On the stability of a strongly stabilizing control for degenerate systems in Hilbert spaces

Definition 2.1

[11] A pencil of matrices λ L − M is polynomial matrix whose coefficients are polynomials of degree less than or equal to 1, λ is a variable indeterminate, where L and M are any two matrices of the same order m × n .

Definition 2.2

[11] A pencil of matrices λ L − M is called regular if

1. L and M are square matrices of the same order n , and

2. The determinant ∣ λ L − M ∣ does not vanish identically.

Definition 2.3

[12,13] The complex number λ ∈ C is called a regular value of the pencil λ L − M , if the resolvent ( λ L − M ) − 1 exists and is bounded. The set of all regular values is denoted by ρ ( L , M ) , and its complement σ ( L , M ) = C \ ρ ( L , M ) is called the spectrum of the pencil λ L − M . The set of all eigenvalues of the pencil λ L − M is denoted by

where ρ ( L , M ) indicates the pencil resolvent set

Definition 3.1

[12]. If there are two constants K > 0 and α > 0 such that for each solution ξ ( t ) , ( t ≥ 0 ) , the system (2) is called exponentially stable

Definition 3.2

[12,14]. If the system (2) satisfies the following qualities:

1. for any solution ξ ( ⋅ ) such that ξ ( 0 ) = ξ 0 = 0 then ξ ( t ) = 0 , for all t ≥ 0 ,

2. S M ( t ) is strongly stable semigroup if for all

   ξ ∈ ℋ , S M ( t ) ξ ⟶ 0 as t ⟶ ∞ ,

   then it is considered to be well posed.

Theorem 3.3

[1]. The strongly exponentially stable of { e M t } t ≥ 0 occurs if and only if there is a norm ‖ ⋅ ‖ 1 equivalent to the initial norm ‖ ⋅ ‖ of ℋ such that M is dissipative in ‖ ⋅ ‖ 1 , i.e.,

Remark 3.4

[1]. An operator M is said to be dissipative if

We call an operator M uniformly dissipative if Re ( M ) ≪ 0 .

Theorem 3.5

[12]. If the system (2) is exponentially stable, then all eigenvalues of the pencil λ A − B are in the half-plane Re λ ≤ α , where α is the constant defined in (4).

Theorem 3.6

Assume that the pencil operator of bounded operators L and M have a spectrum σ ( L , M ) in the left half-plane. Then G ≫ 0 is used for any nonnegative uniform operator ^[1], and there exists an operator W ≫ 0 such that.

Proof

We suppose that

Then i is a regular point, and T = i ( i L + M ) ( i L − M ) − 1 is a bounded operator. By using the conformal mapping method, we obtain

then

The operator μ I − T is invertible. As a result, σ ( T ) = σ ( I , T ) = φ σ ( L , M ) is the spectrum of T . Hence, σ ( T ) is contained in the unit disk. We have the following as a consequence of the general Lyapunov theorem [1]. For each operator H ≫ 0 , there is an operator W ≫ 0 such that:

It is the same as

where − G = ( i L ∗ + M ∗ ) H ( i L − M ) ≪ 0 . In fact,

Therefore, the equality is proved.□

Theorem 3.7

Assume that i is a regular point for the bounded pencil operator λ L − M , there is an operator W ≫ 0 such that

Then the pencil operator λ L − M spectrum σ ( L , M ) is in the left half-plane.

Proof

If i ∈ ρ ( L , M ) the operator

is bounded, and the equality becomes

Therefore,

By using the general Lyapunov theorem [1], the spectrum σ ( T ) will be in the unit disk, then we conclude that

where λ = φ − 1 ( μ ) = i μ + 1 μ + i .

Consequently, Theorem 3.7 is proved.□

Theorem 3.8

[12] If (5) is satisfied for the couple of positive uniform operators ( W , G ) , then λ = i is not an eigenvalue for the pencil λ L − M .

Remark 3.9

The necessary and sufficient conditions of stability for stationary degenerate systems can be obtained by using Theorems 3.6–3.8. We obtain the following crucial result in finite dimensional spaces using the elementary divisors of the pencil of matrices [11].

Corollary 3.10

[12] The following assertions are equivalents in finite dimensional spaces:

1. the system (2) is exponentially stable;

2. σ ( L , M ) = σ p ( L , M ) ⊂ { λ ∈ C : Re λ < 0 } ;

3. there is nonnegative definite matrix W ≫ 0 that has the property

Example 3.11

Considering the system (2) in the finite-dimensional spaces.

