On a Class of Self-Adjoint Compact Operators in Hilbert Spaces and Their Relations with Their Finite-Range Truncations

Lemma 1. Let V be an inner product space of inner product 〈·, ·〉:H × H → C (or R) endowed with a norm ∥·∥ : V → R₀₊ defined by ∥x∥ = 〈x, x〉 ^1/2 for any x ∈ H, where R₀₊ = {z ∈ R : z ≥ 0}, let $$ and $$ be a finite orthonormal system in V and a given finite or numerable sequence of scalars, respectively, and let M and N be given integers fulfilling 1 ≤ M ≤ N ≤ ∞. If N = ∞ then $$ is, in addition, assumed to be numerable. Then, the following properties hold for any x ∈ H.

• (i)

  $$.

• (ii)

  $$.

• (iii)

  $$.

• (iv)

  $$ any integers $$.

• (v)

  If V = H is a finite-dimensional Hilbert space of dimension N and a_n = 〈x, e_n〉, for all $$ and N = M, then

  $$ ()

• (vi)

  If V = H is a finite-dimensional Hilbert space of dimension N and a_n = 〈x, e_n〉, for $$, then

  $$ ()

• (vii)

  If V = H is a separable infinite-dimensional Hilbert space and a_n = 〈x, e_n〉, for n ∈ N, then

  $$ ()

If, in addition, $$, then a_n → 0  as  n(∈N) → ∞. If, furthermore, there is some integer α ≥ M such that the real sequence $$ converges to zero exponentially according to |a_n| ≤ ρⁿ ≤ ρ < 1, for n ∈ N, then $$ for any given x ∈ H with ρ ∈ (0, 1) being some real constant and C(α) being a bounded constant dependent on α satisfying C(M) = 0.

Proof. Properties (i)-(ii) follow from the best approximation lemma since

Property (iii) is a direct consequence of subtracting both sides of the relations in Properties (i)-(ii). Property (iv) is Pythagoras theorem in inner product spaces. Property (v) (Bessel’s inequality) follows directly from Property (i) with the orthonormal system $$ in the Hilbert space H being a basis of H. Property (vi) follows from Properties (ii)–(iii) with a_n = 〈x, e_n〉; $$ and the orthonormal system $$ in H being an orthonormal basis of H since one gets from Property (i) and from (5), Property (ii), and a_n = 〈x, e_n〉, $$ Hence, Property (vi). Property (vii) follows from the assumption that the infinite-dimensional Hilbert space is separable and Property (vi) leads to which holds under, perhaps, eventual reordering of the elements of the orthonormal basis of H which is a complete orthonormal system for the separable Hilbert space H. If there is some integer α ≥ M such that the real sequence $$ converges to zero exponentially, then where |a_n| ≤ ρⁿ ≤ ρ < 1, for all n(∈N) ≥ α with $$ being dependent on α such that C(α) = 0. Hence, Property (vii).

Lemma 2. Let T : H → H be a linear, closed, and compact self-adjoint operator in an infinite-dimensional separable Hilbert space H with a numerable orthonormal basis of generalized eigenvectors $$. Then, the following properties hold:

• (i)

  $$,

for all x ∈ H for any N ∈ N, where λ_n(T) ∈ σ(T); the spectrum of the operator T is defined by λ_n(T) = 〈Te_n, e_n〉, for all n ∈ N and $$ with $$ as n → ∞, for all N ∈ N.

If P_n is the orthogonal projection operator of H on the one-dimensional subspace D_n generated by the eigenvector e_n then

If $$ is the orthogonal projection operator of H on the $$-dimensional eigensubspace Ω_i, then with $$ where $$, for all n, N ∈ N, for all x ∈ H.

• (ii)

  If, in addition, ∥T∥ ≤ α < 1, then

  $$ ()

Proof. Note that there is a numerable orthonormal basis for H since H is separable and infinite dimensional. Such a basis $$ can be chosen as the set of generalized eigenvectors of the linear self-adjoint T : H → H since it is closed and compact and then bounded

