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

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

Then, the minimum modulus μ(G) of G satisfies and Property (i) has been proven. Now, if G≽0, then there is a self-adjoint operator $$ on X such that $$ so that, since $$ from Property (i), 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

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.

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, $$ is self-adjoint, and since $$, one has from Property (i) of Proposition 1 and Definition 2 that G⁻¹≻0 since

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 → ∞.

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∣dom K, φ_Γ : Γ × H_e → ran  φ_Γ, u : Γ → (H_e∣ ran  K) + ran  φ_Γ; and  y = y_h + y_f, with y_h : Γ ×   C^q → H and y_f : Γ → H_e∣ ran  G. 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

since u = Kr − φ_Γ(t, y) and $$ for any nonzero control from Proposition 1. One gets in the same way that Since 〈P_ty, P_t(φ_Γ(y))〉≥−γ_t > −∞; ∀t ∈ Γ, one gets also that One gets from (7), (9), (8), and (10) that where 0 < γ = sup _t∈Γ γ_t < ∞. Now, since K is a bounded operator, r is a bounded function, P_t  y_h → 0 as t → ∞, and G≻0 is bounded and one-to-one with closed range so that $$ is also bounded and one-to-one with closed range implying from Proposition 1 that $$, and one gets from (12) that Assume that there is some unbounded u : Γ → H. Then, the subsequent contradiction follows from (13) for some λ ∈ R₀₊ since $$. 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) $$, and, since $$, 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.

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:

which makes (13) to remain valid, and Theorem 5 still holds.

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 → ran  φ_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 → ran  φ_Γ that If, in addition, G∣H is a pseudocontraction, then 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

Since G≻0 then G = G^*, and one gets for u = u₁ − u₂ and  $$. Assume that G≻0 is, furthermore, a pseucontraction on H. Then, and, equivalently, implies that and the following cases can occur.

• (a)

  $$ if the controls u₁ and u₂ fulfil $$.

• (b)

  $$if the controls u₁ and u₂ fulfil 〈u₁, u₂〉+〈u₁, u₂〉 ^* < 0.

• (c)

  $$if the controls u₁ and u₂ fulfil $$. This case is only feasible with equality.

Combining the three cases one gets that with the lower-bound equating zero if and only if u₁ = u₂; that is u = u₁ − u₂ = 0.

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,

  $$ ()
   for some real convergent sequence $$ in [ α, ∞) such that α_t → α ∈ (0,  1] as t(∈Γ) → ∞ and zero initial conditions and exogenous reference in (1), where
  $$ ()
  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:

  $$ ()
   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

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 $$ in [ α, ∞) such that α_t → α ∈ (0,  1) as t(∈Γ) → ∞. Since then one has so that, if $$, then $$ 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.

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.

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:

  $$ ()
  $$ ()
  with $$ being a real sequence subject to 0 ≤ χ_t ≤ σ < ∞ and $$ 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

since nβ ∈ Γ; ∀n ∈ Z₊, and 0 ∈ Γ is the first element of Γ. Thus, since 0 ≤ χ < 1, one gets and also 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.

Example 1. Define the truncated function within the time interval [0, t]; ∀t ∈ R₀₊ of f : R → R as follows:

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 ∈ Rⁿ 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 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: 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 $$ being the complex unit. Note that, in the previous expressions, the integral expressions have been also denoted by inner products 〈·, ·〉 _t ≡ 〈·_t, ·〉≡〈·, ·_t〉 on the time interval [0, t] for the given t ∈ R₀₊, all of them being equivalent to inner products of truncated functions for the given t ∈ R₀₊on the Hilbert space L²[0, ∞). Equivalently, integrals of complex Fourier transforms on the whole imaginary axis are got through Parseval’s theorem and denoted by 〈GU_t,  U_t〉 involving the impulse response (i.e., the transfer function evaluated on the imaginary complex axis) of the system and the Fourier transform of the truncated input. Now, assume that the controller is r : R → R is an exogenous reference signal which is piecewise continuous on R₀₊, and φ : [0,  t] × R → R is any piecewise continuous nonlinear time-varying function which satisfies the following integral-type constraint: then Note that any hodograph G(iω) has the symmetry rules ReG(iω) = ReG(−iω) and Im  G(iω) = −Im  G(−iω). Also, $$. Thus, one gets by combining (37) and (40) Decompose [0,  t] = I_1ε(t) ∪ I_2ε(t) for each t ∈ R₀₊, where 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: Assume that $$ and $$. Then, one gets from (41) and (43) that This relation leads to the following result.

