Controllability and Observability of Nonautonomous Riesz-Spectral Systems

Definition 1. Let $$ be the set of all bounded linear operators on Hilbert space X. A two-parameter commutative family {R(t, s)} _s,t≥0 in $$ is called a strongly continuous quasi-semigroup, in short C₀-quasi-semigroup, on X if, for each r, s, t ≥ 0 and x ∈ X,

• (a)

  R(t, 0) = I, identity operator on X,

• (b)

  R(t, s + r) = R(t + r, s)R(t, r),

• (c)

  $$,

• (d)

  there exists a continuous increasing function M : [0, ∞)→[0, ∞) such that

  $$ (3)

Definition 2. For every t ≥ 0, let A(t) be an operator of form (4) on a Hilbert space X. A(t) is called a generalized Riesz-spectral operator if A is a Riesz-spectral operator.

Theorem 3. For every t ≥ 0, let A(t) be an operator of (4) where A is a Riesz-spectral operator with simple eigenvalues $$ and corresponding eigenvectors $$. If $$ are the eigenvectors of A^∗, the adjoint of A, such that 〈ϕ_n, φ_m〉 = δ_mn, then

• (a)

  ρ(A(t)) = {λa(t) : λ ∈ ρ(A)}, σ(A(t)) = {λa(t) : λ ∈ σ(A)}, and for λ ∈ ρ(A(t)), the resolvent operator $$ is given by

  $$ (5)

• (b)

  A(t) has representation

  $$ (6)

•  

  for $$, where

  $$ (7)

• (c)

  if $$, then, for every t ≥ 0, A(t) is the infinitesimal generator of a C₀-quasi-semigroup R(t, s) given by

  $$ (8)

•  

  where $$;

• (d)

  the growth bound of the quasi-semigroup at t is given by

  $$ (9)

Proof. Proofs of (a) and (b) follow the proofs of Theorem 2.3.5 of [21] replacing (λI − A) ⁻¹ and y_N with

respectively, for every t ≥ 0. In this context $$.

(c) Let ω = sup_n≥1Re(λ_n). Given t ≥ 0 fixed, for λ such that Re(λ) > a(t)ω, from (a)

and by iteration we have So by the condition b of Lemma 2.3.2 of [21] for m = h and M = H, we have Theorem 3.7 of [15] implies that A(t) is an infinitesimal generator of a C₀-quasi-semigroup R(t, s) with We verify that the operators R(t, s), t, s ≥ 0, given by where $$ and $$, are a C₀-quasi-semigroup on X satisfying (15) with the infinitesimal generator on domain

(d) By (10) we have

On the other hand, taking x = ϕ_n in (16) we get It implies Therefore

Corollary 4. If, for every t ≥ 0, A(t) is the generalized Riesz-spectral generator of a C₀-quasi-semigroup R(t, s) on a Hilbert space X, then for any $$ and r ≥ 0 the initial value problem

admits a unique solution.

Proof. It follows from Theorem 2.2 of [13] that (23) admits a unique solution.

Definition 5. Assume that the state linear system (A(t), B(t), −) holds for all initial state x₀ ∈ X and for all input $$. The state

is defined to be a mild solution of (1).

Definition 6. The linear system (A(t), B(t), −) is said to be

• (a)

  exactly controllable on [0, τ] if for each x₀, x₁ ∈ X there exists a control u ∈ L_p([0, τ], U) such that the mild solution x(·) of (1) corresponding to u(·) satisfies x(τ) = x₁;

• (b)

  approximately controllable on [0, τ] if for each x₀, x₁ ∈ X and any ϵ > 0 there exists a control u ∈ L_p([0, τ], U) such that the mild solution x(·) of (1) corresponding to u(·) satisfies ‖x(τ) − x₁‖ < ϵ.

Lemma 7. The controllability map in (26) satisfies the following conditions.

• (a)

  The operator $$ and $$ for 0 ≤ t ≤ τ.

• (b)

  ($$ on [0, τ].

