;∀𝑢(≠0)∶Γ⟶𝐻. (5) Thus, ⟨𝐺𝑃_𝑡𝑢,𝑃_𝑡𝑢⟩>0 for some 𝑡∈Γ if 𝑢≠0. Hence, Property (iii) follows.

Definition 2. The operator 𝐺∶𝐻→𝐻 is said to be strictly positive (denoted as 𝐺≻0) if it is positive (i.e., 𝐺≽0) and 𝜇(𝐺)>0.

Note from Proposition 1 that if 𝐺≽0 is a one-to-one operator on 𝐻 with closed range, then it is invertible and 𝐺≻0.

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

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

Proposition 4. If 𝐺∶𝐻→H is a one-to-one linear bounded strictly positive operator with closed range, then it is invertible and 𝐺⁻¹∶𝐻→𝐻 is also strictly positive with closed range and bounded and 𝜇(𝐺⁻¹)>0 so that ⟨𝐺⁻¹𝑢,𝑢⟩>0 for any nonzero 𝑢∶Γ→𝐻.

Proof. Note that 1/‖𝐺⁻¹‖=𝜇(𝐺)>0 from Proposition 1. Thus, ‖𝐺⁻¹‖<∞ so that 𝐺⁻¹ is bounded. Since 𝐺 is bounded, then ‖𝐺‖<∞ and 1/‖𝐺‖=𝜇(𝐺⁻¹)>0. Thus, 𝐺⁻¹∶𝐻→𝐻 is also one-to-one with closed range from Proposition 3. Then, 𝐺⁻¹=𝐺⁻¹𝐺^∗−1=𝐺⁻² is self-adjoint, and since 𝜇(𝐺⁻¹)>0, one has from Property (i) of Proposition 1 and Definition 2 that 𝐺⁻¹≻0 since ⟨𝐺⁻¹𝑢,𝑢⟩^1/2=⟨𝐺⁻²𝑢,𝑢⟩^1/2=⟨𝐺⁻¹𝑢,𝐺⁻¹𝑢⟩^1/2=𝐺⁻¹𝑢=𝜇𝐺⁻¹‖𝑢‖()>0,∀𝑢≠0∶Γ⟶𝐻. (6) Note that, if 𝐺≽0, then ⟨𝐺𝑢,𝑢⟩ can be zero for some nonzero 𝑢∶Γ→𝐻. The following result refers to the fulfilment of relationships (1) for all 𝑡∈Γ, that is, on the space 𝐻_𝑒={𝑓_𝑡=𝑃_𝑡𝑓∶(𝑓∶Γ→𝑃𝐶(Γ));∀𝑡∈Γ} provided that 𝐺≻0 and bounded. Under some additional weak boundedness conditions, it is proven the stability of (1) with 𝑢 and 𝑦 belonging to 𝐻. Note that 𝐻_𝑒 is not a Hilbert space (even though 𝐻 is a Hilbert space) since it is not ensured that, for any 𝑓∶Γ→𝑃𝐶(Γ), 𝑓_𝑡∶Γ→𝐻 as (Γ∋)𝑡→∞.

An important result follows.

Theorem 5. Assume that (1) holds for all 𝑡∈Γ, that is, 𝐺∶𝐻_𝑒→𝐻_𝑒 and 𝐾∶𝐻_𝑒→𝐻_𝑒, where 𝐻_𝑒={𝑓_𝑡=𝑃_𝑡𝑓∶(𝑓∶Γ→𝑃𝐶(Γ));∀𝑡∈Γ}, 𝑦_ℎ=𝑦_ℎ(𝑝) and 𝑝∈𝐂^𝑞 is some given complex parameterizing vector, 𝑟∶Γ→𝐻_𝑒∣dom𝐾, 𝜑_Γ∶Γ×𝐻_𝑒→ran𝜑_Γ, 𝑢∶Γ→(𝐻_𝑒∣ran𝐾)+ran𝜑_Γ; and𝑦=𝑦_ℎ+𝑦_𝑓, with 𝑦_ℎ∶Γ×𝐂^q→𝐻 and 𝑦_𝑓∶Γ→𝐻_𝑒∣ran𝐺. Assume also that(1)𝑦_ℎ=𝑦_ℎ(𝑝) is bounded and 𝑃_𝑡𝑦_ℎ→0 as (Γ∋)𝑡→∞,(2)𝐺∶𝐻_𝑒→𝐻_𝑒 is stable (or, equivalently, 𝐺∶𝐻_𝑒∣𝐻→𝐻 is bounded and causal), one-to-one, and with closed range,(3)𝐺≻0,(4)𝐾∶𝐻→𝐻 is bounded,(5)𝑟∶Γ→𝐻 is bounded,(6)⟨𝑃_𝑡𝑦,𝑃_𝑡(𝜑_Γ(𝑦))⟩≥−𝛾_𝑡≥−𝛾>−∞; ∀𝑡∈Γ.Then, 𝑢, 𝑦_𝑓, 𝑦∶Γ→𝐻, and they are bounded. Also, if 𝑟≡0, then 𝑢(𝑡)→0, 𝑦_𝑓(𝑡)→0, 𝑦(𝑡)→0 as (Γ∋)𝑡→∞.