Proposition 11. Assume that

• (1)

  k, r ∈ L_∞(R₀₊) so that $$ and $$,

• (2)

  the transfer function G(s) is strongly strictly positive real; that is, ReG(s) > 0 for all complex s with Res ≥ 0,

• (3)

  +∞ > lim  sup _t→∞(γ(t,  x₀)) ≥ liminf _t→∞  (γ(t,  x₀)) > 0; ∀x₀ ∈ Rⁿ.

Then, one gets the following properties for any given initial state x₀ ∈ Rⁿ.

• (i)

  u, y ∈ L_∞(R₀₊).

• (ii)

  $$; $$.

• (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 $$ and $$.

• (iv)

  if  r ≡ 0, then u(t) → 0 and y(t) → 0 as t → ∞ and are both square-integrable on R₀₊; ∀x₀ ∈ Rⁿ. 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

Note that $$ since lim _t→∞|c(t,  x₀)| = 0 at exponential rate since the dynamic system is strictly stable. Since ε ∈ R₊ can be chosen arbitrarily to build the disjoint union I_1ε(t) ∪ I_2ε(t) equalizing [0, t]; ∀t ∈ R₀₊, then choose $$. Now, assume that u : R → R 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, 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 $$, and the above contradiction holds. Then, liminf _t→∞(|u(t)|² − y(t)k(t) r(t)) ≤ 0. Note also that if $$, then the subsequent contradiction follows Then, $$. 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

leads to lim _t→ ∞u(t) = 0 exponentially and the lim _t→ ∞y(t) = 0; ∀x₀ ∈ Rⁿ 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.

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 $$ as follows:

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 ∈ Rⁿ under a piecewise continuous control with eventual isolated bounded discontinuities u : R → R∩H, now the Hilbert space being H = ℓ²(Z₀₊), is where Z₀₊ = {  z ∈ Z : z ≥ 0}, “∘” stands for the discrete convolution operator, c_k = c_k( x₀) ≡ c_k(kT,  x₀) and $$ is the impulse response sequence since the dynamic system is realizable g^d(k,  j) = 0 for j > k. If this dynamic system is the same system as in the previous example subject to a piecewise control sequence $$, 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 r : R → R is an exogenous reference sequence, and φ : [0,  k] × R → R 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 ℓ²(Z₀₊) 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)

  k, r ∈ ℓ_∞(Z₀₊) so that $$ and $$,

• (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)

  +∞ > lim  sup _k→∞(γ_k( x₀)) ≥ liminf _t→∞(γ_k(x₀)) > 0; ∀x₀ ∈ Rⁿ.

Then, one gets the following properties for any given initial state x₀ ∈ Rⁿ.

• (i)

  u, y ∈ ℓ_∞(Z₀₊).

• (ii)

  $$; $$.

• (iii)

  If $$; ∀j ∈ Z₀₊, then $$. If, in addition, k_j = k and r_j = r are nonzero constants; j ∈ Z₀₊, then $$ and $$ and $$.

• (iv)

  If r ≡ 0, then u_j → 0 and y_j → 0 as j → ∞, and they are both square-summable on Z₀₊; ∀x₀ ∈ Rⁿ. Thus, the closed-loop discrete dynamic system (50)-(51) is asymptotically hyperstable for any controller device of output sequence $$ satisfying the discrete summation inequality $$; ∀k ∈ Z₀₊.

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 $$; ∀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

so that the closed-loop system can be described by the operator Q : H_e|ℓ²(Z₀₊) → H_e|ℓ²(Z₀₊) represented as or, equivalently, as 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 $$ is positive, bounded and of closed range, invertible and of nonzero minimum modulus; and Since E(t) is nonnegative, bounded, and nondecreasing, y_t → 0 as t(∈Z₀₊) → ∞, and then u_t = −φ_t(t, y_t−1)y_t−1 → 0 as t(∈Z₀₊) → ∞. One gets for any given finite integer T > 0 that lim _Γ∋t→∞(P_t+Tu − P_ru) = lim _Γ∋t→∞(P_t+Ty − P_ty) = 0. Thus, $$ is the unique fixed point of Q : H_e|ℓ²  [t, t + T] → H_e|ℓ²  [t + T, t + 2T]; ∀t ∈ Γ.