In this section, we discuss some problems of the singular fractional timedelay system (1.1).
Let {D}^{\alpha} be the Caputo fractional differential operator of order 0<\alpha \le 1, \mathbb{T} and \mathbb{T}={(N{D}^{\alpha}I)}^{1}. It is not difficult to verify the following:
\mathbb{T}=(I+N{D}^{\alpha}+{N}^{2}{D}^{2\alpha}+\cdots +{N}^{h1}{D}^{(h1)\alpha}),
(3.1)
where I\in {R}^{{n}_{2}\times {n}_{2}} is an identity matrix.
Theorem 3.1 The fractional differential operator \mathbb{T} is bounded i.e. there exists a positive constant M such that for \mathrm{\forall}{x}_{2}(t) we have
\parallel \mathbb{T}{x}_{2}(t)\parallel \le M\parallel {x}_{2}(t)\parallel .
(3.2)
Proof Obviously, the fractional differential operator \mathbb{T} is linear. According to Lemma 2.3, we are only necessary to show that \mathbb{T} is continuous at 0. Let any sequences {x}_{n}(t)\to 0, {y}_{n}(t)\to {y}_{0}(t), and {y}_{n}(t)=\mathbb{T}{x}_{n}(t), all we finally need to do is to show that {y}_{0}(t)=0.
According to {y}_{n}(t)=\mathbb{T}{x}_{n}(t)={(N{D}^{\alpha}I)}^{1}{x}_{n}(t), we have
(N{D}^{\alpha}I){y}_{n}(t)={x}_{n}(t),
(3.3)
and for n\to \mathrm{\infty},
\begin{array}{r}(N{D}^{\alpha}I){y}_{0}(t)=0\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}N{D}^{\alpha}{y}_{0}(t)={y}_{0}(t)\\ \phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}{y}_{0}(t)=N{D}^{\alpha}{y}_{0}(t)={N}^{2}{D}^{2\alpha}{y}_{0}(t)=\cdots ={N}^{h1}{D}^{(h1)\alpha}{y}_{0}(t)={N}^{h}{D}^{h\alpha}{y}_{0}(t).\end{array}
(3.4)
Combining Remark 2.3 and (3.4) yields
Therefore, the operator \mathbb{T} is bounded. □
To give the solution of systems (2.12), let us define a new function.
Definition 3.1 (see [25])
Let α obey (0\le \alpha <1), the function
{\delta}^{\alpha}(t)=\frac{1}{\mathrm{\Gamma}(1\alpha )}{\int}_{0}^{t}\frac{\delta (\theta )}{{(t\theta )}^{\alpha}}\phantom{\rule{0.2em}{0ex}}d\theta
(3.6)
is called an \alpha \delta function, where \delta (t) is the Dirac delta function.
Remark 3.1 (see [25])
The Laplace transformation of the \alpha \delta function is L\{{\delta}^{\alpha}(t);s\}={s}^{\alpha 1}.
Theorem 3.2 If the system (1.1) is regular, the solution for the system (2.12) exists uniquely.
Proof From Remark 2.3, we know that the pencil (E,A) is regular if the system (1.1) is regular. By the coordinate transformation, the system (2.12) is equivalent to the system (1.1). For t\in [0,\tau ], then t\tau \in [\tau ,0], the system (2.12)(a) may be written as
{x}_{1}^{(\alpha )}(t)={A}_{1}{x}_{1}(t)+{B}_{11}{\phi}_{1}(t\tau )+{B}_{12}{\phi}_{2}(t\tau ),\phantom{\rule{1em}{0ex}}t\in [0,\tau ].
(3.7)
Let {f}_{1}(t)={B}_{11}{\phi}_{1}(t\tau )+{B}_{12}{\phi}_{2}(t\tau ). Obviously, if {f}_{1}(t) is the known function, then (3.7) may be written as
{x}_{1}^{(\alpha )}(t)={A}_{1}{x}_{1}(t)+{f}_{1}(t).