Proof. Direct calculations yield 𝑦,𝑢=⟨𝐺𝑢+𝑦_ℎ,𝐾𝑟−𝜑_Γ𝑦⟩=⟨𝐺𝑢,𝐾𝑟⟩+⟨𝑦_ℎ−𝑦+𝑦,𝐾𝑟⟩𝐺𝑢,𝜑_ℎ,𝜑_Γ𝑦=⟨𝐺𝑢,𝐾𝑟⟩+⟨𝑦_ℎ,𝐾𝑟⟩+⟨𝐺𝑢,𝑢⟩−⟨𝑦_ℎ,𝜑_Γ𝑦⟩−⟨𝐺𝑢,𝐾𝑟⟩=⟨𝐺𝑢,𝑢⟩+⟨𝑦_ℎ,𝐾𝑟−𝜑_Γ𝑦⟩≥𝜇²𝐺‖𝑢‖²+⟨𝑦_ℎ,𝐾𝑟−𝜑_Γ𝑦⟩=𝜇²𝐺sup_𝑡∈Γ𝑃_𝑡𝑢²+⟨𝑦_ℎ,𝐾𝑟−𝜑_Γ𝑦⟩≥𝜇²𝐺𝑃_𝑡𝑢²+⟨𝑦_ℎ,𝐾𝑟−𝜑_Γ𝑦⟩;∀𝑡∈Γ (7) since 𝑢=𝐾𝑟−𝜑_Γ(𝑡,𝑦) and ⟨𝐺𝑢,𝑢⟩≥𝜇²(𝐺)‖𝑢‖²≥𝜇²(𝐺)‖𝑃_𝑡𝑢‖²>0 for any nonzero control from Proposition 1. One gets in the same way that 𝑃_𝑡𝑦,𝑃_𝑡𝑢≥𝜇²𝐺𝑃_𝑡𝑢²+𝑃_𝑡𝑦_ℎ,𝑃_𝑡𝐾𝑟−𝜑_Γ𝑦,∀𝑡∈Γ. (8) Since ⟨𝑃_𝑡𝑦,𝑃_𝑡(𝜑_Γ(𝑦))⟩≥−𝛾_𝑡>−∞; ∀𝑡∈Γ, one gets also that 𝑦,𝑢=⟨𝑦,𝐾𝑟−𝜑_Γ𝑦⟩=⟨𝑦,𝐾𝑟⟩−⟨𝑦,𝜑_Γ𝑦⟩≤𝑦,𝐾𝑟+sup_𝑡∈Γ𝛾_𝑡𝑃;∀𝑡∈Γ,_𝑡𝑦,𝑃_𝑡𝑢≤𝑃_𝑡𝑦,𝑃_𝑡(𝐾𝑟)+𝛾_𝑡;∀𝑡∈Γ. (9)(10) One gets from (7), (9), (8), and (10) that 𝑦,𝐾𝑟+𝛾≥𝜇²𝐺‖𝑢‖²+𝑦_ℎ,𝐾𝑟−𝜑_Γ𝑦≥𝜇²𝐺𝑃_𝑡𝑢²+𝑦_ℎ,𝐾𝑟−𝜑_Γ𝑦,𝑃∀𝑡∈Γ,_𝑡𝑦,𝑃_𝑡(𝐾𝑟)𝛾_𝑡≥𝜇²𝐺𝑃_𝑡𝑢²⟨𝑃_𝑡𝑦_ℎ,𝑃_𝑡𝐾𝑟−𝜑_Γ𝑦⟩;∀𝑡∈Γ, (11)(12) where 0<𝛾=sup_𝑡∈Γ𝛾_𝑡<∞. Now, since 𝐾 is a bounded operator, 𝑟 is a bounded function, 𝑃_𝑡𝑦_ℎ→0 as 𝑡→∞, and 𝐺≻0 is bounded and one-to-one with closed range so that 𝐺≻0 is also bounded and one-to-one with closed range implying from Proposition 1 that 𝜇(𝐺)=𝜇²(𝐺)>0, and one gets from (12) that liminf_{Γ∋𝑡→∞}𝛾_𝑡+⟨𝑃_𝑡𝐺𝑃_𝑡𝑢⟩,𝑃_𝑡(𝐾𝑟)>−𝜇²𝐺𝑃_𝑡𝑢²≥liminf_{Γ∋𝑡→∞}𝛾_𝑡+⟨𝑃_𝑡𝐺𝑃_𝑡𝑢⟩,𝑃_𝑡(𝐾𝑟)>−𝜇²𝐺𝑃_𝑡𝑢²≥0. (13) Assume that there is some unbounded 𝑢∶Γ→𝐻. Then, the subsequent contradiction 0≤liminf_{Γ∋𝑡→∞}𝛾_𝑡+𝜇²𝐺𝜆−𝑃_𝑡𝑢𝑃_𝑡𝑢=−∞ (14) follows from (13) for some 𝜆∈𝐑₀₊ since 𝜇(𝐺)>0. Then any 𝑢∶Γ→𝐻 is bounded. Since the operator 𝐺 on 𝐻_𝑒 is bounded, it is stable, and then 𝐺∶𝐻_𝑒∣𝐻→𝐻 is also bounded and causal, and, since the function 𝑢∶Γ→𝐻 is bounded, then 𝑦_𝑓∶Γ→𝐻 is also bounded with ‖𝑦_𝑓‖≤‖𝐺‖‖𝑢‖ and ‖𝑃_𝑡𝑦_𝑓‖≤‖𝑃_𝑡𝐺𝑃_𝑡‖‖𝑢‖≤‖𝐺‖‖𝑢‖; ∀𝑡∈Γ, and 𝑦∶Γ→𝐻 is also bounded since 𝑦=𝑦_ℎ+𝑦_𝑓. On the other hand, if 𝑟∶Γ→𝐻 is identically zero, then one gets from (13) 0≤𝜇(𝐺)limsup_{Γ∋𝑡→∞}‖𝑃_𝑡𝑢‖≤𝛾<∞, and, since 𝜇(𝐺)>0, then ∃lim_{Γ∋𝑡→∞}𝑢(𝑡)=0.
Also, it is clear that, since 𝑢∶Γ→𝐻_𝑒, and since 𝑢∶Γ→𝐻 is bounded and converges asymptotically to zero and ‖𝑦_𝑓‖≤‖𝐺‖‖𝑢‖, then 𝑢∶Γ→𝐻, 𝑦_𝑓 is bounded, 𝑦_𝑓∶Γ→𝐻, and then ∃lim_{Γ∋𝑡→∞}𝑦(𝑡)=lim_{Γ∋𝑡→∞}𝑦_𝑓(𝑡)=0 since 𝑦_ℎ∶Γ→𝐻 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 liminf_𝑡→∞⟨𝑃_𝑡𝑦,𝑃_𝑡(𝜑_Γ(𝑦))⟩≥−𝛾>−∞.

Proof. Note that (7) still holds since it is independent of assumption 6. The constraint (12) is modified as follows: liminf_{Γ∋𝑡→∞}⟨𝑃_𝑡𝑦,𝑃_𝑡(𝐾𝑟)⟩+𝛾_𝑡−𝜇²𝐺𝑃_𝑡𝑢²−⟨𝑃_𝑡𝑦_ℎ,𝑃_𝑡𝐾𝑟−𝜑_Γ𝑦⟩≥0 (15) which makes (13) to remain valid, and Theorem 5 still holds.

In a physical context, 𝐸=⟨𝑦,𝑢⟩ is the whole input-output energy of (1), 𝐸(𝑡)=⟨𝑃_𝑡𝑦,𝑃_𝑡𝑢⟩ is the input-output energy dissipated on [0,𝑡]∩Γ, and (𝑢∗𝑦)(𝑡) is the instantaneous input-output power at 𝑡∈Γ while ⟨𝐺𝑢+𝑦_ℎ,𝐾𝑟⟩ is the energy supplied by the external source. Particular cases of interest in control engineering are (a) if the reference input 𝑟≡0, then the feedback control system is a regulator evolving only from its initial conditions, (b) if such reference is a constant real level, then the control system is a position servomechanism, (c) if the reference 𝑟(𝑡)=𝐾𝑡 for 𝑡∈Γ, then the control system is a velocity servomechanism and so forth.

On the other hand, the extended Popov-type control inequality of the controller ⟨𝑃_𝑡𝑦,𝑃_𝑡(𝜑_Γ(𝑦))⟩≥−𝛾>−∞ and 𝐺≻0 implies that 0≤𝐸(𝑡)≤𝛾<∞, (0<𝐸≤𝛾<∞ for any nonzero control input with compact support); ∀𝑡∈Γ and all 𝜑_Γ(𝑦) satisfying the assumption 6 of Theorem 5; that is the input-output energy is nonnegative and bounded; ∀𝑡∈Γ. The use of such a constraint allows the simultaneous investigation of the maintenance of the positivity and stability properties of (1) under a class of nonlinear time-varying controllers (defined by such a Popov constraint itself) rather than for a particular controller device belonging to such a class.

Note that 𝐺 on 𝐻_𝑒 is stable since ‖𝐺𝑢‖_𝑡≤𝑀‖𝑢‖_𝑡 for some finite ∈𝐑₊; ∀𝑡∈Γ and, equivalently, 𝐺∶𝐻_𝑒∣𝐻→𝐻 is bounded. Now, one concludes from Proposition 4 for the system defined by the inverse operator 𝐺⁻¹ that 0≤⟨𝑃_𝑡(𝐺⁻¹𝑢),𝑃_𝑡𝑢⟩<∞ for any admissible control input 𝑢 since 𝐺⁻¹≻0, bounded and causal.

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

Theorem 7. Assume that the relationships of (1) hold for all 𝑡∈Γ with 𝑟≡0, 𝑦_ℎ≡0, 𝐺∶𝐻_𝑒→𝐻_𝑒, 𝜑_Γ∶Γ×𝐻_𝑒→ran𝜑_Γ and, furthermore,(1)𝐺∶𝐻_𝑒∣𝐻→𝐻 is bounded and causal, one-to-one, and with closed range.(2)𝐺≻0.(3)⟨𝑃_𝑡𝑦,𝑃_𝑡(𝜑_Γ(𝑦))⟩≥−𝛾_𝑡≥−𝛾>−∞; ∀𝑡∈Γ.Then, 𝑢,𝑦_𝑓,𝑦∶Γ→𝐻 and are bounded, and 𝑢(𝑡)→0, 𝑦(𝑡)→0 as (Γ∋)𝑡→∞. Furthermore, one gets for any, 𝑢₁,𝑢₂∶Γ→ran𝜑_Γ that 𝑢₁,𝑢₂+𝑢₁,𝑢₂^∗≤2𝛾𝜇²𝐺. (16) If, in addition, 𝐺∣𝐻 is a pseudocontraction, then 0≤𝐺𝑢₁−𝐺𝑢₂,𝑢₁−𝑢₂4≤𝛾min1,𝜇²𝐺 (17) with the lower-bound equating zero if and only if 𝑢₁=𝑢₂.