So,

it is a regular pencil.

• Finite elementary dividers (FED)

  D 2 ( λ ) = λ + 1 , D 1 ( λ ) = 1 , D 0 ( λ ) = 1 .

• Invariant polynomials are given by

  i 1 ( λ ) = D 2 ( λ ) D 1 ( λ ) = λ + 1 , i 2 ( λ ) = D 1 ( λ ) D 0 ( λ ) = 1 .

Since

then the system (2) is exponentially stable.

Remark 4.1

If a vector c ∈ ℋ such that ⟨ c , Φ k ⟩ ≤ 1 k 2 , then c ∈ H .

Theorem 4.2

Let ‖ c ‖ ⋅ ‖ q ‖ < Δ σ 2 , where Δ σ = 1 2 min i ≠ j ∣ λ i − λ j ∣ > 0 . Then the eigenvectors Φ n of the pencil operator λ ˜ n L − M ˜ , where M ˜ = M + c q ⋆ construct a Riesz basis ℋ .

Proof

Let us consider the spectral equation for the eigenvectors Φ n :

or

and apply the pencil resolvent ( λ n L − M ) − 1 of the operator λ L − M . We obtain

where θ n = − ⟨ Φ n , q ⟩ and λ n is eigenvalue of pencil operator λ L − M .

We obtain

with the property ∣ λ n − λ ˜ j ∣ > Δ σ , for all n ≠ j , the resolvent

we have

Theorem 4.3

Let { λ ˜ n } n = 1 ∞ any set of complex numbers such that:

1. ∣ λ n − λ ˜ n ∣ < Δ σ , n ∈ N ∗ ,

2. ∑ n = 1 ∞ ∣ λ n − λ ˜ n ∣ 2 ∣ c n ∣ 2 < Δ σ ‖ c ‖ 2 ,

Definition 5.1

The controlled degenerate system (7) is said to be exponentially stabilizable by means of a direct feedback u ( t ) = q ∗ ξ ( t ) , q ∗ ∈ ℋ if the corresponding system

is strongly exponentially stable.

Theorem 5.2

Let system (8) is strongly exponentially stable and ‖ c ‖ ⋅ ‖ q ‖ < Δ σ 2 . Then there exists a Hilbert norm ‖ ⋅ ‖ W = ⟨ W ⋅ , ⋅ ⟩ 1 ∕ 2 with positive definite W such that the operator M ˜ is dissipative then:

1. the system (8) is well posed,^[2]

2. for any solution ξ ( t ) of the system (8), one has

   d d t ‖ ξ ( t ) ‖ W 2 = d d t ⟨ W ξ ( t ) , ξ ( t ) ⟩ = − ⟨ G 0 ξ ( t ) , ξ ( t ) ⟩ ,

   where G 0 is self-adjoint nonnegative compact operator G 0 = ∑ i = 1 N ζ i ω i ω i ∗ , { ω i } i = 1 ∞ is orthonormal basis of eigenvectors corresponding to eigenvalues ζ i ≥ 0 ; i > 1 , such that:

   1. ∑ i = 1 N ζ i < ∞ ,

   2. there exist α 1 , α 2 > 0 such that for any normed eigenelement Φ n of pencil operator λ ˜ n L − M ˜ and ( λ ˜ n L − M ˜ ) Φ n = 0 ; n ≥ 1 , the following estimate holds

      α 1 ≤ ∑ i = 1 N ζ i ∣ ⟨ ω i , Φ i ⟩ ∣ 2 ∣ Re λ ˜ i ∣ ≤ α 2 , i = 1 , … , N ,

3. the operator W is given by

   W = ∫ 0 ∞ ∑ i = 1 N ζ i e L 1 − 1 M ˜ ∗ t ω i ω i ∗ e L 1 − 1 M ˜ t d t , t ≥ 0 .

