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

and Applied Analysis 3 Thus, ⟨GPtu, Ptu⟩ > 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 : H → H is also strictly positive with closed range and bounded and μ(G) > 0 so that ⟨Gu, u⟩ > 0 for any nonzero u : Γ → H. Proof. Note that 1/‖G‖ = μ(G) > 0 from Proposition 1. Thus, ‖G‖ < ∞ so thatG is bounded. SinceG is bounded, then ‖G‖ < ∞ and 1/‖G‖ = μ(G) > 0. Thus, G : H → H is also one-to-one with closed range from Proposition 3. Then, G = GG = G is self-adjoint, and since μ (G −1 ) > 0, one has from Property (i) of Proposition 1 and Definition 2 that G ≻ 0 since ⟨G −1 u, u⟩ 1/2 = ⟨G −2 u, u⟩ 1/2 = ⟨G −1 u, G −1 u⟩ 1/2 = 󵄩󵄩󵄩󵄩 G −1 u 󵄩󵄩󵄩󵄩 = μ (G −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 He = {ft = Ptf : (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 He is not a Hilbert space (even thoughH is a Hilbert space) since it is not ensured that, for any f : Γ → PC(Γ), ft : Γ → H as (Γ ∋)t → ∞. An important result follows. Theorem 5. Assume that (1) holds for all t ∈ Γ, that is, G : He → He and K : He → He, where He = {ft = Ptf : (f : Γ → PC(Γ)); ∀t ∈ Γ}, yh = yh(p) and p ∈ C is some given complex parameterizing vector, r : Γ → He | domK, φΓ : Γ × He → ran φΓ, u : Γ → (He | ran K) + ran φΓ; and y = yh + yf, with yh : Γ × Cq → H and yf : Γ → He | ran G. Assume also that (1) yh = yh(p) is bounded and Ptyh → 0 as (Γ ∋) t → ∞, (2) G : He → He is stable (or, equivalently,G : He | 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) ⟨Pty, Pt(φΓ(y))⟩ ≥ −γt ≥ −γ > −∞; ∀t ∈ Γ. Then, u, yf, y : Γ → H, and they are bounded. Also, if r ≡ 0, then u(t) → 0, yf(t) → 0, y(t) → 0 as (Γ ∋) t → ∞. Proof. Direct calculations yield ⟨y, u⟩ = ⟨Gu + yh, Kr − φΓ (y)⟩ = ⟨Gu,Kr⟩ + ⟨yh, Kr⟩ − (⟨Gu, φ (y)⟩ + ⟨yh, φΓ (y)⟩) = ⟨Gu,Kr⟩ + ⟨yh, Kr⟩ + ⟨Gu, u⟩ − ⟨yh, φΓ (y)⟩ − ⟨Gu,Kr⟩ = ⟨Gu, u⟩ + ⟨yh, Kr − φΓ (y)⟩ ≥ μ 2 (G) ‖u‖ 2 + ⟨yh, Kr − φΓ (y)⟩ = μ 2 (G) sup t∈Γ 󵄩󵄩󵄩󵄩Ptu 󵄩󵄩󵄩󵄩 2 + ⟨yh, Kr − φΓ (y)⟩ ≥ μ 2 (G) 󵄩󵄩󵄩󵄩Ptu 󵄩󵄩󵄩󵄩 2 + ⟨yh, Kr − φΓ (y)⟩; ∀t ∈ Γ (7) since u = Kr − φΓ(t, y) and ⟨Gu, u⟩ ≥ μ 2 (G)‖u‖ 2 ≥ μ 2 (G)‖Ptu‖ 2 > 0 for any nonzero control from Proposition 1. One gets in the same way that ⟨Pty, Ptu⟩ ≥ μ 2 (G) 󵄩󵄩󵄩󵄩Ptu 󵄩󵄩󵄩󵄩 2 + ⟨Pt (yh) , Pt (Kr − φΓ (y))⟩ , ∀t ∈ Γ. (8) Since ⟨Pty, Pt(φΓ(y))⟩ ≥ −γt > −∞; ∀t ∈ Γ, one gets also that ⟨y, u⟩ = ⟨y,Kr − φΓ (y)⟩ = ⟨y,Kr⟩ − ⟨y, φΓ (y)⟩ ≤ ⟨y,Kr⟩ + sup t∈Γ γt; ∀t ∈ Γ, (9) ⟨Pty, Ptu⟩ ≤ ⟨Pty, Pt (Kr)⟩ + γt; ∀t ∈ Γ. (10) One gets from (7), (9), (8), and (10) that ⟨y,Kr⟩ + γ ≥ μ 2 (G) ‖u‖ 2


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 Popovtype integral inequality, then implying global Lyapunov's stability [1][2][3][4][5][6][7][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 oneto-one operators with closed range on Hilbert spaces linked to contractive, pseudocontractive, asymptotically pseudocontractive, and asymptotically pseudocontractive in the intermediate sense mappings.See [9][10][11][12][13][14][15][16][17][18][19][20][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][23][24][25][26][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][2][3][4][5][6][7][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 squaresummable 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.