Proof. Take the relation proved in Theorem 5  ⟨𝑦,𝑢⟩=⟨𝐺𝑢+𝑦_ℎ,𝐾𝑟−𝜑_Γ(𝑦)⟩under zero exogenous reference and initial conditions in (1) to yield 0<𝜇²𝐺⟨𝑢,𝑢⟩≤⟨𝐺𝑢,𝑢⟩=𝐺𝑢,𝐺𝑢≤𝛾<∞. (18) Since 𝐺≻0 then 𝐺=𝐺^∗, and one gets for 𝑢=𝑢₁−𝑢₂0≤⟨𝐺𝑢₁−𝐺𝑢₂,𝑢₁−𝑢₂⟩=⟨𝐺𝑢₁−𝐺𝑢₂,𝐺𝑢₁−𝐺𝑢₂⟩=⟨𝐺𝑢₁,𝐺𝑢₁⟩+⟨𝐺𝑢₂,𝐺𝑢₂⟩−⟨𝐺𝑢₁,𝐺𝑢₂⟩−⟨𝐺𝑢₂,𝐺𝑢₁⟩=⟨𝐺𝑢₁,𝑢₁⟩+⟨𝐺𝑢₂,𝑢₂⟩−⟨𝐺𝑢₁,𝑢₂⟩−⟨𝐺𝑢₂,𝑢₁⟩≤2𝛾−⟨𝐺𝑢₁,𝑢₂⟩−⟨𝑢₂,𝐺𝑢₁⟩=2𝛾−⟨𝐺𝑢₁,𝑢₂⟩−⟨𝐺𝑢₁,𝑢₂⟩^∗⟹0≤max0,⟨𝐺𝑢₁,𝑢₂⟩+⟨𝐺𝑢₁,𝑢₂⟩^∗≤⟨𝐺𝑢₁−𝐺𝑢₂,𝑢₁−𝑢₂⟩−⟨𝐺𝑢₁,𝑢₂⟩−⟨𝐺𝑢₁,𝑢₂⟩^∗𝑢≤2𝛾<∞0≤₁−𝑢₂²=⟨𝑢₁−𝑢₂,𝑢₁−𝑢₂⟩=𝑢₁²+𝑢₂²−⟨𝑢₁,𝑢₂⟩−⟨𝑢₁,𝑢₂⟩^∗≤2𝛾𝜇²𝐺−⟨𝑢₁,𝑢₂⟩−⟨𝑢₁,𝑢₂⟩^∗ (19) and  ⟨𝑢₁,𝑢₂⟩+⟨𝑢₁,𝑢₂⟩^∗≤2𝛾/𝜇²(𝐺). Assume that 𝐺≻0 is, furthermore, a pseucontraction on 𝐻. Then, 0≤⟨𝐺𝑢₁−𝐺𝑢₂,𝑢₁−𝑢₂⟩=⟨𝐺𝑢₁−𝐺𝑢₂,𝐺𝑢₁−𝐺𝑢₂⟩≤𝑢₁−𝑢₂² (20) and, equivalently, 𝐺𝑢₁−𝐺𝑢₂²≤𝑢₁−𝑢₂²+(𝐼−𝐺)𝑢₁−(𝐼−𝐺)𝑢₂²0≤⟨𝐺𝑢₁−𝐺𝑢₂,𝑢₁−𝑢₂⟩≤⟨𝐺𝑢₁−𝐺𝑢₂,𝐺𝑢₁−𝐺𝑢₂⟩≤𝑢₁−𝑢₂²≤2𝛾𝜇²𝐺−⟨𝑢₁,𝑢₂⟩−⟨𝑢₁,𝑢₂⟩^∗ (21) implies that ⟨𝑢₁,𝑢₂⟩+⟨𝑢₁,𝑢₂⟩^∗<⟨𝐺𝑢₁−𝐺𝑢₂,𝑢₁−𝑢₂⟩+⟨𝑢₁,𝑢₂⟩+⟨𝑢₁,𝑢₂⟩^∗≤2𝛾𝜇²𝐺, (22) and the following cases can occur.(a)0≤⟨𝐺𝑢₁−𝐺𝑢₂,𝑢₁−𝑢₂⟩≤min(𝛾,2𝛾/𝜇²(𝐺)−⟨𝑢₁,𝑢₂⟩−⟨𝑢₁,𝑢₂⟩^∗) if the controls 𝑢₁ and 𝑢₂ fulfil 0<⟨𝑢₁,𝑢₂⟩+⟨𝑢₁,𝑢₂⟩^∗<2𝛾/𝜇²(𝐺).(b)0≤⟨𝐺𝑢₁−𝐺𝑢₂,𝑢₁−𝑢₂⟩≤min(𝛾,2𝛾/𝜇²(𝐺)−|⟨𝑢₁,𝑢₂⟩+⟨𝑢₁,𝑢₂⟩^∗|)if the controls 𝑢₁ and 𝑢₂ fulfil ⟨𝑢₁,𝑢₂⟩+⟨𝑢₁,𝑢₂⟩^∗<0. (c)0<2𝛾/𝜇²(𝐺)≤⟨𝐺𝑢₁−𝐺𝑢₂,𝑢₁−𝑢₂⟩≤𝛾if the controls 𝑢₁ and 𝑢₂ fulfil ⟨𝑢₁,𝑢₂⟩+⟨𝑢₁,𝑢₂⟩^∗≥2𝛾/𝜇²(𝐺). This case is only feasible with equality.Combining the three cases one gets that 0≤⟨𝐺𝑢₁−𝐺𝑢₂,𝑢₁−𝑢₂4⟩≤𝛾min1,𝜇²𝐺 (23) with the lower-bound equating zero if and only if 𝑢₁=𝑢₂; that is 𝑢=𝑢₁−𝑢₂=0.

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 𝐺 satisfies 𝜇(𝐺)>4. The following result guarantees the fulfilment of Theorem 5 if 𝐺∶𝐻→𝐻 is strictly positive and asymptotically pseudocontractive in the intermediate sense under a modified Popov-type inequality.

Theorem 8. Assume that(1)𝐺≻0 is one-to-one, bounded, causal, and of closed range with minimum modulus 𝜇(𝐺)>𝛼,(2)𝐺∶𝐻→𝐻 is asymptotically pseudocontractive in the intermediate sense satisfying the constraint, 0≤𝑃_𝑡𝑦₁−𝑃_𝑡𝑦₂,𝑃_𝑡𝑢₁−𝑃_𝑡𝑢₂≤𝛼_𝑡𝑃_𝑡𝑢₁−𝑃_𝑡𝑢₂²;∀𝑡∈Γ (24)  for some real convergent sequence {𝛼_𝑡}_𝑡∈Γ in [𝛼,∞) such that 𝛼_𝑡→𝛼∈(0,1] as 𝑡(∈Γ)→∞ and zero initial conditions and exogenous reference in (1), where 𝑦_𝑡=𝑦_𝑡+𝑇−𝑦_𝑡=𝑃_𝑡+𝑇𝑦−𝑃_𝑡𝑦,𝑢_𝑡=𝑢_𝑡+𝑇−𝑢_𝑡=𝑃_𝑡+𝑇𝑢−𝑃_𝑡𝑢 (25) are incremental values of 𝑦 and 𝑢 with 𝑡,𝑡+𝑇(≻𝑡)∈Γ being adjacent elements in the strict ordering on Γ if such an indexing set is discrete and [𝑡,𝑡+𝑇] being a closed interval of nonzero constant Lebesgue measure 𝑇 in Γ if such an indexing set is real,(3)the following inequality holds: liminf_{Γ∋𝑡→∞}𝑃_𝑡𝑦,𝑃_𝑡𝜑_Γ𝑦+𝛼_𝑡‖𝑢‖²_𝑡≥0. (26)  Then, 𝑢,𝑦∶Γ→𝐻, and they are bounded, and, furthermore, 𝑢(𝑡)→0, 𝑦(𝑡)→0 as (Γ∋)𝑡→∞ under a zero exogenous input and initial conditions.