Proof. (a) Since R(t, s) is strongly continuous and u ∈ L_p([0, τ], U), then the map s ↦ 〈x, R(s, τ − s)B(s)u(s)〉 is measurable on [0, τ] for every x ∈ X. Moreover,

Lemma A.5.5 of [21] states that the integral in (26) is well-defined. We verify easily that $$ is linear. Now, for 0 ≤ t ≤ τ we have This shows that $$ is a bounded mapping from L_p([0, τ], U) to X and $$, 0 ≤ t ≤ τ, is a bounded mapping from L_p([0, τ], U) to L_p([0, τ], X).

(b) The definition of adjoint operator shows that $$ is bounded. Moreover,

This proves that $$ on [0, τ].

Theorem 8. For u ∈ L_p([0, τ], U), the system (A(t), B(t), −) is exactly controllable on [0, τ] if and only if any one of the following conditions holds for some γ > 0 and all x ∈ X:

• (a)

  $$.

• (b)

  $$ and $$ is closed.

Proof. (a) We set V = L_p([0, τ], U), so $$. It is enough to prove that $$. By similarity of adjoint and dual operator in Hilbert space, Corollary 3.5 of [20] states

if and only if there exists γ > 0 such that for all x ∈ X. So, by condition (b) of Lemma 7 the assertion is confirmed.

(b) The condition $$ shows that $$ is injective, and so $$. Next, let $$ be a Cauchy sequence in L_p([0, τ], U). Condition (a) shows that (x_n) is a Cauchy sequence in X. However, Lemma 7 (a) forces $$ for some $$. Thus, $$ has a closed range.

Theorem 9. The linear system (A(t), B(t), −) is approximately controllable on [0, τ] if and only if any one of the following conditions holds:

• (a)

  B^∗(s)R^∗(s, τ − s)x = 0, 0 ≤ s ≤ τ, implies x = 0.

• (b)

  $$.

Proof. (a) We see that the system (A(t), B(t), −) is approximately controllable on [0, τ] if and only if $$. According to Lemma VI 2.8 of [24], this is equivalent to the fact that the mapping $$ is injective. The similarity between adjoint and dual operator gives

which implies x = 0 almost everywhere. Therefore, if this verifies that x = 0.

(b) Condition (a) and condition (b) of Lemma 7 give the desired result.

Definition 10. The linear system (A(t), B(t), C(t)) is said to be

• (a)

  exactly observable on [0, τ] if the initial state can be uniquely constructed from the knowledge of the output in L_q([0, τ], Y);

• (b)

  approximately observable on [0, τ] if the knowledge of the output in L_q([0, τ], Y) determines the initial state uniquely.

Lemma 11. For the linear system (A(t), B(t), C(t)) one has the following duality:

• (a)

  The linear system (A(t), −, C(t)) is approximately observable on [0, τ] if and only if the dual (A^∗(t), C^∗(t), −) is approximately controllable on [0, τ].

• (b)

  The linear system (A(t), −, C(t)) is exactly observable on [0, τ] if and only if the dual (A^∗(t), C^∗(t), −) is exactly controllable on [0, τ].

Proof. As a consequence of Proposition 1.2 and Theorem 1.6 of [14], if A(t) generates a C₀-quasi-semigroup R(t, s) on a Hilbert space, then A^∗(t) generates the C₀-quasi-semigroup R^∗(t, s). Furthermore, we verify that

This implies that the range of $$ equals that of the controllability map for the dual system (A^∗(t), C^∗(t), −). If $$ denotes the controllability map of the dual system, then $$ or $$.

(a) We see that (A(t), −, C(t)) is approximately observable on [0, τ] if and only if $$. Condition (b) of Theorem 9 implies that $$ if and only if (A^∗(t), C^∗(t), −) is approximately controllable on [0, τ]. This proves the equivalence.

(b) Suppose that (A(t), −, C(t)) is exactly observable on [0, τ]. There exists an inverse $$ on $$ and a constant κ > 0 such that

The exact controllability of (A^∗(t), C^∗(t), −) follows from Theorem 8.