Corollary 5.3

If ‖ c ‖ ⋅ ‖ q ‖ < Δ σ 2 , then the following assertions are equivalents

1. the system (8) is strongly exponentially stable;

2. all the eigenvalues λ ˜ n of pencil operator λ ˜ n L − M ˜ have a negative real part Re λ ˜ n < 0 , i.e.,

   σ ( L , M ˜ ) = σ p ( L , M ˜ ) = { λ ˜ n ∈ C , n ≥ 1 : Re λ ˜ n < 0 } ;

3. operator L 1 − 1 M ˜ is uniformly W-dissipative in some Hilbert norm and has not pure imaginary eigenvalues;

4. there exists a nonnegative matrix W ≫ 0

   Q 1 ≡ L ∗ W M ˜ + M ˜ ∗ W L = − G 0 ≪ 0 .

Theorem 6.1

Suppose that

1. the system (8) is strongly exponentially stable and ‖ c ‖ ⋅ ‖ q ‖ < Δ σ 2 ;

2. for any vector p for which there exist a finite or infinite orthonormal system { ω i } i = 1 N ⊂ H , and { ζ i } i = 1 N ⊂ H , ζ i ≥ 0 such that

   p , W , c ∈ { ω i } i = 1 N , L 1 = L ∣ H

   and the condition

   λ + ∣ ⟨ ξ + , ω i ⟩ ∣ < ζ i ‖ ξ + ‖ , i = 1 , 2 , … , N .

Proof

The norm ‖ ⋅ ‖ 1 satisfies ‖ ξ ( t ) ‖ 1 ≤ ‖ ξ ( 0 ) ‖ 1 , t ≥ 0 for all solutions of (9) and the compact nonnegative operator G 1 , constructed by { ω i } i = 1 N , { ς i } i = 1 N and the operator G 0 (Theorem 5.2). This yields for the eigenelements Φ n of pencil operator λ ˜ n L − M ˜ :

The existence of such constants α 1 , α 2 > 0 with α 1 2 ≤ ⟨ W Φ n , Φ n ⟩ ≤ α 2 2 , and since Re λ ˜ n = − ∣ Re λ ˜ n ∣ (see property ii of Theorem 5.2), by Corollary 5.3, equation (10) is equivalent to

or

We denote by λ ± and ξ ± the eigenvalues and eigenvectors of the two-dimensional self-adjoint operator

given by

with λ + ≤ ε λ ˜ n , we obtain

then, for all ξ ∈ ℋ , we have

The eigenvalues λ 1 < 0 and λ 2 < 0 of the self-adjoint nonpositive operator Q 2 ≡ − L 1 ∗ W c p ∗ − p c ∗ W L 1 given by

1. if ⟨ W , c ⟩ ≠ 0 , then

   λ 1 , 2 = − ‖ c ‖ 2 − ‖ W ‖ 2 2 ± ‖ c ‖ 2 − ‖ W ‖ 2 2 2 + 4 ⟨ W , c ⟩ ;

2. if ⟨ W , c ⟩ = 0 , then

   λ 1 = − 2 ‖ c ‖ 2 and λ 2 = − ‖ W ‖ 2 .

It is used to prove that M ˜ + c p ∗ has no imaginary eigenvalues. Assume that exists λ ˜ n ∈ i R such that

It is easy to deduce from (12) that ⟨ ( Q 1 + Q 2 ) ξ ˆ , ξ ˆ ⟩ = 0 . If p , W , c ∈ { ω i } i = 1 N , then ξ ˆ is orthogonal to { ω i } i = 1 N . Hence, ⟨ ξ ˆ , p ⟩ = 0 implies that M ˜ ξ ˆ = λ ˜ n L ξ ˆ , which contradicts the exponential stability of M ˜ since λ ˜ n is a pure imaginary eigenvalue. Consequently, Theorem 6.1 is proved.□

Example 6.2

Considering system (1) in the infinite-dimensional Hilbert spaces.

Let us denote by S n − the set

It can be shown that system (8) is strongly exponentially stable if and only if

We study system (8) with the feedback control u ( ξ ) = ⟨ ξ , q + p ⟩ , q , p ∈ ℋ . Let

where λ ˜ n → 0 ; c = ( c n ) ∈ H , c n ≠ 0 for n ∈ N ∗ , and ∑ n = 1 ∞ 1 λ ˜ n c n 2 < + ∞ , with c n ≠ 0 for n ∈ N ∗ , that system (7) is approximately controllable for any feedback law, on the other hand, the system

is not exponentially stable, and

or

Finally, ( ϖ n ) ∈ ℋ is the solution only for some ( δ n ) ∈ ℋ , such that