Proof. Since 𝐺∶𝐻→𝐻 is asymptotically pseudocontractive in the intermediate sense 0≤⟨𝑃_𝑡𝑦₁−𝑃_𝑡𝑦₂,𝑃_𝑡𝑢₁−𝑃_𝑡𝑢₂⟩=⟨𝑃_𝑡𝐺𝑃_𝑡𝑢₁−𝑃_𝑡𝐺𝑃_𝑡𝑢₂,𝑃_𝑡𝑢₁−𝑃_𝑡𝑢₂⟩≤𝛼_𝑡𝑃_𝑡𝑢₁−𝑃_𝑡𝑢₂²;∀𝑡∈Γ (27) 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 {𝛼_𝑡}_𝑡∈Γ in [𝛼,∞) such that 𝛼_𝑡→𝛼∈(0,1) as 𝑡(∈Γ)→∞. Since liminf_{Γ∋𝑡→∞}⟨𝑃_𝑡𝑦,𝑃_𝑡𝜑_Γ𝑦⟩+𝛼_𝑡‖𝑢‖²_𝑡=liminf_{Γ∋𝑡→∞}𝛼_𝑡𝑃_𝑡𝑢²−𝑃_𝑡𝑦,𝑃_𝑡𝑢≥0, (28) then one has 0≤𝜇²𝐺𝑢_𝑡²=𝜇²𝐺‖𝑢‖²_𝑡≤𝛼_𝑡‖𝑢‖²_𝑡⟹−𝛼_𝑡‖𝑢‖²_𝑡≤𝜇²𝐺−𝛼_𝑡‖𝑢‖²_𝑡≤0,∀𝑡∈Γ (29) so that, if 𝛼=lim_{Γ∋𝑡→∞}𝛼_𝑡<𝜇²(𝐺), then limsup_{Γ∋𝑡→∞}(𝜇²(𝐺)−𝛼_𝑡)‖𝑢‖²_𝑡≤0 and 𝑢(𝑡)→0 as Γ∈𝑡→∞. 𝑢∶Γ→𝐻 is bounded since it is piecewise continuous with eventual bounded discontinuities, and 𝑇>0 and finite. Since 𝐺 on 𝐻_𝑒 and 𝐺 restricted to 𝐻 are stable, 𝑦∶Γ→𝐻 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 𝐺∶𝐻→𝐻 being a pseudocontraction.

Proof. It follows since Theorem 8 holds, in particular, under the condition 𝛼_𝑡=𝛼=1; ∀𝑡∈Γ.

If 𝐺∶𝐻→𝐻 is strictly positive and contractive, we obtain the subsequent result.

Theorem 10. Assume that (1)𝐺≻0 is one-to-one, bounded, causal, and of closed range.(2)𝐺∶𝐻→𝐻 satisfies the following positive-bounded and contractive constraints for some given 𝛽∈Γ and 𝑢=𝑢₁−𝑢₂: 0≤𝑃_𝛽𝐺𝑃_𝛽𝑢₁−𝑃_𝛽𝐺𝑃_𝛽𝑢₂,𝑃_𝛽𝑢₁−𝑃_𝛽𝑢₂−𝑃₀GP₀u₁−P₀GP₀u₂,P₀u₁−P₀u₂≤M<∞0≤⟨𝑃_𝑡+2𝛽𝐺𝑃_𝑡+2𝛽𝑢₁−𝑃_𝑡+2𝛽𝐺𝑃_𝑡+2𝛽𝑢₂,𝑃_𝑡+2𝛽𝑢₁−𝑃_𝑡+2𝛽𝑢₂⟩−⟨𝑃_𝑡+𝛽𝐺𝑃_𝑡+𝑇𝑢₁−𝑃_𝑡+𝛽𝐺𝑃_𝑡+𝛽𝑢₂,𝑃_𝑡+𝛼𝑢₁−𝑃_𝑡+𝛽𝑢₂⟩≤𝜒_𝑡⟨𝑃_𝑡+𝛽𝐺𝑃_𝑡+𝜙𝑢₁−𝑃_𝑡+𝛽𝐺𝑃_𝑡+𝛽𝑢₂,𝑃_𝑡+𝛽𝑢₁−𝑃_𝑡+𝛽𝑢₂⟩−𝜒_𝑡𝑃_𝑡𝐺𝑃_𝑡𝑢₁−𝑃_𝑡𝐺𝑃_𝑡𝑢₂,𝑃_𝑡𝑢₁−𝑃_𝑡𝑢₂;∀𝑡∈Γ (30)(31) with {𝜒_𝑡}_𝑡∈Γ being a real sequence subject to 0≤𝜒_𝑡≤𝜎<∞ and ∏^𝑡+𝛽_Γ∋𝑡[𝜒_𝑡]=𝜒<1 with 𝛽, (𝑡+𝑛𝛽)∈Γ;  ∀𝑛∈𝐙₊,  ∀𝑡∈Γ provided that 0∈Γ is the first element of Γ, and(3)⟨𝑃_𝑡𝑦,𝑃_𝑡(𝜑_Γ(𝑦))⟩≥−𝛾_𝑡≥−𝛾=𝑀/(𝜒−1)>−∞; ∀𝑡∈Γ.Then, 𝑢,𝑦∶Γ→𝐻, and they are bounded, and, furthermore, 𝑢(𝑡)→0, 𝑦(𝑡)→0 as (Γ∋)𝑡→∞ under a zero exogenous input and initial conditions.

Proof. Note from (30) that 0≤⟨𝑃_𝑛𝛽𝐺𝑃_𝑛𝛽𝑢₁−𝑃_𝑛𝛽𝐺𝑃_𝑛𝛽𝑢₂,𝑃_{(𝑛−1)𝛽}𝑢₁−𝑃_{(𝑛−1)𝛽}𝑢₂⟩≤𝜒^𝑛−1⟨𝑃_𝛽𝐺𝑃_𝛽𝑢₁−𝑃_𝛽𝐺𝑃_𝛽𝑢₂,𝑃_𝛽𝑢₁−𝑃_𝛽𝑢₂⟩−𝑃₀𝐺𝑃₀𝑢₁−𝑃₀𝐺𝑃₀𝑢₂,𝑃₀𝑢₁−𝑃₀𝑢₂=𝑀𝜒^𝑛−1≤𝑀<∞ (32) since 𝑛𝛽∈Γ; ∀𝑛∈𝐙₊, and 0∈Γ is the first element of Γ. Thus, since 0≤𝜒<1, one gets 0≤_𝑛∈𝐙+⟨𝑃_𝑛𝛽𝐺𝑃_𝑛𝛽𝑢₁−𝑃_𝑛𝛽𝐺𝑃_𝑛𝛽𝑢₂,𝑃_{(𝑛−1)𝛽}𝑢₁−𝑃_{(𝑛−1)𝛽}𝑢₂⟩≤⟨𝐺𝑢₁−𝐺𝑢₂,𝑢₁−𝑢₂𝑀⟩≤𝛾=1−𝜒<∞ (33) and also 0≤^𝑁_𝑛=1⟨𝑃_𝑛𝛽𝐺𝑃_𝑛𝛽𝑢₁−𝑃_𝑛𝛽𝐺𝑃_𝑛𝛽𝑢₂,𝑃_{(𝑛−1)𝛽}𝑢₁−𝑃_{(𝑛−1)𝛽}𝑢₂⟩≤⟨𝐺𝑢₁−𝐺𝑢₂,𝑢₁−𝑢₂𝑀⟩≤𝛾=1−𝜒<∞,∀𝑁∈𝐙₊. (34) Thus, 𝑢(𝑡)=𝑢₁(𝑡)−𝑢₂(𝑡)→0 as Γ∈𝑡→∞. Since 𝑢≡0 is an admissible control, one concludes that any admissible control is bounded and it converges asymptotically to zero. The output 𝑦(𝑡) has a similar property.