Also, since the linear operator T : H → H is closed and compact, the spectrum σ(T) of T : H → H is a proper nonempty (since T : H → H is infinite dimensional and bounded since it is compact) subset of C and numerable and it satisfies σ(T) = σ_p(T) ∪ {0}, with σ_c(T) ∪ σ_r(T) = {0}, where σ_p(T), σ_c(T), and σ_r(T) are the punctual, continuous, and residual spectra of T : H → H, respectively. Note that {0} ∈ σ(T) is also an accumulation point of the spectrum σ(T) since H is infinite dimensional and T : H → H is compact. Also, since H is separable, the spectrum of T : H → H is numerable, and 〈e_j, e_n〉 = δ_jn; for all j, n ∈ N, one gets

where λ_n(T) = 〈Te_n, e_n〉 is an eigenvalue of T : H → H; that is, λ_n(T) ∈ σ(T), associated with the eigenvector e_n since so that so that, except perhaps for reordering, |λ_n(T)| ≥ |λ_n+1(T)|, for all n ∈ N with {λ_n(T)} → 0 since H is separable and σ(T) is numerable. Assume that for any positive integer N the following identity is true: Then, since $$ is an orthonormal basis of generalized eigenvectors, where δ_jn is the Kronecker delta. Then, $$. Furthermore, T^N : H → H is compact as it follows by complete induction as follows. Assume that T^N : H → H is compact, then it is bounded. Note also that T^N : H → H is self-adjoint by construction and then normal. Thus, T^N+1 = T(T^N) : H → H is compact since it is a composite operator of a bounded operator T^N : H → H with a compact operator T : H → H. Then, by complete induction, $$  (∈σ(T^N)) as n → ∞, for any N ∈ N since T^N : H → H is compact and H is infinite dimensional. Also, where P_n is the projection operator of H on the one-dimensional subspace D_n generated by the eigenvector e_n so that P_nx = 〈x, x_n〉x_n → 0 as n → ∞, for all x ∈ H. Thus, Property (i) has been proved. To prove Property (ii), take an orthonormal basis associated with the set of finite-dimensional eigenspaces of the respective eigenvalues. Note from Cauchy-Schwarz inequality that for some real constant α ∈ (0, 1), where $$ is a nondecreasing sequence of finite nonnegative integers defined by $$ being built such that each q_n for n ∈ N accounts for the total of the dimensions p_j of the eigenspaces Ω_j associated with the set of eigenvalues {λ₁(T), λ₂(T), …, λ_n−1(T)}  previous to  λ_n(T) for n ∈ N after eventual reordering by decreasing moduli. Then, $$, for all n ∈ N, and where $$ is now a set of p_i linearly independent elements belonging to the orthonormal basis of H that generate the eigenspace Ω_i associated with λ_i(T) with $$ being an eigenvector and $$ is a set of complex coefficients. Then, α^Ne_i → 0 as →∞, for all i ∈ N from (20), so that lim _N→∞(P_i(T^Nx)) = {0}(∈D_i). If there are some multiple eigenvalues, with all being of finite multiplicity since the operator T : H → H is compact, the above expression may be reformulated with projections on the finite-dimensional eigenspaces associated to each of the eventually repeated eigenvalues leading to $$, for all i ∈ N. Note that $$  ×  $$ is the finite p_i(≥1)-dimension of the eigenspace Ω_i associated with λ_i(T), where p_i is one-dimensional if λ_i ∈ σ(T) is single. Finally, it follows from (19) that and Property (ii) has been proved.

Lemma 3. Let T : H → H be a linear closed and compact self-adjoint operator in a finite-dimensional Hilbert space H of finite dimension p with a finite orthonormal basis of eigenvectors $$ of T : H → H. Then, the following properties hold.

• (i)

  $$

for any N ∈ N, where λ_n(T) ∈ σ(T); the spectrum of the operator T is defined by λ_n(T) = 〈Te_n, e_n〉, for all $$ and $$, for all N ∈ N.

• (ii)

  If, in addition, ∥T∥^N ≤ ηα^N for some real constants α ∈ (0, 1  ) and η ≥ 1, then

  $$ ()

Remark 4. It turns out that Lemma 2 (ii) and Lemma 3 (ii) also hold if T : H → H is not self-adjoint since the corresponding mathematical proofs are obtained by using an orthonormal basis formed by all linearly independent vectors generating each of the subspaces. However, if the operator is not self-adjoint or if it is infinite dimensional while being self-adjoint, the set of (nongeneralized) eigenvectors is not always an orthogonal basis of the Hilbert space.

Theorem 5. Let H be a separable Hilbert space and let T(p) : H → H be a linear degenerated p-finite-dimensional approximating operator of the linear closed and compact self-adjoint operator T : H → H. Then, the following properties hold.