Problem Statement and Main Results
Through this paper, one considers the complex Hilbert space  on C and operators  :  →  and  :  →  which define the following associated relations: where  ℎ =  ℎ () and  ∈ C  is some given complex parameterizing vector, and 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   is a projection operator being a truncation operator so that   =    and for each  ∈ Γ, and we define the seminorm on  by ‖‖  = ‖  ‖ = ‖  ‖; ∀ ∈ Γ with   ̸ =  and the family {‖‖  :  ∈ Γ} of seminorms defines the resolution topology on  since {  :  ∈ } is a resolution of the identity [28].Note that 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 ⟨ −1 , ⟩ 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  ⪰ 0), then⟨, ⟩ > 0 for any nonzero  : Γ → , (iii) there is  ∈ Γ such that ⟨  ,   ⟩ > 0 for any nonzero  ∈ dom().
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  ∈ R + such that Then, the minimum modulus () of  satisfies and Property (i) has been proven.Now, if  ⪰ 0, then there is a self-adjoint operator G = G * ⪰ 0 on  such that  = G * G = GG * = G2 so that, since ( G) > 0 from Property (i), ⟨, ⟩ 1/2 = ⟨ G2 , ⟩ Thus, ⟨  ,   ⟩ > 0 for some  ∈ Γ if  ̸ = 0. Hence, Property (iii) follows.
Note from Proposition 1 that if  ⪰ 0 is a one-to-one operator on  with closed range, then it is invertible and  ≻ 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  :  → H is a one-to-one linear bounded strictly positive operator with closed range, then it is invertible and  −1 :  →  is also strictly positive with closed range and bounded and ( −1 ) > 0 so that ⟨ −1 , ⟩ > 0 for any nonzero  : Γ → .
An important result follows.
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 [0, ] ∩ Γ, and ( * )() is the instantaneous input-output power at  ∈ Γ while ⟨ +  ℎ , ⟩ is the energy supplied by the external source.Particular cases of interest in control engineering are (a) if the reference input  ≡ 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 () =  for  ∈ 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 ⟨  ,   ( Γ ())⟩ ≥ − > −∞ and  ≻ 0 implies that 0 ≤ () ≤  < ∞, (0 <  ≤  < ∞ for any nonzero control input with compact support); ∀ ∈ Γ and all  Γ () satisfying the assumption 6 of Theorem 5; that is the input-output energy is nonnegative and bounded; ∀ ∈ 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.
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.
This case is only feasible with equality.
Combining the three cases one gets that with the lower-bound equating zero if and only if 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 () > 4.
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 then one has ≤ 0 and () → 0 as Γ ∈  → ∞.  : Γ →  is bounded since it is piecewise continuous with eventual bounded discontinuities, and  > 0 and finite.Since  on   and  restricted to  are stable,  : Γ →  is also bounded and converges to zero.