3. Application Examples

Example 1. Define the truncated function within the time interval [0,𝑡]; ∀𝑡∈𝐑₀₊ of 𝑓∶𝐑→𝐑 as follows: 𝑓_𝑡(𝜏)=𝑃_𝑡𝑓(𝜏)[]𝑓=for𝜏∈0,𝑡0otherwise. (35) Thus, the output of a single-input single-output linear time-invariant continuous-time dynamic system of 𝑛th order and initial state 𝑥(0)=𝑥₀∈𝐑^𝑛 under a piecewise continuous control with eventual isolated bounded discontinuities 𝑢∶𝐑→𝐑∩𝐻, where 𝐻=𝐿²(𝐑₀₊)≡𝐿²[0,∞) the Hilbert space of the square-integrable functions on 𝐑₀₊ is 𝑦(𝑡)=^𝑡₀𝑔(𝑡,𝜏)𝑢(𝜏)𝑑𝜏+𝑐𝑡,𝑥₀=^∞_−∞𝑔(𝑡,𝜏)𝑢_𝑡(𝜏)𝑑𝜏+𝑐𝑡,𝑥₀=𝑔∗𝑢(𝑡)+𝑐𝑡,𝑥₀,∀𝑡∈𝐑₀₊, (36) where Γ=𝐑₀₊={𝑧∈𝐑∶𝑧≥0}, 𝑔∶𝐑×𝐑→𝐑 is the impulse response, 𝑐(𝑡,𝑥₀) is the zero-input response (i.e., the response contribution due to initial conditions) for initial sate 𝑥₀, and “∗” stands for the convolution integral operator. Since the dynamic system is realizable, 𝑔(𝑡,𝜏)=0 for 𝜏>𝑡. The complex function 𝐺∶𝐂→𝐂 defined as 𝐺(𝑠)=𝐋(𝑔(𝑡)) is the transfer function, where 𝐋 stands for the Laplace transform of the impulse response where it exists. After defining 𝑦(𝑡)=0 for 𝑡<0, the input-output energy obeys the following relations by using twice Parseval theorem: 𝐸(𝑡)=^𝑡₀𝑦(𝜏)𝑢(𝜏)𝑑𝜏=^∞_−∞𝑦(𝜏)𝑢_𝑡(𝜏)=1𝑑𝜏2𝜋^∞_−∞𝑌(𝐢𝜔)𝑈_𝑡(−𝐢𝜔)𝑑𝜔=⟨𝑦,𝑢⟩_𝑡=⟨𝑦,𝑢_𝑡⟩=⟨𝑦_𝑡,𝑢_𝑡⟩=⟨𝑦_𝑡1,𝑢⟩=2𝜋⟨𝑌,𝑈_𝑡⟩=12𝜋^∞_−∞𝐺(𝐢𝜔)𝑈_𝑡(𝐢𝜔)𝑈_𝑡()+−𝐢𝜔𝑑𝜔_𝑡0𝑐𝜏,𝑥₀𝑢(𝜏)=1𝑑𝜏2𝜋^∞_−∞𝐺(𝐢𝜔)𝑈_𝑡(𝐢𝜔)²+𝑑𝜔_𝑡0𝑐𝜏,𝑥₀𝑢(𝜏)=1𝑑𝜏2𝜋⟨𝐺𝑈_𝑡,𝑈_𝑡⟩+^𝑡₀𝑐𝜏,𝑥₀𝑢(𝜏)=1𝑑𝜏𝑈2𝜋⟨𝐺,_𝑡²⟩+^𝑡₀𝑐𝜏,𝑥₀𝑢(𝜏)≥1𝑑𝜏2𝜋min_{𝜔∈𝐑0+}Re𝐺(𝐢𝜔)^∞_−∞𝑈_𝑡(𝐢𝜔)²=𝑑𝜔min_{𝜔∈𝐑0+}Re𝐺(𝐢𝜔)^∞_−∞𝑢_𝑡(𝜏)²+𝑑𝜏_𝑡0𝑐𝜏,𝑥₀𝑢(𝜏)=𝑑𝜏min_{𝜔∈𝐑0+}Re𝐺(𝐢𝜔)^𝑡₀|𝑢(𝜏)|²+𝑑𝜏_𝑡0𝑐𝜏,𝑥₀𝑢(𝜏)=𝑑𝜏min_{𝜔∈𝐑0+}Re𝐺(𝐢𝜔)⟨𝑢_𝑡,𝑢_𝑡⟩+^𝑡₀𝑐𝜏,𝑥₀𝑢(𝜏)=𝑑𝜏min_{𝜔∈𝐑0+}Re𝐺(𝐢𝜔)⟨𝑢,𝑢⟩_𝑡+^𝑡₀𝑐𝜏,𝑥₀𝑢(𝜏)𝑑𝜏;∀𝑡∈𝐑₀₊, (37) where 𝐹(𝐢𝜔) is the pointwise value at frequency 𝜔 of 𝐹∶𝐂→𝐂, the Fourier transform of 𝑓∶𝐑→𝐑 provided that it exists with 𝐢=√−1 being the complex unit. Note that, in the previous expressions, the integral expressions have been also denoted by inner products ⟨⋅,⋅⟩_𝑡≡⟨⋅_𝑡,⋅⟩≡⟨⋅,⋅_𝑡⟩ on the time interval [0,𝑡] for the given 𝑡∈𝐑₀₊, all of them being equivalent to inner products of truncated functions for the given 𝑡∈𝐑₀₊on the Hilbert space 𝐿²[0,∞). Equivalently, integrals of complex Fourier transforms on the whole imaginary axis are got through Parseval’s theorem and denoted by ⟨𝐺𝑈_𝑡,𝑈_𝑡⟩ 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 𝑢(𝑡)=𝑘(𝑡)𝑟(𝑡)−𝜑𝑡,𝑦(𝑡);∀𝑡∈𝐑₀₊. (38)𝑟∶𝐑→𝐑 is an exogenous reference signal which is piecewise continuous on 𝐑₀₊, and 𝜑∶[0,𝑡]×𝐑→𝐑 is any piecewise continuous nonlinear time-varying function which satisfies the following integral-type constraint: ^𝑡₀𝜑𝜏,𝑦(𝜏)𝑦(𝜏)𝑑𝜏≥−𝛾𝑡,𝑥₀,∀𝑡∈𝐑₀₊, (39) then 𝐸(𝑡)=^𝑡₀𝑦(𝜏)𝑢(𝜏)=𝑑𝜏_𝑡0𝑦(𝜏)𝑘(𝜏)𝑟(𝜏)𝑑𝜏−^𝑡₀𝑦(𝜏)𝜑𝜏,𝑦(𝜏)𝑑𝜏≤𝛾𝑡,𝑥₀+^𝑡₀𝑦(𝜏)𝑘(𝜏)𝑟(𝜏)𝑑𝜏. (40) Note that any hodograph 𝐺(𝐢𝜔) has the symmetry rules Re𝐺(𝐢𝜔)=Re𝐺(−𝐢𝜔) and Im𝐺(𝐢𝜔)=−Im𝐺(−𝐢𝜔). Also, 𝜇(𝐺)≥min_{𝜔∈𝐑0+}Re𝐺(𝐢𝜔)>0. Thus, one gets by combining (37) and (40) min_{𝜔∈𝐑0+}Re𝐺(𝐢𝜔)^𝑡₀|𝑢(𝜏)|²𝑑𝜏+^𝑡₀𝑐𝜏,𝑥₀𝑢(𝜏)𝑑𝜏≤𝛾𝑡,𝑥₀+^𝑡₀𝑦(𝜏)𝑘(𝜏)𝑟(𝜏)𝑑𝜏;∀𝑡∈𝐑₀₊. (41) Decompose [0,𝑡]=𝐼_1𝜀(𝑡)∪𝐼_2𝜀(𝑡) for each 𝑡∈𝐑₀₊, where 𝐼_1𝜀(𝑡)={[]∶|𝑢(𝜏},𝐼𝜏∈0,𝑡)|≤𝜀_2𝜀(𝑡)={𝜏∈[0,𝑡]∶|𝑢(𝜏)|>𝜀}, (42) for some given prefixed 𝜀∈𝐑₊. Note that one (but not both) of the disjoint sets 𝐼_𝑖𝜀(𝑡) for 𝑖=1,2 can be empty. Then, by direct calculations one gets the following: min_{𝜔∈𝐑0+}Re𝐺(𝐢𝜔)^𝑡₀|𝑢(𝜏)|²𝑑𝜏+^𝑡₀𝑐𝜏,𝑥₀𝑢(𝜏)≥𝑑𝜏min_{𝜔∈𝐑0+}Re𝐺(𝐢𝜔)^𝑡₀|𝑢(𝜏)|²𝑑𝜏−𝜀_𝐼1(𝑡)𝑐𝜏,𝑥₀+𝑑𝜏_𝐼2(𝑡)𝑐𝜏,𝑥₀𝑢(𝜏)|𝑢(𝜏)|²≥𝑑𝜏min_{𝜔∈𝐑0+}Re𝐺(𝐢𝜔)^𝑡₀|𝑢(𝜏)|²𝑑𝜏−𝜀_𝐼1(𝑡)𝑐𝜏,𝑥₀−𝑑𝜏_𝐼2(𝑡)𝑐𝜏,𝑥₀𝑢(𝜏)|𝑢(𝜏)|²≥𝑑𝜏min_{𝜔∈𝐑0+}Re𝐺(𝐢𝜔)−𝜀⁻¹max_{𝑡∈𝐑0+}𝑐𝑡,𝑥₀×^𝑡₀|𝑢(𝜏)|²𝑑𝜏−𝜀_𝐼1(𝑡)𝑐𝜏,𝑥₀𝑑𝜏;∀𝑡∈𝐑₀₊. (43) Assume that sup_{𝑡∈𝐑0+}|𝑘(𝑡)|≤𝑘 and sup_{𝑡∈𝐑0+}|𝑟(𝑡)|≤𝑟. Then, one gets from (41) and (43) that |𝑢^𝑡₀(𝜏)|−𝑘𝑟max_{𝜔∈𝐑0+}|𝐺(𝑢(𝜏≤1𝐢𝜔)||)|𝑑𝜏min_{𝜔∈𝐑0+}Re𝐺(𝐢𝜔)−𝜀⁻¹max_{𝑡∈𝐑0+}𝑐𝑡,𝑥₀×𝛾𝑡,𝑥₀+𝜀_𝐼1(𝑡)𝑐𝜏,𝑥₀𝑑𝜏,∀𝑡∈𝐑₀₊. (44) This relation leads to the following result.