• (i)

  Assume that ∥T∥^N ≤ ηα^N, for all N ∈ N for some real constants α ∈ (0, 1) and η ≥ 1, where $$ is a numerable orthonormal basis of generalized eigenvectors of  T : H → H. Then,

  $$ ()

• (ii)

  Assume that there is a finite n₀ ∈ N such that $$ for some positive real constant M₀ = M₀(n₀). Thus, for any given positive real constant ε ≤ 1, there are nonnegative finite integers p₀ = p₀(ε, n₀) > n₀ and N₀ = N₀(p₀, ε) such that for any finite p(≥p₀)-dimensional degenerated approximating operator T(p) : H → H of T : H → H, the following inequality holds

  $$ ()

Furthermore, for any T(p) : H → H linear degenerated p(≥p₀)-finite-dimensional approximating operator of the linear closed and compact self-adjoint operator T : H → H and some finite p₀ ∈ N.

• (iii)

  If {T^Nx} → z as N → ∞ for some x, z ∈ H such that lim _N→∞(∥T^Nx − T^N(p)x∥) = 0, then {T^N(p)x} → z as N → ∞. Furthermore, such a z is a fixed point of both T : H → H and T(p) : H → H.

Proof. The operator T : H → H is represented as follows:

The associated degenerated p-finite-dimensional operator is so that Thus, assume that $$. Then, so that the assumption $$ is true as it has been proved from (30) by complete induction. The following properties are also direct for any x ∈ H if ∥T∥^N ≤ ηα^N < 1 for some real constants α ∈ (0, 1) and η ≥ 1; for all N ≥ N₀ and some finite N₀ ∈ N, we have Property (i) has been proved. On the other hand, if $$ for some finite n₀ ∈ N and some M₀ ∈ R₊, then for any given real ε(≤1) ∈R₊, there is a positive finite integer p₀ = p₀(ε) > n₀ such that for any ν(∈R₊) ≤ ε/M₀ and any p(≥p₀) ∈N, the following inequalities hold: since $$, for all n ∈ N, $$ as n → ∞, 0 ∈ σ(T), and $$, for all n(≥n₀) ∈N. Note that since M₀ ∈ R₊ exists such that $$ for some finite n₀ ∈ N, then, for any given ε(≤1) ∈R₊, (32) holds for any p(≥p₀)∈N and some p₀ = p₀(ε) > n₀. Then, one gets via complete induction for any N(>N₀) ∈ N and $$ as N → ∞ if ε < 1, for all p(≥p₀) ∈N. Thus, one gets from Lemma 1 (iv) for any p(≥p₀) ∈N and for all N(>N₀) ∈N. Furthermore, note from (32) that ν → 0 and p₀ → ∞ as ε → 0 and the function ν = ν(ε) is nonincreasing. Also, a strictly monotone decreasing positive real sequence ν_n = ν(ε_n) can be built with {ε_n} → 0 since there are infinite many values of the spectrum σ(T) such that the inequality $$ is strict since, otherwise, the convergence of the sequence {|λ_n(T)|} to zero would be impossible. Then, from (34) and $$ as N → ∞ if ε < 1, for all p(≥p₀) ∈N, there are subsequences of positive real and positive integers $$ and $$, respectively, as N → ∞ such that the following subsequent relation holds: for all N(>N₀) ∈ N. Then, and Property (ii) follows directly.

If {T^Nx} → z and lim _N→∞(∥T^Nx − T^N(p)x∥) = 0 as N → ∞ for some x, z ∈ H, then $$ which converges to zero such that

and then ∃  lim _N→∞(∥z − T^N(p)x∥) = 0. Also, T : H → H is bounded, since it is compact, and it is then continuous since it is linear and bounded. Also, T(p) : H → H is of finite-dimensional and closed image, then compact, and then bounded and continuous since it is linear. Thus, ∥z − T^Nx∥ → 0, ∥z − T^N(p)x∥ → 0 as N → ∞  leads to and Property (iii) has been proved.

Lemma 6. Let H be a separable Hilbert space and let T : H → H be a nonnull and nondegenerated (i.e., of infinite-dimensional image) linear closed and compact operator and let T(p) : H → H be the linear degenerated p-finite-dimensional approximating operator of T : H → H. Then, there is an operator $$ such that T(p) can be represented by $$, $$, and $$ with the following properties.

• (i)

  There exists an (in general, nonunique) operator $$, restricted to $$ for each approximating T(p) : H → H of given dimension p.

• (ii)

  The operator $$ is nondegenerated, unique, and compact.

• (iii)

  The minimum modulus of  T : H → H is $$ so that if it is invertible, its inverse is not bounded. If T : H → H is degenerated, that is, finite dimensional of dimension q > p, injective with closed image then its minimum modulus is positive and finite. If, furthermore, T : H → H is invertible then $$ is a compact operator with bounded minimum modulus $$.