(3.8)
Applying the Laplace transformation on both sides of (3.8) and using (2.5) yield
\begin{array}{r}{s}^{\alpha}{X}_{1}(s){s}^{\alpha 1}{x}_{1}(0)={A}_{1}{X}_{1}(s)+{F}_{1}(s),\\ {X}_{1}(s)={({s}^{\alpha}I{A}_{1})}^{1}{s}^{\alpha 1}{x}_{1}(0)+{({s}^{\alpha}I{A}_{1})}^{1}{F}_{1}(s).\end{array}
(3.9)
Applying the Laplace inverse transformation on both sides of (3.9) and using (2.9) yield
{x}_{1}(t)={E}_{\alpha ,1}\left({A}_{1}{t}^{\alpha}\right){x}_{1}(0)+{\int}_{0}^{t}{(t\theta )}^{\alpha 1}{E}_{\alpha ,\alpha}\left({A}_{1}{(t\theta )}^{\alpha}\right){f}_{1}(\theta )\phantom{\rule{0.2em}{0ex}}d\theta ,\phantom{\rule{1em}{0ex}}t\in [0,\tau ].
(3.10)
As for the system (2.12)(b), it may be rewritten as
N{x}_{2}^{(\alpha )}(t)={x}_{2}(t)+{B}_{21}{\phi}_{1}(t\tau )+{B}_{22}{\phi}_{2}(t\tau ),\phantom{\rule{1em}{0ex}}t\in [0,\tau ].
(3.11)
Similarly, let {f}_{2}(t)={B}_{21}{\phi}_{1}(t\tau )+{B}_{22}{\phi}_{2}(t\tau ), and {f}_{2}(t) is the known sufficiently often differentiable function, then (3.11) may be written as
N{x}_{2}^{(\alpha )}(t)={x}_{2}(t)+{f}_{2}(t).
(3.12)
Taking the Laplace transformation on both sides of (3.12), we have
\begin{array}{c}({s}^{\alpha}NI){X}_{2}(s)={s}^{\alpha 1}N{x}_{2}(0)+{F}_{2}(s),\hfill \\ \begin{array}{rl}{X}_{2}(s)& ={({s}^{\alpha}NI)}^{1}({s}^{\alpha 1}N{x}_{2}(0)+{F}_{2}(s))\\ =\sum _{i=0}^{h1}{N}^{i}{\left({s}^{\alpha}\right)}^{i}({s}^{\alpha 1}N{x}_{2}(0)+{F}_{2}(s))\\ =\sum _{i=1}^{h}{N}^{i1}{s}^{i\alpha 1}{x}_{2}(0)\sum _{i=0}^{h1}{N}^{i}{s}^{i\alpha}{F}_{2}(s).\end{array}\hfill \end{array}
(3.13)
According to Remark 3.1, the inverse Laplace transformation of {X}_{2}(s) yields
{x}_{2}(s)=\sum _{i=1}^{h}{N}^{i1}{\delta}^{i\alpha}{x}_{2}(0)\sum _{i=0}^{h1}{N}^{i}{D}^{i\alpha}{f}_{2}(t),\phantom{\rule{1em}{0ex}}t\in [0,\tau ].
(3.14)
Obviously, by the method of steps, once the solution \overline{x}(t) of the system (2.12) on [0,\tau ] is known, continuing the above process, we can easily obtain the solution \overline{x}(t) of the system (2.12) on [\tau ,2\tau ],[2\tau ,3\tau ],\dots . Thus the solution \overline{x}(t) of the system (2.12) on [0,T] exists uniquely. □
Furthermore, we give the following theorems as regards the MittagLeffler estimation of the solution and finitetime stability for this singular system.
Let us denote by C([a,b]) the space of all continuous real functions defined on [a,b] and by C([a,b],{R}^{n}) the Banach space of continuous functions mapping the interval [a,b] into {R}^{n} with the topology of uniform convergence. Let C=C([\tau ,0],{R}^{n}), [a,b]=[\tau ,0], and designate the norm of an element φ in C by
\parallel \phi \parallel =\underset{\tau \le t\le 0}{sup}\parallel \phi (t)\parallel .
(3.15)
Let X=C([\tau ,T],{R}^{n}) and x(t)=\phi (t), t\in [\tau ,0] be equipped with the norm
\parallel x(t)\parallel :=\underset{0\le t\le T}{sup}x(t),\phantom{\rule{2em}{0ex}}\parallel {x}_{t}\parallel :=\parallel x(t+\theta )\parallel :=\underset{\tau \le \theta \le 0}{sup}\parallel x(t+\theta )\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall}x\in X.
(3.16)
Definition 3.2 (see [15])
The system given by (1.1) satisfying the initial condition x(t)=\phi (t), for t\in [\tau ,0] is finitetime stable w.r.t. \{{t}_{0},\delta ,\u03f5,J\}, \delta <\u03f5, J=[{t}_{0},{t}_{0}+T] if and only if
\parallel \phi \parallel <\delta
(3.17)
implies
\parallel x(t)\parallel <\u03f5,\phantom{\rule{1em}{0ex}}\mathrm{\forall}t\in J.