Proposition 11. Assume that(1)𝑘, 𝑟∈𝐿_∞(𝐑₀₊) so that sup_{𝑡∈𝐑0+}|𝑘(𝑡)|≤𝑘<∞ and sup_{𝑡∈𝐑0+}|𝑟(𝑡)|≤𝑟<∞,(2)the transfer function 𝐺(𝑠) is strongly strictly positive real; that is, Re𝐺(𝑠)>0 for all complex 𝑠 with Re𝑠≥0,(3)+∞>limsup_𝑡→∞(𝛾(𝑡,𝑥₀))≥liminf_𝑡→∞(𝛾(𝑡,𝑥₀))>0; ∀𝑥₀∈𝐑^𝑛.Then, one gets the following properties for any given initial state x₀∈𝐑^𝑛.(i)𝑢, 𝑦∈𝐿_∞(𝐑₀₊).(ii)liminf_𝑡→∞∫^𝑡₀𝑦(𝜏)𝑘(𝜏)𝑟(𝜏)𝑑𝜏>−∞; liminf_𝑡→∞(𝑢²(𝑡)−𝑦(𝑡)𝑘(𝑡)𝑟(𝑡))≤0.(iii)If   |u(t)|²≥𝑦(𝑡)𝑘(𝑡)𝑟(𝑡); ∀𝑡∈𝐑₀₊, then ∃lim_𝑡→∞(|𝑢(𝑡)|²−𝑦(𝑡)𝑘(𝑡)𝑟(𝑡))=0. If, in addition, 𝑘(𝑡)=𝑘 and 𝑟(𝑡)=𝑟 are nonzero constants; ∀𝑡∈𝐑₀₊, then ∃lim_𝑡→∞𝑦(𝑡)=𝑦_∞ and ∃lim_𝑡→∞𝑢(𝑡)=𝑢_∞=𝑘𝑟∫^∞₀𝑔(𝜏)𝑑𝜏 and ∃lim_𝑡→∞𝑦(𝑡)=𝑦_∞=𝑘𝑟(∫^∞₀𝑔(𝜏)𝑑𝜏)². (iv)if  𝑟≡0, then 𝑢(𝑡)→0 and 𝑦(𝑡)→0 as 𝑡→∞ and are both square-integrable on 𝐑₀₊; ∀𝑥₀∈𝐑^𝑛. 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 𝐑₀₊, and it converges asymptotically to zero as time tends to infinity for any controller device 𝜑∶[0,𝑡]×𝐑→𝐑 satisfying (39).