Conversely, assume that (A^∗(t), C^∗(t), −) is exactly controllable on [0, τ]. Theorem 8 (a) gives that $$ is injective and $$ is closed. Since $$, then $$ is injective and $$ is closed. This states that (A(t), −, C(t)) is exactly observable on [0, τ].

Corollary 12. For the linear system (A(t), −, C(t)), one has the following necessary and sufficient conditions for exact and approximate observability:

• (a)

  (A(t), −, C(t)) is exactly observable on [0, τ] if and only if any one of the following conditions holds for some γ > 0 and all x ∈ X:

  • (i)

    $$.

  • (ii)

    $$ and $$ is closed.

• (b)

  (A(t), −, C(t)) is approximately observable on [0, τ] if and only if any one of the following conditions holds:

  • (i)

    C(s)R(0, s)x = 0, 0 ≤ s ≤ τ, implies x = 0.

  • (ii)

    $$.

Theorem 13. Consider the linear system (A(t), B(t), C(t)) of (37)-(38), where A is a Riesz-spectral operator with simple eigenvalues $$ such that $$ and corresponding eigenvectors $$. Let $$ be the eigenvectors of A^∗ such that 〈ϕ_n, φ_m〉 = δ_mn. Then

• (a)

  (A(t), B(t), −) is approximately controllable on [0, τ] if and only if for all n

  $$ (45)

•  

  for all t ∈ [0, τ];

• (b)

  (A(t), −, C(t)) is approximately observable on [0, τ] if and only if for all n

  $$ (46)

•  

  for all t ∈ [0, τ].

Proof. (a) We consider the matrix B_n:

on [0, τ]. By Lemma 3.14 of [20] and (42), (A(t), B(t), −) is approximately controllable on [0, τ] if and only if for all n implies x = 0. Suppose that (A(t), B(t), −) is not approximately controllable on [0, τ], there exists an $$ such that 〈x, ϕ_n〉≠0 and This gives 〈b_i(t), φ_n〉 = 0 for all i = 1, …, m and t ∈ [0, τ], and so rank B_n ≠ 1.

Conversely, suppose that rank $$ for some n₀, then $$, for all i = 1, …, m and t ∈ [0, τ]. So we can find a nonzero x ∈ X such that

Thus, (42) is satisfied for x ≠ 0. This is equivalent to the fact that (A(t), B(t), −) is not approximately controllable on [0, τ].

(b) We can have similar proof to (a) for the matrix

on [0, τ].

Definition 14. The state linear system (A(t), B(t), C(t)) of (1)-(2) is called a nonautonomous Sturm-Liouville system if A(t) is the negative of a nonautonomous Sturm-Liouville operator of form (54).

Corollary 15. For every t ≥ 0, let A(t) be the negative of a nonautonomous Sturm-Liouville operator of the form (54) on its domain $$ given by (52). Then

• (a)

  A(t) is generalized Riesz-spectral operator;

• (b)

  A(t) is the infinitesimal generator of a C₀-quasi-semigroup on X.

Proof. (a) Lemma 1 of [27] gives the fact that A(t) is generalized Riesz-spectral operator.

(b) If $$ is the set of eigenvalues of $$, then $$. Hence, Theorem 3 concludes that, for every t ≥ 0, A(t) is the infinitesimal generator of a C₀-quasi-semigroup on X.

Example 1. Consider the boundary condition problem of the PDE:

We are ready to show that the problem has a unique solution. Let X be a Hilbert space of L₂[1, b]. Problem (55) can be written as

on X, where A(t)≔a(t)A, a(t) = 1/(t + 1), and on $$ with

We verify that operator A is not self-adjoint on $$. Furthermore, we obtain the eigenvalues and corresponding eigenvectors of A as

respectively. It is obvious that every eigenvalue λ_n is simple and the set $$ forms Riesz basis of X. Moreover, $$ is totally disconnected, that is, for $$, $$.

The adjoint of A is

on $$. The eigenvalues and corresponding eigenvectors of A^∗ are for all 1 ≤ ξ ≤ b, $$, and satisfy

Next, since the adjoint of any operator is always closed, then A^∗ is closed. But in this case we have A = (A^∗) ^∗, so A is closed. Thus, A is a Riesz-spectral operator. In other words, A(t), t ≥ 0, is a generalized Riesz-spectral operator.

