On the Stability and Controllability of Degenerate Differential Systems in Banach Spaces ^†

Abstract

The aim of this research is to generalize the famous Lyapunov Theorem of the classical explicit differential systems given by two abstract forms: $x^{\prime }\left(t\right)=Px\left(t\right)$ and $x_{n+1}=P{x}_n$ where P is a linear operator, or a matrix when the space has finite dimension in order to study the spectrum of a degenerate differential system. We also use some properties of the spectral theory for the corresponding pencil and an appropriate conformal mapping. The achieved results can be applied to study the stability and controllability of those degenerate systems.

1. Introduction

The fundamental challenges of mathematical problems about differential systems appear precisely in control theory. Many researchers use linear systems of two types continuous or discrete.

Since 1970, many mathematicians have become interested in another wide class of differential systems, which have real practical and physical applications can be found in [1,2,3,4].

In the present paper, we consider the differential system of implicit stationary equations, described as follows:

$$A{x}^{\prime }\left(t\right)=Bx\left(t\right)+\phi (t,x\left(t\right)),t\ge 0;$$

(1)

with initial condition:

$$x\left(t_0\right)=x_0.$$

Here, we assume that A and B are bounded linear operators acting in the same complex Hilbert space $\mathcal{H}$ and $\phi$ is a continuous function from $[0,+\infty [\times \mathcal{H}$ into $\mathcal{H}$, the operator A is not invertible.

In the following we need these notations: $\parallel .\parallel$, $<.,.>$ are the norm and the inner product in $\mathcal{H}$ respectively. Now, we consider the homogeneous case of the system (1) and we suppose that it has a solution then, we have:

• System (1) is said to be exponential, if for any solution $x\left(t\right)$ with $t\ge 0$, we have

  $$\parallel x\left(t\right)\parallel \le M{e}^{\alpha t}\parallel x_0\parallel ,t\ge 0,$$

  (2)

  where the constants $\alpha$ and M are not dependent on the solution $x\left(t\right)$.
• For $\alpha <0$, the system (1) is exponentially stable. In particular, if $\alpha =0$ then (1) is uniformly bounded.
• The system (1) is well-posed if it satisfies the following conditions:
  • If for any solution $x(.)$ such that $x\left(0\right)=x_0$, we get $x\left(t\right)=0$ for all $t\ge 0$.
  • It generates an evolution semigroup of bounded operators $S\left(t\right):x_0⟼x\left(t\right)$ for all $t\ge 0$.
  The operators $S\left(t\right)$ are defined on the set $D_0=\left\{x_0\right\}$ of the admissible initial vectors $x_0$.

2. Lyapunov Theorem Generalization

In this section, we investigate the homogeneous problem of (1) as follows:

$$\begin{array}{c}A{x}^{\prime }\left(t\right)=Bx\left(t\right),t\ge 0,\end{array}x\left(t_0\right)=x_0.$$

(3)

Our main obective is to extend the general Lyapunov Theorem [1] of the linear differential systems to the operator pencil $\lambda A-B$ then, we apply the achieved results to affirm the stability and controllability of some degenerate systems as (1) and (3).

Definition 1 

([3]). The complex parameter λ is said to be a regular point of the pencil $\lambda A-B$ if the operator ${(\lambda A-B)}^{-1}$ exists and is bounded on $\mathcal{H}$.

We denote by $\rho (A,B)$ the set of all regular points and its complement by $\sigma (A,B)=\mathbb{C}\backslash {\sigma }_p(A,B)$, which is also called the spectrum of the operator pencil $\lambda A-B$. The set of all eigen-values is denoted by ${\sigma }_p$, such that:

$${\sigma }_p(A,B)=\{\lambda \in \mathbb{C}|\exists v\ne 0,(\lambda A-B)v=0\}.$$

Theorem 1. 

Consider problem (3). If the spectrum $\sigma (A,B)$ of the linear bounded operators A and B lies in the left half plane ($Rel\left(\lambda \right)<0$), then for any uniform positive operator $G\gg 0$ there exists a uniform positive operator $W\gg 0$ such that

$$B^{\ast }WA+A^{\ast }WB=G.$$

(4)

Proof. 