Proof. Since the transfer function 𝐺(𝑠) is strictly positive real then it is strictly stable (i.e. all its poles are in Re𝑠≤−𝜌<0 for some 𝜌∈𝐑₊) and Re𝐺(𝑠)>0 for all complex 𝑠 with Re𝑠≥0. Since it is, furthermore, strongly positive real (i.e., a strictly positive operator on 𝐿²(𝐑₀₊)), and it is associated to a dynamic system, so that it is realizable, then it is rational with pole-zero excess is zero (otherwise, if the pole-zero excess was +1, then it could not be strongly strictly positive real since lim_𝜔→±∞Re𝐺(𝐢𝜔)=0, and if the pole-zero excess was −1 then it would not be realizable.) Since it has the same number of zeros and poles, and it is strongly strictly positive real, then its modulus is everywhere bounded in its definition domain, invertible, and of bounded inverse, so that one has 0<min_{𝜔∈𝐑0+}Re𝐺(𝐢𝜔)≤max_{𝜔∈𝐑0+}|𝐺(𝐢𝜔)|<+∞. (45) Note that ∫_𝐼1(𝑡)|𝑐(𝜏,𝑥₀)|𝑑𝜏≤∫^∞₀|𝑐(𝜏,𝑥₀)|𝑑𝜏=𝑣(𝑥₀)<+∞ since lim_𝑡→∞|𝑐(𝑡,𝑥₀)|=0 at exponential rate since the dynamic system is strictly stable. Since 𝜀∈𝐑₊ can be chosen arbitrarily to build the disjoint union 𝐼_1𝜀(𝑡)∪𝐼_2𝜀(𝑡) equalizing [0,𝑡]; ∀𝑡∈𝐑₀₊, then choose 𝜀>|𝑐(𝑡,𝑥₀)|/(min_{𝜔∈𝐑0+}Re𝐺(𝐢𝜔)). Now, assume that 𝑢∶𝐑→𝐑 is unbounded. Since, it is piecewise continuous with eventual bounded discontinuities, then lim_𝑡→∞∫^𝑡₀|𝑢(𝜏)|𝑑𝜏=∞ which implies that ∫^𝑡₀(|𝑢(𝜏)|−𝑘𝑟max_{𝜔∈𝐑0+}|𝐺(𝐢𝜔)|)|𝑢(𝜏)|𝑑𝜏 is strictly increasing so that the subsequent contradiction follows +∞=lim_𝑡→∞^𝑡₀|𝑢(𝜏)|−𝑘𝑟max_{𝜔∈𝐑0+}|𝐺(𝐢𝜔)||𝑢(𝜏)|𝑑𝜏≤lim_𝑡→∞^𝑡₀|𝑢(𝜏)|²(𝜏)𝑟(𝑡)𝑦(𝜏≤1−𝑘)𝑑𝜏min_{𝜔∈𝐑0+}Re𝐺(𝐢𝜔)−𝜀⁻¹max_{𝑡∈𝐑0+}𝑐𝑡,𝑥₀×𝛾𝑡,𝑥₀+𝜀_𝐼1(𝑡)𝑐𝜏,𝑥₀𝑑𝜏<+∞,∀𝑡∈𝐑₀₊. (46) Thus, 𝑢∈𝐿_∞(𝐑₀₊). Since 𝐺(𝑠) is strictly stable and 𝑢∈𝐿_∞(𝐑₀₊), then 𝑦∈𝐿_∞(𝐑₀₊). Property (i) has been proved. On the other hand, if liminf_𝑡→∞(|𝑢(𝑡)|²−𝑦(𝑡)𝑘(𝑡)𝑟(𝑡))>0, then lim_𝑡→∞∫^𝑡₀(|𝑢(𝜏)|²−𝑘(𝜏)𝑟(𝑡)𝑦(𝜏))𝑑𝜏=+∞, and the above contradiction holds. Then, liminf_𝑡→∞(|𝑢(𝑡)|²−𝑦(𝑡)𝑘(𝑡)𝑟(𝑡))≤0. Note also that if lim_𝑡→∞∫^𝑡₀𝑦(𝜏)𝑘(𝜏)𝑟(𝜏)𝑑𝜏=−∞, then the subsequent contradiction follows 0≤lim_𝑡→∞^𝑡₀|𝑢(𝜏)|²≤1𝑑𝜏min_{𝜔∈𝐑0+}Re𝐺(𝐢𝜔)−𝜀⁻¹max_{𝑡∈𝐑0+}𝑐𝑡,𝑥₀×𝛾𝑡,𝑥₀+^𝑡₀𝜀𝑐𝜏,𝑥₀+𝑘(𝜏)𝑟(𝑡)𝑦(𝜏)𝑑𝜏=−∞,∀𝑡∈𝐑₀₊. (47) Then, liminf_𝑡→∞∫^𝑡₀𝑦(𝜏)𝑘(𝜏)𝑟(𝜏)𝑑𝜏>−∞. Property (ii) has been proven.
Note that ∃lim_𝑡→∞(|𝑢(𝑡)|²−𝑦(𝑡)𝑘(𝑡)𝑟(𝑡))=0 if |𝑢(𝑡)|²≥𝑦(𝑡)𝑘(𝑡)𝑟(𝑡); ∀𝑡∈𝐑₀₊ is a direct consequence of liminf_𝑡→∞(|𝑢(𝑡)|²−𝑦(𝑡)𝑘(𝑡)𝑟(𝑡))≤0 from Property (ii). This proves the first part of Property (iii). Also, if 𝑘(𝑡)=𝑘 and 𝑟(𝑡)=𝑟 are nonzero constants; ∀𝑡∈𝐑₀₊, then lim_𝑡→∞(|𝑢(𝑡)|²−𝑦(𝑡)𝑘𝑟)=0.
Now, if 𝑟(𝑡) is identically zero in 𝐑₀₊, then lim_𝑡→∞^𝑡₀|𝑢(𝜏)|²≤1𝑑𝜏min_{𝜔∈𝐑0+}Re𝐺(𝐢𝜔)−𝜀⁻¹max_{𝑡∈𝐑0+}𝑐𝑡,𝑥₀×𝛾𝑡,𝑥₀+𝜀_𝐼1(𝑡)𝑐𝜏,𝑥₀𝑑𝜏<+∞,∀𝑡∈𝐑₀₊ (48) leads to lim_𝑡→∞𝑢(𝑡)=0 exponentially and the lim_𝑡→∞𝑦(𝑡)=0; ∀𝑥₀∈𝐑^𝑛 since 𝐺(𝑠) is strongly strictly positive real so that the internal state of any minimal state-space realization is uniformly bounded, and it converges asymptotically to zero as time tends to infinity. Thus, asymptotic hyperstability follows for any 𝜑∶[0,𝑡]×𝐑→𝐑 satisfying (38). As a result, Property (iv) has been proven.

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

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

Example 2. Example 1 has a direct parallel discrete-time counterpart as discussed in the sequel. Define the truncated sequence on [0,𝑘𝑇]; ∀𝑘∈Γ=𝐙₀₊ of the real sequence {𝑓_𝑘≡𝑓(𝑘𝑇)}_{𝑘∈𝐙0+} as follows: 𝑓_𝑗(𝑘)=𝑃_𝑗𝑓()=𝑓𝑘𝑇^𝑗≡𝑓𝑗𝑇for𝑗∈[0,𝑘]∩𝐙0otherwise;∀𝑘∈𝐙₀₊, (49) where 𝑇>0 is the sampling period. Thus, the output of a single-input single-output linear continuous-time dynamic system of 𝑛th order and initial state 𝑥(0)=𝑥₀∈𝐑^𝑛 under a piecewise continuous control with eventual isolated bounded discontinuities 𝑢∶𝐑→𝐑∩𝐻, now the Hilbert space being 𝐻=ℓ²(𝐙₀₊), is 𝑦_𝑘=^𝑘_𝑗=0𝑔^𝑑𝑘,𝑗𝑢_𝑗+𝑐𝑘,𝑥₀=^∞_𝑗=−∞𝑔^𝑑𝑘,𝑗𝑢_𝑘𝑗+𝑐_𝑘𝑥₀=𝑔^𝑑∘𝑢(𝑘)+𝑐_𝑘𝑥₀,∀𝑘∈𝐙₀₊, (50) where 𝐙₀₊={𝑧∈𝐙∶𝑧≥0}, “∘” stands for the discrete convolution operator, 𝑐_𝑘=𝑐_𝑘(𝑥₀)≡𝑐_𝑘(𝑘𝑇,𝑥₀) and {𝑔^𝑑(𝑘,𝑗)}_{𝑘,𝑗∈𝐙0+} is the impulse response sequence since the dynamic system is realizable 𝑔^𝑑(𝑘,𝑗)=0 for 𝑗>𝑘. If this dynamic system is the same system as in the previous example subject to a piecewise control sequence {𝑢_𝑘}_{𝑘∈𝐙0+}, with 𝑢_𝑘=𝑢(𝑘𝑇); ∀𝑘∈𝐙₀₊, then 𝑔^𝑑(𝑘,0)=(1−𝑞⁻¹)𝐋⁻¹(𝐺(𝑠)/𝑠)_{𝑡=𝑘𝑇}; ∀𝑘∈𝐙₀₊ where 𝑞⁻¹ is the one-step delay operator such that 𝑓_𝑘=𝑞⁻¹𝑓_𝑘+1. In this case, the discrete controller is 𝑢_𝑘=𝑘_𝑘𝑟_𝑘−𝜑_𝑘𝑘,𝑦_𝑘;∀𝑘∈𝐙₀₊. (51)𝑟∶𝐑→𝐑 is an exogenous reference sequence, and 𝜑∶[0,𝑘]×𝐑→𝐑 are the elements of any nonlinear time-varying real sequence {𝜑_𝑘=𝜑_𝑘(𝑘,𝑦_𝑘)}_{𝑘∈𝐙0+} which satisfies the following summation-type constraint: ^𝑘_𝑗=0𝜑_𝑗𝑗,𝑦_𝑗𝑦_𝑗≥−𝛾_𝑘𝑥₀;∀𝑘∈𝐙₀₊. (52) A close discussion to that of the former example by using the discrete Parseval theorem and inner products on the Hilbert space of square-summable sequences ℓ²(𝐙₀₊) yields 𝐸_𝑘=^𝑘_𝑗=0𝑦_𝑗𝑢_𝑗=^∞_𝑗=−∞𝑦_𝑗𝑢_𝑗(𝑘)=12𝜋_|𝑧|=1𝑧⁻¹𝑌(𝑧)𝑈𝑘,𝑧⁻¹=1𝑑𝑧2𝜋₀^2𝜋𝑌𝑒^𝐢𝜑𝑈𝑘,𝑒^−𝐢𝜑𝑒^−𝐢𝜑(𝑘)1𝑑𝜑=⟨𝑦,𝑢⟩=(𝑘)⟩=12𝜋⟨𝑌,𝑈2𝜋⟨𝐺^𝑑𝑈(𝑘),𝑈(𝑘)⟩+^𝑘_𝑗=0𝑐_𝑗𝑥₀𝑢_𝑗=12𝜋⟨𝐺^𝑑,|𝑈(𝑘)|²⟩+^𝑘_𝑗=0𝑐_𝑗𝑥₀𝑢_𝑗≥12𝜋min_{[]𝜑∈0,2𝜋}Re𝐺^𝑑𝑒^𝐢𝜑₀^2𝜋𝑈𝑘,𝑒^𝐢𝜑2=𝑑𝜑min_{[]𝜑∈0,2𝜋}Re𝐺^𝑑𝑒^𝐢𝜑∞_𝑗=−∞𝑢²_𝑗(𝑘)+^𝑘_𝑗=0𝑐_𝑗𝑥₀𝑢_𝑗=min_{[]𝜑∈0,2𝜋}Re𝐺^𝑑𝑒^𝐢𝜑𝑘_𝑗=0𝑢²_𝑗(𝑘)+^𝑘_𝑗=0𝑐_𝑗𝑥₀𝑢_𝑗,∀𝑘∈𝐙₀₊. (53) By using (53), one gets a discrete counterpart of (44) as follows: ^𝑘_𝑗=0𝑢_𝑗−𝑘𝑟max_{[]𝜑∈0,2𝜋}𝐺^𝑑𝑒^𝐢𝜑𝑢_𝑗≤1min_{𝜑∈[0,2𝜋]}Re𝐺^𝑑𝑒^𝐢𝜑−𝜀⁻¹max_{𝑘∈𝐙0+}𝑐_𝑘𝑥₀×𝛾_𝑘𝑥₀+𝜀^𝑘_𝑗=0𝑐_𝑘𝑥₀,∀𝑘∈𝐙₀₊ (54) which leads to the subsequent result which is the discrete-time counterpart of Proposition 11, whose proof is close to that of Proposition 11, and it is then omitted.