(3.18)
Theorem 3.3 If \overline{x}(t)=[{x}_{1}(t)/{x}_{2}(t)] is a solution of the system (2.12), then there exist positive constants a and b such that

(i)
a=1+M\parallel {B}_{21}\parallel +M\parallel {B}_{22}\parallel;

(ii)
b>\parallel {A}_{1}\parallel +\parallel {B}_{11}\parallel +\parallel {B}_{12}\parallel;

(iii)
\parallel \overline{x}(t)\parallel \le a\parallel \overline{\phi}\parallel {E}_{\alpha}(b{t}^{\alpha}), \mathrm{\forall}t\in J=[0,T].
Proof According to Lemma 2.1, the system (2.12)(a) may be rewritten in the form of the equivalent Volterra integral equation
\begin{array}{rl}{x}_{1}(t)=& {x}_{1}(0)+\frac{1}{\mathrm{\Gamma}(\alpha )}{\int}_{0}^{t}{(ts)}^{\alpha 1}[{A}_{1}{x}_{1}(s)+{B}_{11}{x}_{1}(s\tau )+{B}_{12}{x}_{2}(s\tau )]\phantom{\rule{0.2em}{0ex}}ds\\ =& {\phi}_{1}(0)+\frac{1}{\mathrm{\Gamma}(\alpha )}{\int}_{0}^{t}{(ts)}^{\alpha 1}{A}_{1}{x}_{1}(s)\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{1}{\mathrm{\Gamma}(\alpha )}{\int}_{0}^{t}{(ts)}^{\alpha 1}[{B}_{11}{x}_{1}(s\tau )+{B}_{12}{x}_{2}(s\tau )]\phantom{\rule{0.2em}{0ex}}ds,\phantom{\rule{1em}{0ex}}t\ge 0.\end{array}
(3.19)
Using the appropriate property of the norm \parallel \cdot \parallel on (3.19), it follows that
\begin{array}{rl}\parallel {x}_{1}(t)\parallel \le & \parallel {\phi}_{1}(0)\parallel +\frac{1}{\mathrm{\Gamma}(\alpha )}{\int}_{0}^{t}{ts}^{\alpha 1}\parallel {A}_{1}\parallel \parallel {x}_{1}(s)\parallel \phantom{\rule{0.2em}{0ex}}ds\\ +\frac{1}{\mathrm{\Gamma}(\alpha )}{\int}_{0}^{t}{ts}^{\alpha 1}[\parallel {B}_{11}\parallel \parallel {x}_{1}(s\tau )\parallel +\parallel {B}_{12}\parallel \parallel {x}_{2}(s\tau )\parallel ]\phantom{\rule{0.2em}{0ex}}ds\\ \le & \parallel \overline{\phi}\parallel +\frac{1}{\mathrm{\Gamma}(\alpha )}{\int}_{0}^{t}{ts}^{\alpha 1}\parallel {A}_{1}\parallel \parallel \overline{x}(s)\parallel \phantom{\rule{0.2em}{0ex}}ds\\ +\frac{1}{\mathrm{\Gamma}(\alpha )}{\int}_{0}^{t}{ts}^{\alpha 1}[\parallel {B}_{11}\parallel \parallel \overline{x}(s\tau )\parallel +\parallel {B}_{12}\parallel \parallel \overline{x}(s\tau )\parallel ]\phantom{\rule{0.2em}{0ex}}ds,\phantom{\rule{1em}{0ex}}t\ge 0.\end{array}
(3.20)
As for the system (2.12)(b), we have
\begin{array}{c}N{D}^{\alpha}{x}_{2}(t)={x}_{2}(t)+{B}_{21}{x}_{1}(t\tau )+{B}_{22}{x}_{2}(t\tau ),\hfill \\ \begin{array}{rl}{x}_{2}(t)& ={(N{D}^{\alpha}I)}^{1}[{B}_{21}{x}_{1}(t\tau )+{B}_{22}{x}_{2}(t\tau )]\\ =\mathbb{T}{B}_{21}{x}_{1}(t\tau )+\mathbb{T}{B}_{22}{x}_{2}(t\tau ).\end{array}\hfill \end{array}
(3.21)
Applying the appropriate property of the norm \parallel \cdot \parallel and Theorem 3.1, we have
\begin{array}{rl}\parallel {x}_{2}(t)\parallel & \le M\parallel {B}_{21}\parallel \parallel {x}_{1}(t\tau )\parallel +M\parallel {B}_{22}\parallel \parallel {x}_{2}(t\tau )\parallel \\ \le (M\parallel {B}_{21}\parallel +M\parallel {B}_{22}\parallel )\parallel \overline{x}(t\tau )\parallel ,\phantom{\rule{1em}{0ex}}t\ge 0.\end{array}
(3.22)
Combining (3.20) and (3.22) yields