We suppose that $\sigma (A,B)\subset \{\lambda :Re(\lambda )<0\}$ then, i is a regular point and the operator $T=i(iA+B){(iA-B)}^{-1}$ is bounded. Now, we use the conformal mapping $\mu =\varphi \left(\lambda \right)={\displaystyle \frac{-i\lambda +1}{\lambda -i}}$ and we obtain:

$$T-\mu I={\displaystyle \frac{-2}{z-i}}(\lambda A-B){(iA-B)}^{-1}.$$

So, the operator $T-\mu I$ is invertible if and only if the pencil $\lambda A-B$ is also invertible. Hence,

$$\sigma \left(T\right)=\sigma (I,T)=\varphi \left(\sigma \right(A,B\left)\right).$$

Using the classical Lyapunov Theorem [3], we have:

for any operator $H\gg 0$, there exists an operator $W\gg 0$ such that:

$$\begin{array}{ccc}\hfill Re\left(WT\right)& =& {\displaystyle \frac{WT+T^{\ast }W}{2}}\hfill \\ & =& {\displaystyle \frac{1}{2}}(iW(iA+B){(iA-B)}^{-1}-i{(-i{A}^{\ast }-B^{\ast })}^{-1}(-i{A}^{\ast }+B^{\ast })W)\hfill \\ & =& {(-i{A}^{\ast }-B^{\ast })}^{-1}(A^{\ast }WB+B^{\ast }WA){(iA-B)}^{-1}\hfill \\ & =& -H.\hfill \end{array}$$

We put $G=B^{\ast }WA+A^{\ast }WB$ with $G=-(i{A}^{\ast }+B^{\ast })H(iA-B)\gg 0.$

In fact, $G=G^{\ast }$ and $<Gx,x>\ge {c\parallel x\parallel }^2,\mathrm{for}c>0.$ Thus, our theorem is proved. □

Theorem 2. 

If the Equation (4) is satisfied for the pair($W,G$) of bounded positive uniformly operators, then i is not an eigen-value for the operator pencil $\lambda A-B$.

Proof. 

Suppose that $i\in {\sigma }_p(A,B)$ and v is its eigen-vector then, $(iA-B)v=0$ and $Bv=iAv$. After the computing of the inner product, we obtain:

$$<Gv,v>=<(B^{\ast }WA+A^{\ast }WB)v,v>=0,\forall v\in \mathcal{H}.$$

(5)

Since $G\gg 0$, $<Gv,v>\ge {c\parallel v\parallel }^2>0$ then, we have a contradiction with the main hypothesis above. □

We recall some notes concerning uncontrollable system as (3) before summarizing our results.

Let $D_0=\left\{x_0\right\}$ be the initial subspace of $\mathcal{H}$; we denote by $A_0=A\backslash D_0$ the invertible restriction of the operator A in $D_0$ (the operator $A_0$ is invertible if system (3) is well-posed).

Lemma 1. 

Let $A_0$ be an invertible operator. If $\phi (\tau ,x\left(\tau \right))\in A{D}_0$ for any $\tau \ge {\tau }_0$ and the function $S(t-\tau )A_0^{-1}\phi (\tau ,x\left(\tau \right))$ is integrable (with respect to τ) where ${\left\{S\left(t\right)\right\}}_{t\ge 0}$ is the semigroup of the operators of (3), then the system (3) is equivalent to the following:

$$x\left(t\right)=S\left(t\right)x_0+{\int }_0^{t}S(t-\tau )A_0^{-1}\phi (\tau ,x\left(\tau \right))d\tau .$$

(6)

Lemma 2 

(See [3]). If $g\left(t\right)\le c+{\int }_0^{t}g\left(\tau \right)h\left(\tau \right)$ for all $t\ge 0$, where h is a continuous positive real function and $c>0$ is an arbitrary constant, then

$$g\left(t\right)⩽c.exp({\int }_0^{t}h\left(\tau \right)d\tau ).$$

Theorem 3. 