Proof. The existence of such an operator $$ is proved by construction. Let $$ be an orthonormal basis of generalized eigenvectors of T : H → H and $$ an orthonormal basis of $$, respectively. Then, one gets for some sequences of complex coefficients $$, for all n ∈ N,

Then, $$ is a unique nondegenerated compact operator from its representation (40). It follows that the operator identity $$ holds on H if and only if $$; for all x ∈ H and, equivalently, since T and $$ are linear,

Since the vectors in $$ form an orthonormal basis, (41), if the following constraints defining the operator $$, restricted as $$, hold for a nonnull operator T : H → H

so that (42) holds if and only if since the elements of {e_n} are linearly independent. Then (43) holds under infinitely many combinations of constraints on the spectrum of $$. In particular, (43) holds if

Equations (43) are also satisfied with γ_kn = 0, for all $$, for all k ∈ N, and $$ for n > p which holds, for instance, if $$ for all n > p. Thus, $$ is then non-unique, in general. Properties (i)-(ii) have been proved.

Now, let μ(Γ) = {inf ∥Γx∥ : x ∈ H, ∥x∥ = 1} be the minimum modulus of the linear operator Γ : H → H. If ∥x∥ = 1, then if T : H → H is injective with closed image (this implies that such an image is finite dimensional), then μ(T) > 0 and since $$ are both bounded since they are compact, one gets

If T : H → H is infinite dimensional, then μ(T) = 0 and it cannot then have bounded inverse. If T : H → H is degenerated of dimension q = p, then $$ is the null operator with $$. If T : H → H is degenerated of dimension q > p and invertible, then μ⁻¹(T) = μ⁻¹(T^*) = ∥T⁻¹∥ < ∞ and $$ so that $$ is bounded and compact since it is a composite operator of a compact operator (T − T(p)) on H and a bounded operator T⁻¹ on H. Property (iii) has been proved.

Example 7. Assume that T, T(p) : H → H are two degenerated finite-dimensional operators on a separable Hilbert space H of, respectively, dimensions two and one defined by Tx = λ₁(T)〈x, e₁〉e₁ + λ₂(T)〈x, e₂〉e₂; for all x ∈ H and T(p)x = λ₁(T)〈x, e₁〉e₁; for all x ∈ H. Thus, the constraints (42) hold for an incremental operator $$ of spectrum defined by $$, $$ with γ₂₂ ≠ 0 and γ₂₁ = 0. Then,

Remark 8. If T : H → H is infinite dimensional and invertible, then $$ is not compact, since T⁻¹ : H → H is unbounded, since μ(T) = 0⇔μ⁻¹(T) = ∥T⁻¹∥ = ∞.

Example 1. Consider the forced linear time-invariant differential system of real coefficients and nth as

under a piecewise continuous square-integrable forcing function u : R₀₊ → R; that is, u ∈ L²(0, ∞), with α_n ≠ 0. The unique solution for any given initial conditions (dyⁱ(0))/dtⁱ for i = 0,1, …, s − 1 is where the superscript Tstands for transposition, c, b ∈ R^s are Euclidean vectors of, respectively, first and last components being unity and the remaining ones being zero $$, and

The matrix function e^At is a C₀-semigroup generated by the infinitesimal generators A, respectively [17, 19]. Using a sampling period of length θ, we can write from (48) for time instants being integer multiples of the sampling period