\begin{array}{rl}\parallel \overline{x}(t)\parallel \le & \parallel {x}_{1}(t)\parallel +\parallel {x}_{2}(t)\parallel \\ \le & \parallel \overline{\phi}\parallel +(M\parallel {B}_{21}\parallel +M\parallel {B}_{22}\parallel )\parallel \overline{x}(t\tau )\parallel \\ +\frac{1}{\mathrm{\Gamma}(\alpha )}{\int}_{0}^{t}{ts}^{\alpha 1}\parallel {A}_{1}\parallel \parallel \overline{x}(s)\parallel \phantom{\rule{0.2em}{0ex}}ds\\ +\frac{1}{\mathrm{\Gamma}(\alpha )}{\int}_{0}^{t}{ts}^{\alpha 1}(\parallel {B}_{11}\parallel +\parallel {B}_{12}\parallel )\parallel \overline{x}(s\tau )\parallel \phantom{\rule{0.2em}{0ex}}ds,\phantom{\rule{1em}{0ex}}t\ge 0.\end{array}
(3.23)
For 0\le t\le \tau, \parallel \overline{x}(t\tau )\parallel \le \parallel \overline{\phi}\parallel, (3.23) can be written as
\begin{array}{rl}\parallel \overline{x}(t)\parallel \le & (1+M\parallel {B}_{21}\parallel +M\parallel {B}_{22}\parallel )\parallel \overline{\phi}\parallel \\ +\frac{1}{\mathrm{\Gamma}(\alpha )}{\int}_{0}^{t}{ts}^{\alpha 1}\parallel {A}_{1}\parallel \parallel \overline{x}(s)\parallel \phantom{\rule{0.2em}{0ex}}ds\\ +\frac{1}{\mathrm{\Gamma}(\alpha )}{\int}_{0}^{t}{ts}^{\alpha 1}(\parallel {B}_{11}\parallel +\parallel {B}_{12}\parallel )\parallel \overline{x}(s\tau )\parallel \phantom{\rule{0.2em}{0ex}}ds,\phantom{\rule{1em}{0ex}}0\le t\le \tau .\end{array}
(3.24)
From Definition 2.2, we know that {I}^{\alpha}f(t) is an increasing function of t, if f(t)>0. So \frac{1}{\mathrm{\Gamma}(\alpha )}{\int}_{0}^{t}{ts}^{\alpha 1}\parallel {A}_{1}\parallel \parallel \overline{x}(s)\parallel \phantom{\rule{0.2em}{0ex}}ds and \frac{1}{\mathrm{\Gamma}(\alpha )}{\int}_{0}^{t}{ts}^{\alpha 1}[\parallel {B}_{11}\parallel +\parallel {B}_{12}\parallel ]\parallel \overline{x}(s\tau )\parallel \phantom{\rule{0.2em}{0ex}}ds are both increasing functions with regard to t. Taking into account (3.24) and (3.16) yields
\begin{array}{rl}\parallel {\overline{x}}_{t}\parallel \le & (1+M\parallel {B}_{21}\parallel +M\parallel {B}_{22}\parallel )\parallel \overline{\phi}\parallel \\ +\frac{1}{\mathrm{\Gamma}(\alpha )}{\int}_{0}^{t}{ts}^{\alpha 1}\parallel {A}_{1}\parallel \parallel {\overline{x}}_{s}\parallel \phantom{\rule{0.2em}{0ex}}ds\\ +\frac{1}{\mathrm{\Gamma}(\alpha )}{\int}_{0}^{t}{ts}^{\alpha 1}(\parallel {B}_{11}\parallel +\parallel {B}_{12}\parallel )\parallel {\overline{x}}_{s}\parallel \phantom{\rule{0.2em}{0ex}}ds\\ \le & (1+M\parallel {B}_{21}\parallel +M\parallel {B}_{22}\parallel )\parallel \overline{\phi}\parallel \\ +\frac{1}{\mathrm{\Gamma}(\alpha )}{\int}_{0}^{t}{ts}^{\alpha 1}(\parallel {A}_{1}\parallel +\parallel {B}_{11}\parallel +\parallel {B}_{12}\parallel )\parallel {\overline{x}}_{s}\parallel \phantom{\rule{0.2em}{0ex}}ds,\phantom{\rule{1em}{0ex}}0\le t\le \tau .\end{array}
(3.25)
Let a=1+M\parallel {B}_{21}\parallel +M\parallel {B}_{22}\parallel and {b}_{0}=\parallel {A}_{1}\parallel +\parallel {B}_{11}\parallel +\parallel {B}_{12}\parallel, we can see the function a(t) in Lemma 2.5 to be
a(t)=(1+M\parallel {B}_{21}\parallel +M\parallel {B}_{22}\parallel )\parallel \overline{\phi}\parallel =a\parallel \overline{\phi}\parallel ,
(3.26)
obviously, it is nondecreasing.