Suppose that:

1. 

the system (3) is well-posed,

2. 

the linear operator $\phi (t,x(t\left)\right)$ transforms $D_0$ into $A{D}_0$ such that

$${\int }_0^{+\infty }\parallel A_0^{-1}\phi (t,x\left(t\right))\parallel dt<\infty ,\forall t\ge 0$$

then, the system (3) is exponentially stable.

Proof. 

Assuming that the first condition of this theorem is verified, we have

$$\parallel S\left(t\right)x_0\parallel \le M{e}^{\alpha t}\parallel x_0\parallel ,$$

and

$$\parallel S(t-\tau )A_0^{-1}\phi (\tau ,x\left(\tau \right))\parallel \le M{e}^{t-\tau }\parallel A_0^{-1}\phi (\tau ,x\left(\tau \right))\parallel ,$$

with $A_0^{-1}\phi (\tau ,x\left(\tau \right))\in D_0.$ Using (6), we get:

$$\parallel x\left(t\right)\parallel \le M{e}^{\alpha t}\parallel x_0\parallel + M{\int }_0^t{e}^{\alpha (t-\tau )}\parallel A_0^{-1}\phi (\tau ,x\left(\tau \right))\parallel \parallel x\left(\tau \right)\parallel d\tau ,$$

which is equivalent to

$$e^{-\alpha t}\parallel x\left(t\right)\parallel \le M\parallel x_0\parallel + M{\int }_0^t{e}^{-\alpha \tau }\parallel A_0^{-1}\phi (\tau ,x\left(\tau \right))\parallel \parallel x\left(\tau \right)\parallel d\tau .$$

Applying Lemma 2, where $g\left(t\right)=e^{-\alpha t}\parallel x\left(t\right)\parallel$, and $h\left(\tau \right)=M\parallel A_0^{-1}\phi (\tau ,x\left(\tau \right))$, $c=M\parallel x_0\parallel$, then

$$e^{-\alpha t\parallel x\left(t\right)}\parallel \le M\parallel x_0\parallel exp[M{\int }_0^t\parallel A_0^{-1}\phi (\tau ,x\left(\tau \right))\parallel d\tau ] \le M\parallel x_0\parallel exp[M{\int }_0^{\infty }\parallel A_0^{-1}\phi (\tau ,x\left(\tau \right))\parallel d\tau ].$$

Therefore, $\parallel x\left(t\right)\parallel \le M_1{e}^{\alpha t}{x}_0$. □

Particularly, in finite dimensional spaces, we can apply the theory of elementary divisors for the matrix pencil $\lambda A-B$ (for an example, see [5]) and we establish the next important result.

Theorem 4. 

The following statements are equivalent:

1. 

the system (3) is exponentially stable;

2. 

$\sigma (A,B)=\sigma (A,B)\subset \{\lambda :Re(\lambda )<0\}$;

3. 

there exists a positive definite matrix $W\gg 0$ such that $B^{\ast }WA+A^{\ast }WB\gg 0.$

Remark 1. 

In finite dimensional spaces, exponential stability is characterized by the fact that the spectrum of matrices A and B lies in the left half plane, but the situation in infinite dimensional spaces is much more complicated.

3. Relation between Stabilizability and Controllability

We provide here some definitions and basic results about the exact controllability and complete stabilizability of an implicit differential control system governed by the general form:

$$A{x}^{\prime }\left(t\right)=Bx\left(t\right)+Cu\left(t\right),x\left(0\right)=x_0.$$

(7)

where C is also a linear bounded operator and $u\left(t\right)$ is a function takes values in the Hilbert space $U\subset \mathcal{H}$, which is supposed to be Bockner integrable. $x\left(t\right)$ is the mild solution for restrictions on a class of controls u given by

$$x\left(t\right)=x(t,u(.),x_0)=S\left(t\right)x_0+{\int }_0^{t}S(t-\tau )A_0^{-1}Cu\left(\tau \right)d\tau .$$

(8)

The famous relation between exact controllability and complete stabilizability was first established by Slemrod, who proved that the controllability from any state to any state implies the exponential stabilizability.

In [6], Zabczyk showed that the implication in the opposite way is possible, some authors proposed that in the case of Hilbert spaces with bounded operators the previous idea is not available.