Proposition 12. Assume that(1)𝑘, 𝑟∈ℓ_∞(𝐙₀₊) so that sup_{𝑗∈𝐙0+}|𝑘_𝑗|≤𝑘<∞ and sup_{𝑘∈𝐙0+}|𝑟_𝑘|≤𝑟<∞,(2)the discrete function 𝐺^𝑑(𝑧) is strongly strictly positive real; that is, Re𝐺(𝑧)>0 for all complex 𝑧 with |𝑧|≥1.(3)+∞>limsup_𝑘→∞(𝛾_𝑘(𝑥₀))≥liminf_𝑡→∞(𝛾_𝑘(𝑥₀))>0; ∀𝑥₀∈𝐑^𝑛.Then, one gets the following properties for any given initial state 𝑥₀∈𝐑^𝑛.(i)𝑢, 𝑦∈ℓ_∞(𝐙₀₊).(ii)liminf_𝑘→∞(∑^𝑘_𝑗=0𝑦_𝑗𝑘_𝑗𝑟_𝑗)>−∞; liminf_𝑗→∞(𝑢²_𝑗−𝑦_𝑗𝑘_𝑗𝑟_𝑗)≤0.(iii)If 𝑢²_𝑗≥𝑦_𝑗𝑘_𝑗𝑟_𝑗; ∀𝑗∈𝐙₀₊, then ∃lim_𝑗→∞(𝑢²_𝑗−𝑦_𝑗𝑘_𝑗𝑟_𝑗)=0. If, in addition, 𝑘_𝑗=𝑘 and 𝑟_𝑗=𝑟 are nonzero constants; 𝑗∈𝐙₀₊, then ∃lim_𝑗→∞𝑦_𝑗=𝑦_∞ and ∃lim_𝑗→∞𝑢_𝑗=𝑢_∞=𝑘𝑟(∑^∞_𝑗=0𝑔_𝑗) and ∃lim_𝑗→∞𝑦_𝑗=𝑦_∞=𝑘𝑟(∑^∞_𝑗=0𝑔_𝑗)².(iv)If 𝑟≡0, then 𝑢_𝑗→0 and 𝑦_𝑗→0 as 𝑗→∞, and they are both square-summable on 𝐙₀₊; ∀𝑥₀∈𝐑^𝑛. Thus, the closed-loop discrete dynamic system (50)-(51) is asymptotically hyperstable for any controller device of output sequence {𝜑_𝑘=𝜑_𝑘(𝑘,𝑦_𝑘)}_{𝑘∈𝐙0+} satisfying the discrete summation inequality ∑^𝑘_𝑗=0𝜑_𝑗(𝑗,𝑦_𝑗)𝑦_𝑗≥−𝛾_𝑘(𝑥₀); ∀𝑘∈𝐙₀₊.

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

Example 3. Assume that, in Example 2, a feedback stabilizing discrete control law 𝑢_𝑡=−𝜑_𝑡(𝑡,𝑦_𝑡−1)𝑦_𝑡−1 satisfying the constraint ∑^𝑡_𝑗=0𝜑_𝑗(𝑗,𝑦_𝑗)𝑦_𝑗≥−𝛾_𝑡(𝑥₀)≥−𝛾>∞; ∀𝑗, 𝑡∈𝐙₀₊ is injected to the system (1), neglecting initial conditions, and equivalently if the initial conditions are zero (this assumption does not affect the stability study,) we get 𝑃_𝑡𝑦=𝑦_𝑡=−𝑃_𝑡𝐺𝑢=−𝑃_𝑡𝐺𝜑_𝑡𝑡,𝑦_𝑡−1𝑃_𝑡−1𝑦=−𝑃_𝑡𝐺𝜑_𝑡𝑡,𝑃_𝑡−1𝑦𝑃_𝑡−1𝑦;∀𝑡∈𝐙₀₊ (55) so that the closed-loop system can be described by the operator 𝑄∶𝐻_𝑒|ℓ²(𝐙₀₊)→𝐻_𝑒|ℓ²(𝐙₀₊) represented as 𝑦_𝑡=𝑃_𝑡𝑄𝑃_𝑡−1𝑦=−𝑃_𝑡𝐺𝜑_𝑡𝑡,𝑦_𝑡−1𝑦_𝑡−1;∀𝑡∈𝐙₀₊ (56) or, equivalently, as 𝑃_𝑡𝐼+𝐺𝜑_𝑡𝑡,𝑃_𝑡−1𝑦𝑃_𝑡−1𝑦=0;∀𝑡∈𝐙₀₊. (57) Assume that 𝑄∶ℓ²[0,𝑧)→ℓ²[0,𝑧) for any 𝑧∈𝐙₊ is stable, positive, one-to-one, and of closed range. Then, 𝑄∶𝐻_𝑒|ℓ²(𝐙₀₊)→𝐻_𝑒|ℓ²(𝐙₀₊), where 𝐻_𝑒=⋃_𝑧∈𝐙+ℓ²[0,𝑧) is positive, bounded and of closed range, invertible and of nonzero minimum modulus; and 0≤𝐸(𝑡)=𝑃_𝑡𝑄𝑃_𝑡−1𝑦,𝑃_𝑡𝑦≤𝛾<∞;∀𝑡∈𝐙₀₊. (58) Since 𝐸(𝑡) is nonnegative, bounded, and nondecreasing, 𝑦_𝑡→0 as 𝑡(∈𝐙₀₊)→∞, and then 𝑢_𝑡=−𝜑_𝑡(𝑡,𝑦_𝑡−1)𝑦_𝑡−1→0 as 𝑡(∈𝐙₀₊)→∞. One gets for any given finite integer 𝑇>0 that lim_{Γ∋𝑡→∞}(𝑃_𝑡+𝑇𝑢−𝑃_𝑟𝑢)=lim_{Γ∋𝑡→∞}(𝑃_𝑡+𝑇𝑦−𝑃_𝑡𝑦)=0. Thus, 𝑦(𝑇)≡0(∈ℓ²[𝑡,𝑡+𝑇]) is the unique fixed point of 𝑄∶𝐻_𝑒|ℓ²[𝑡,𝑡+𝑇]→𝐻_𝑒|ℓ²[𝑡+𝑇,𝑡+2𝑇]; ∀𝑡∈Γ.

Acknowledgments

The author is very grateful to the Spanish Government for its support of this research through Grant DPI2012-30651 and to the Basque Government for its support of this research through Grants IT378-10 and SAIOTEK S-PE12UN015. The author is also grateful to the University of the Basque Country for its financial support through Grant UFI 2011/07.