Since $$, condition (c) of Theorem 3 forces that A(t) is the infinitesimal generator of a C₀-quasi-semigroup R(t, s) given by

Corollary 4 guarantees that for each $$ problem (56) admits a unique solution Thus, boundary condition problem (55) has a solution

Example 2. Consider the controlled wave equation

where $$ is bounded uniformly continuous and u(t, ·) ∈ L₂[0,1] is a distributed control.

We shall analyze the approximate controllability and approximate observability of the system. Problem (66) can be formulated as a linear system:

on the Hilbert space $$ with the inner product where $$ and $$ In this context $$, $$, $$, and A₀h = −d²h/dξ² for $$ with We verify that A(t) has the eigenvalues λ_n = inπ, n = ±1, ±2, … and the corresponding Riesz basis of eigenvectors $$, where $$. We see that ϕ_n(ξ) = ψ_n(ξ) for every n. Moreover, Example 2.3.8 of [21] shows that A(t) is a Riesz-spectral operator on X. Hence, A(t) is the generalized Riesz-spectral operator generating a C₀-quasi-semigroup R(t, s).

We assume that the system is controlled around the point ξ_c. So, we may set

where χ is an indicator function. Theorem 13 shows that system (67) is approximately controllable on [0, τ] if and only if Equation (71) demonstrates that the control points ξ_c for which sin⁡(nπξ_c) = 0 affect the loss of approximate controllability. This is also the case when b(t) = 0, that is, at the zeros of b on interval [0, τ].

Next, consider the observation

where $$ is an output function around the sensing point ξ_s. Following Example 4.2.5 of [21] we can reformulate the observation map as an inner product on X: Condition (b) of Theorem 13 gives the fact that (A(t), −, C(t)) is approximately observable on [0, τ] if and only if for all n We verify that Therefore, the system is approximately observable on [0, τ] if and only if for all n This shows that the system loses the approximate observability at points for which sin⁡(nπξ_s) = 0 for some n or c(t) = 0. Specially, for c(t) = sin⁡(f_s(t)π), where f_s(t) is the sampling frequency, then the system loses the approximate observability at sampling frequencies of f_s = k, for k = 0,1, 2, … discrete measurement.

Example 3. Consider the controlled nonautonomous heat equation on interval [1, b]:

where $$ are bounded uniformly continuous and u(t, ·) ∈ L₂[1, b] is a distributed control.

We shall analyze the approximate controllability and approximate observability of the system. Let X be a Hilbert space of L₂[1, b]. System (77) can be formulated as a nonautonomous Sturm-Liouville system

on X, where $$, B(t)≔β(t)I, and $$ is a Sturm-Liouville operator (53) with w(ξ) = 1, p(ξ) = ξ², and q(ξ) = 0 on its domain $$: The eigenvalues and corresponding eigenvectors of $$ are for 1 ≤ ξ ≤ b, respectively. Moreover, A(t) is the infinitesimal generator of a C₀-quasi-semigroup R(t, s) given by

As the previous example, we assume that the system is controlled around the point ξ_c and

Condition (a) of Theorem 13 shows that system (78) is approximately controllable on [0, τ] if and only if, for all n ≥ 1, Equation (83) confirms that the zeros of b on interval [0, τ] affect the loss of approximate controllability.

Next, we locate the measurement y(t) at the system output with the point measurement:

where c(t) is a continuous function. Condition (b) of Theorem 13 implies that the system is approximately observable on [0, τ] if and only if, for all n ≥ 1, for some ϵ > 0. The Mean Value Theorem for integral implies that (85) is equivalent to for some ξ_n ∈ (b − ϵ, b).

Remark 4. In this example we consider the nonautonomous regular Sturm-Liouville problem with the Dirichlet boundary condition. Actually, we can verify that all the results remain valid for which the problem has the Neumann boundary condition. Even, the nonautonomous singular Sturm-Liouville problems can be applied for the results. However, the periodic cases do not hold for the theory due to not simpleness of the related eigenvalues.