The system (7) is exactly controllable if for all $x_0$, $x_1$$\in \mathcal{H}$, there exists a time T and a control $u\in L_p(0,T;U),p\ge 1$ such that $x(T,u(.),x_0)=x_1$. $x_1=0$, here we talk about exact null controllability.

The system (7) is said to be exponentially stabilizable if there exists a linear feedback control $u\left(t\right)=Dx\left(t\right)$, where D is also another linear bounded operator such that

$$\parallel e^{A_0^{-1}(B+CD)t}\parallel \le M_{\omega }{e}^{-\omega t},M_{\omega }\ge 1 ,\omega >0.$$

(9)

The system (7) is said to be completely stabilizable if it is exponentially stabilizable for all $\omega >0.$

Let ${\mathcal{R}}_T$ be the reachability operator defined as

$${\mathcal{R}}_{T}u(.)={\int }_0^{T}S(t-\tau )A_0^{-1}Cu\left(\tau \right)d\tau ,$$

(10)

which is a linear bounded operator acting from $L_p$ to $\mathcal{H}$. System (7) is exactly controllable if $Im\left({\mathcal{R}}_T\right)=\mathcal{H}$; further, it is exactly null controllable if and only if

$$Im\left({\mathcal{R}}_T\right)\subset Im\left(e^{A_0^{-1}(B+CD)T}\right),\forall T>0.$$

We denote by ${\mathcal{R}}_T^{\ast }$ the adjoint operator of ${\mathcal{R}}_T$ satisfies the property:

$$\exists \delta >0,\forall x^{\ast }\in \mathcal{H}, \parallel {\mathcal{R}}_T^{\ast }{x}^{\ast }\parallel \ge \delta \parallel x^{\ast }\parallel .$$

(11)

We can have many implicit conditions on exact controllability summarized by the bellow formulation:

$$\begin{array}{ccc}\hfill \parallel {\mathcal{R}}_T^{\ast }{x}^{\ast }\parallel & =& ({\int }_0^T\parallel C^{\ast }{\left(A_0^{\ast }\right)}^{-1}{S}^{\ast }\left(t\right)x^{\ast }{{\parallel }^{q}dt)}^{\frac{1}{q}},u\in L_p,p>1,\hfill \\ \\ \hfill \parallel {\mathcal{R}}_T^{\ast }{x}^{\ast }\parallel & =& ess\left\{sup\right\{\parallel C^{\ast }{\left(A_0^{\ast }\right)}^{-1}{S}^{\ast }\left(t\right)x^{\ast }\parallel \left\}\right\},0\le t\le T,u\in L_1;\hfill \end{array}$$

(12)

where $ess\left\{sup\right\}$ denotes the essential supremum of the given set.

Remark 2. 

The condition of exact controllability in the class of controls $L_p$, $p>1$ are equivalent but the situation become more complicated to show it in $L_1$.

Theorem 5 

([7,8]). The system (7) is exactly controllable in the class $L_p$ if and only if the following conditions are met:

• $p=1$, if $\underset{n\to \infty }{lim}$$\left(C^{\ast }{\left(A_0^{\ast }\right)}^{-1}{S}^{\ast }\left(t\right)x_n^{\ast }\right)=0$, uniformly almost evrywhere on $t\in [0,T]$; then, $\underset{n\to \infty }{lim}{x}_n^{\ast }=0.$
• $p>1,$ if $\underset{n\to \infty }{lim}\left(C^{\ast }{\left(A_0^{\ast }\right)}^{-1}{S}^{\ast }\left(t\right)x_n^{\ast }\right)=0$, almost evrywhere on $t\in [0,T]$, then $\underset{n\to \infty }{lim}{x}_n^{\ast }=0.$

4. Conclusions

The notion of stability and controllability of a new class of controlled systems defined by implicit stationary differential equations was the purpose of this paper. In the course of this work, we used the spectral theory of the operator pencil $\lambda A-B$ to provide conditions for the stability in the sense of Lyapunov, we also studied the controllability of those systems. Furthermore, most of researches will be directed to open problems for some particular types of systems on special spaces with infinite dimension.