where x_n = x(nθ) and g = α_n provided that the input is u_n = u_n(θ) = u_n(τ), for all τ ∈ [nθ, (n + 1)θ). The matrix function e^Aθ can be expanded as follows: where σ(A) = {λ_k : k = 0,1, …, ϑ − 1} is the spectrum of A, that is, set of ϑ distinct eigenvalues of A with respective multiplicities ν_k for k = 0,1, …, μ − 1 in the minimal polynomial of A where $$ is the degree of the minimal polynomial of A, and then 1 ≤ μ ≤ s and γ_jk are complex constants. The above α_k(t); k = 0,1, …, μ − 1 are everywhere continuous and linearly independent time-differentiable functions on R. Then, the unique solution (or output) of (47) for zero initial conditions is with y ∈ L²(0, ∞) provided that h ∈ L²(0, ∞), guaranteed from (51) if and only if Re(λ) < 0; for all λ ∈ σ(A) and h(t, τ) = h(t − τ) is a convolution operator and Λ_c is a convolution integral operator since the differential system is time-invariant where h(t, τ) = 0, for all τ(>t) ∈ R₀₊. Thus, such an operator is normal, since it is time invariant [1], and then self-adjoint. Now, define the sequence of samples $$ for a sampling period θ as with the operator Λ being defined from Λ_c on the space of square-summable sequences ℓ²(0, ∞), where $$, for all n ∈ N. Assume that the forcing input u(t) = u_n = u(nθ) is piecewise constant, for all n ∈ N, for all t ∈ [nθ, (n + 1)θ). Note that if h₀ = 0, then h_L(s), the unilateral Laplace transform of h(t), is strictly proper; that is, it has more poles than zeros. In the case that h₀ = h(0) ≠ 0,  h_L(s) is proper by not strictly proper; that is, it has the same number of poles and zeros. It turns out that we can define an operator sequence $$: for all n ∈ N, with the second one being a natural projection P_n+1 on ℓ²[0, n + 1] of an operator $$ on ℓ²[0, ∞] so that, by using $$; for all n ∈ N, one gets: with $$; for all n ∈ N, $$, $$, with T₀ being the identity operator. One has from (51) that and h_n → 0 as n → ∞ if Re(λ_i) < 0; ℓ = 0,1, …, ϑ − 1. Some particular cases are discussed below under the assumption {u_n} ⊂ ℓ²[0, ∞) and Re(λ_i) < 0; ℓ = 0,1, …, ϑ − 1 implying {h_n} ⊂ ℓ²[0, ∞), {|h_n|} ⊂ ℓ[0, ∞), so that $$ and $$, since {h_n} is bounded.

Proposition 2 (constant piecewise constant open-loop control). Assume that Re(λ) < 0, for all λ ∈ σ(A), and consider a constant open-loop control u_n = u₀, for all n ∈ N. The following properties hold.

• (i)

  The sequence $$ satisfies y_n+1 = Ty_n = T_n+1y₀, subject to y₀ = u₀h₀, for all n ∈ N₀, where the operator T : N₀ × R → R is defined as the sequence of scalar gains $$, for all n ∈ N in the Banach space (R, |·|) which is the Euclidean Hilbert space for the product of real numbers being an inner product. Furthermore, $$.

• (ii)

  Assume that p(∈N) ≥ p₀ for some given p₀ ∈ N, and $$. Then, $$ and $$ as N → ∞, for all n ∈ N, for all p(≥p₀) ∈N for some finite p₀ ∈ N.

• (iii)

  There is p₀ = p₀(ε, u₀) ∈ N for each given ε ∈ R₊ and u₀ ∈ R such that $$, for all n(∈N₀) ≥ p ≥ p₀. Also, for each given u₀ ∈ R satisfying $$, for all n(∈N) ≥ p − 1, it follows that

  $$ ()

Proof. Property (i) follows from $$, or equivalently, $$, for all n ∈ N₀ subject to an initial condition y₀ = u₀h₀. Since {h_n} is bounded, {h_n} → 0 as n → ∞, and $$, then $$. Thus, the sequence $$ is generated as y_n+1 = Ty_n = Tⁿ⁺¹y₀, subject to y₀ = u₀h₀, for all n ∈ N₀, where the operator T : N₀ × R → R is defined in the Banach space (R, |·|) as the sequence of scalar gains $$, for all n ∈ N which is the Euclidean Hilbert space for the product of real numbers being an inner product. Furthermore, $$. Property (i) has been proved. On the other hand, since $$, it follows that $$ and then

Property (ii) has been proved. Now, note that for any given ε ∈ R₊ and u₀ ∈ R, there is p₀ = p₀(ε, u₀) ∈ N such that for any p(≥p₀) ∈N

for p₀ = p₀(ε, u₀) ∈ N satisfying $$ if u₀ ≠ 0 and such a p₀ exists since {|h_n|} ⊂ ℓ[0, ∞). Note that if u₀ = 0, then $$ so that $$ for any p₀ = p₀(ε) ∈ N. The first part of Property (iii) has been proved. Note that $$. Then, the second part of Property (iii) follows since Property (iii) follows from Theorem 5 with the operator $$, for all n ∈ N of (58) and its degenerated finite truncation $$, for all n ∈ N in the subsequent way where T : R → R maps each element of the sequence $$, which is strictly ordered according to the time occurrence, to its next consecutive one, (then $$in the second part of (62)), for all i ∈ N₀, and $$ is a basis of orthogonal vectors $$ if u₀h_i ≠ 0 and v_i = 0, where e_i is the ith unit vector in Rⁿ with its ith component being one, such that the set $$ is an orthonormal basis so that $$ as