A particular case of Theorem 8 of interest is as follows.
If  :  →  is strictly positive and contractive, we obtain the subsequent result.

Application Examples
Example 1. Define the truncated function within the time interval [0, ]; ∀ ∈ R 0+ of  : R → R as follows: Thus, the output of a single-input single-output linear timeinvariant continuous-time dynamic system of th order and initial state (0) =  0 ∈ R  under a piecewise continuous control with eventual isolated bounded discontinuities  : R → R ∩ , where  =  2 (R 0+ ) ≡  2 [0, ∞) the Hilbert space of the square-integrable functions on R 0+ is where Γ = R 0+ = { ∈ R :  ≥ 0},  : R × R → R is the impulse response, (,  0 ) is the zero-input response (i.e., the response contribution due to initial conditions) for initial sate  0 , and " * " stands for the convolution integral operator.Since the dynamic system is realizable, (, ) = 0 for  > .The complex function  : C → C defined as () = L(()) is the transfer function, where L stands for the Laplace transform of the impulse response where it exists.
After defining () = 0 for  < 0, the input-output energy obeys the following relations by using twice Parseval theorem: where (i) is the pointwise value at frequency  of  : C → C, the Fourier transform of  : 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 ⟨⋅, ⋅⟩  ≡ ⟨⋅  , ⋅⟩ ≡ ⟨⋅, ⋅  ⟩ on the time interval [0, ] for the given  ∈ R 0+ , all of them being equivalent to inner products of truncated functions for the given  ∈ R 0+ on the Hilbert space  2 [0, ∞).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 : R → R is an exogenous reference signal which is piecewise continuous on R 0+ , and  : [0, ] × R → R is any piecewise continuous nonlinear time-varying function which satisfies the following integral-type constraint: Note that any hodograph (i) has the symmetry rules Re (i) = Re (−i) and Im (i) = − Im (−i).Also, () ≥ min ∈R 0+ Re (i) > 0. Thus, one gets by combining (37) and ( 40) Decompose [0, ] =  1 () ∪  2 () for each  ∈ R 0+ , where for some given prefixed  ∈ R + .Note that one (but not both) of the disjoint sets   () for  = 1, 2 can be empty.Then, by direct calculations one gets the following: This relation leads to the following result.
Proposition 11.Assume that (2) the transfer function () is strongly strictly positive real; that is, Re () > 0 for all complex  with Re  ≥ 0, Then, one gets the following properties for any given initial state x 0 ∈ R  .
Proof.Since the transfer function () is strictly positive real then it is strictly stable (i.e.all its poles are in Re  ≤ − < 0 for some  ∈ R + ) and Re () > 0 for all complex  with Re  ≥ 0. Since it is, furthermore, strongly positive real (i.e., a strictly positive operator on  2 (R 0+ )), 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  → ±∞ Re (i) = 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 This proves the first part of Property (iii).Also, if () =  and () =  are nonzero constants; leads to lim  → ∞ () = 0 exponentially and the lim  → ∞ () = 0; ∀ 0 ∈ R  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  : [0, ] × 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 .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][5][6][7][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][30][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.
: Γ →  | dom ,  Γ : Γ ×  → ran  Γ (⊂ ),  : Γ → ( | ran )+ ran  Γ (⊂ ); and  =  ℎ +   , with  ℎ : Γ × C  →  and   : Γ →  | ran .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 = R 0+ being the nonnegative real set for picking up values  ∈ Γ of the continuous-time argument) and linear discrete-time dynamic systems (with Γ = Z 0+ being the nonnegative integer set for values  ∈ Γ of the discrete-time argument) where  is an exogenous, or reference, signal,  is a feedback control,  ℎ is the parameterized response to initial conditions, versus  which is the forced response,  is the measurable output to be controlled, and  Γ () is a nonlinear (and, eventually, timevarying) controller device.
5 if  :  →  is strictly positive and asymptotically pseudocontractive in the intermediate sense under a modified Popov-type inequality.