We study singular differential systems with delay. A general description for the solutions of singular differential systems with delay is given and a necessary and sufficient condition for exact observability of singular differential systems with delay is derived.

In many practical systems, such as economic systems, power systems, biological systems, ecosystems, and so on, due to the transmission of information need time span, usually there are the phenomena of delay. Up to now, many authors have paid much attention to the study for delay, and many excellent results have been given [

We must emphasize that there are a lot of systems in which there exist the phenomena of delay and singular simultaneously, we call such systems as the singular differential systems with delay. These systems have many special characters. If we want to describe them more exactly, more accurately design them, and more effectively control them, we must pay tremendous endeavor to investigate them, but that is obviously very difficult. In [

Generally, the singular differential control systems with delay can be written as

The autonomous linear form of (

Usually, the output functions of (

Singular differential control systems with delay (

The observability concept in Definition

Recently, many authors have discussed the observability of singular differential control systems without delay [

In papers [

In this paper, we study singular differential systems with delay. Firstly, for the singular differential systems with delay, the general solutions will be given. Then, for the observability of singular differential systems with delay, we will give criterion.

In the study of the observability of singular differential systems with delay, we will use some concepts and conclusions. Now, we give some of them, and specially give the general solutions of singular differential systems with delay (

Let

From [

For any square matrix

Consider the following systems

If

Let

Let

For delta function

This Lemma is well known. The proof will be omitted.

For any square matrix

Let

For

If

Using the method of [

Suppose that matrix couple

To prove Theorem

We can easily prove the following.

If

Suppose that matrix couple

Since

That is,

Because

If

Define an auxiliary function

Suppose that matrix couple

Since

From Lemma

Just as the proof of Theorem

Suppose that matrix couple

From Lemma

Suppose that matrix couple

As the illustration in Section

Now, let us discuss observability of singular differential systems with delay (

From (

Singular differential system with delay (

If the observability matrix

If there exist

For

Let

Obviously, for

From (

In this paper, we study linear singular differential systems with delay in the state equations. We give a general description of the solutions and as the use of such result, we derive a necessary and sufficient condition for exact observability in finite time.

In [

For the observability of singular control systems with delay, we study it in two cases. When

This research has been supported by the National Natural Science Foundation of China (no. 11371027 and no. 11071001), Doctoral Fund of the Ministry of Education of China (no. 20093401110001), and the Major Program of Educational Commission of Anhui Province of China (no. KJ2010ZD02).