Example 2. Consider again (47) with Re(λ) < 0, for all λ ∈ σ(A). If one measures some more state variables than just the solution, then an extended solution (48) of the form

is built with $$which is the output; 1 ≤ s₀ ≤ s, z(t) = Cx(t) and x⁰(t) is formed by all or some of the components of x(t) except y(t), $$, and $$ where s_m ≥ 1 is the dimension of the piecewise-continuous input $$ which is in $$. If x⁰(t) is not used to (64), then z(t) = y(t) and s₀ = 1. If z(t) = x(t), then s₀ = s. Equation (64) can be expressed as with x₀ = x(0), T_h(t) = Ce^At and the operators $$ and $$ are defined as so that T_f(t, τ) = 0 for τ > t is a convolution operator, where $$ is piecewise continuous on R and square integrable defined as u_e(t) = u(t) for t ∈ R₀₊ and $$; otherwise, w_t is a truncated multiplicative (or truncated gate) from (−∞, t]  ∩  R to (0,1) for each defined as w_t(τ) = 1 for 0 ≤ τ ≤ t and w_t(τ) = 0, otherwise, for all t ∈ R. Note that w_t(τ) = 1(t − τ)1(τ), for all (t, τ) ∈ R². The multiplicative (or gate) operator w from R to (0,1) is defined as w(t) = 1  for t ≥ 0 and w(t) = 1, otherwise; for all t ∈ R. Now, where 〈·, ·〉 is the inner product on $$, T_f(p, t − τ) is the kernel of $$; {θ_i} and {φ_i}; $$ are two reciprocal orthogonal bases of the pth dimensional subspace M_p of $$ and $$ maps $$ in the orthogonal projection of $$ on M_p, for all t ∈ R₀₊. Note that $$ is a self-adjoint, since it is time invariant (and convolution), compact operator since its image is finite dimensional. On the other hand, note that so that $$ has a square-integrable kernel so that it is a Hilbert-Schmidt operator, then compact, and also self-adjoint since it is time invariant. Note that ∥u_e∥ = ∥u∥. Thus, where {θ_i} and {φ_i}; i ∈ N are two orthogonal complete systems of the infinite-dimensional separable Hilbert space $$ and one has from (67) to (69) that with $$, $$, being nonzero for any finite p, $$; for  all  i ∈ N is a linearly independent set, since the kernel T_f(t − τ) of $$ is bounded and $$, for all i ∈ N, and where the norm is associated with the inner product on $$. Equation (70) describes the truncated error norm on (−∞, t] of $$ in (71), for all t ∈ R₀₊ via the formula (69) while (71) refers to the whole real interval (−∞, ∞). From (73), there is p₀ = p₀(ε) such that $$ for any p ≥ p₀ and any prefixed ε ∈ R₊. Since $$, $$, for all t ∈ R₀₊, and $$ as n → ∞ since $$ is compact and then concluding the following: (a) the true and approximate forced and complete solutions might be made as close as suited, in terms of difference of norms, by using a finite-range operator approximant of sufficiently large range dimension; (b) if the true asymptotic solution is a fixed point $$, then so that z_p(t) → z^* as p(∈N), t(∈R) → ∞, where ∥·∥₂ denotes the spectral norm for vector and matrices. Now, assume that the dynamics is perturbed with a parametrical disturbance $$ in the matrix A, which is nonsingular since Reλ < 0, for all λ ∈ σ(A) to give $$, with I being the nth identity matrix. Thus, A^′ is also a stability matrix if $$ for any matrix norm since from Banach perturbation lemma $$ [7, 19, 21, 22], since $$, exists and its maximum modulus eigenvalues do not cross the imaginary complex axis from the continuity of the eigenvalues with respect to the matrix entries. Thus, the perturbed dynamic system has the following solution:

If the nominal (i.e., unperturbed) solution is a fixed point z^* and, since $$ as t → ∞ since A^′ is a stability matrix, then applying Holder’s inequality to (76a) and (76b), it follows that x ∈ L_∞ with $$, and then

for any δ ∈ R₊ satisfying $$, and K_A ≥ 1 and ρ_A > 0 are real constants such that $$, for all t ∈ R₀₊. In particular, (−ρ_A) is the stability abscissa of the dominant eigenvalue of A if it is either simple or it has an associate diagonal Jordan block, or a number arbitrarily close to it but larger.