An application of the corollary of the generalized Gronwall inequality (2.18) yields
\parallel {\overline{x}}_{t}\parallel \le a\parallel \overline{\phi}\parallel {E}_{\alpha}\left({b}_{0}{t}^{\alpha}\right),\phantom{\rule{1em}{0ex}}0\le t\le \tau .
(3.27)
Similarly, the same argument implies the following estimate:
\parallel {\overline{x}}_{t}\parallel \le a\parallel {\overline{x}}_{{\tau}_{0}}\parallel {E}_{\alpha}\left({b}_{0}{(t{\tau}_{0})}^{\alpha}\right),\phantom{\rule{1em}{0ex}}{\tau}_{0}\le t\le {\tau}_{0}+\tau ,{\tau}_{0}\ge 0.
(3.28)
From Definition 2.5, we know that the MittagLeffler function {E}_{\alpha}(t) is an increasing function with regard to t. Therefore, there exists b>{b}_{0} such that {E}_{\alpha}(b{\tau}^{\alpha})>{E}_{\alpha}({b}_{0}{\tau}^{\alpha}) and \frac{{E}_{\alpha}(b{(t\tau )}^{\alpha}){E}_{\alpha}({b}_{0}{\tau}^{\alpha})}{{E}_{\alpha}(b{t}^{\alpha})}<\frac{1}{a}.
Equations (3.27) and (3.28) suggest the following general expression:
\parallel {\overline{x}}_{t}\parallel \le a\parallel \overline{\phi}\parallel {E}_{\alpha}\left(b{t}^{\alpha}\right),\phantom{\rule{1em}{0ex}}0\le t\le n\tau \le T.
(3.29)
To prove (3.29) by induction we have to show that it holds for n=1 because of (3.27) and if it holds for n=k, then it also holds for n=k+1. Indeed, for t\in [\tau ,(k+1)\tau ], so that t\tau \in [0,k\tau ], on the one hand, using (3.28), we have
\parallel {\overline{x}}_{t}\parallel \le a\parallel {\overline{x}}_{t\tau}\parallel {E}_{\alpha}\left({b}_{0}{\tau}^{\alpha}\right).
(3.30)
On the other hand, using (3.29) we obtain
\parallel {\overline{x}}_{t\tau}\parallel \le a\parallel \overline{\phi}\parallel {E}_{\alpha}\left(b{(t\tau )}^{\alpha}\right).
(3.31)
Taking into account (3.30) and (3.31) we conclude that
\begin{array}{rl}\parallel {\overline{x}}_{t}\parallel & \le a[a\parallel \overline{\phi}\parallel {E}_{\alpha}\left(b{(t\tau )}^{\alpha}\right)]{E}_{\alpha}\left({b}_{0}{\tau}^{\alpha}\right)\\ =a\parallel \overline{\phi}\parallel {E}_{\alpha}\left(b{t}^{\alpha}\right)\frac{a{E}_{\alpha}(b{(t\tau )}^{\alpha}){E}_{\alpha}({b}_{0}{\tau}^{\alpha})}{{E}_{\alpha}(b{t}^{\alpha})}\\ \le a\parallel \overline{\phi}\parallel {E}_{\alpha}\left(b{t}^{\alpha}\right).\end{array}
(3.32)
That is,
\parallel \overline{x}(t)\parallel \le \parallel {\overline{x}}_{t}\parallel \le a\parallel \overline{\phi}\parallel {E}_{\alpha}\left(b{t}^{\alpha}\right).
(3.33)
The proof is completed. □
Theorem 3.4 The fractional singular timedelay system given by (1.1) is finitetime stable w.r.t. \{0,\delta ,\u03f5,J\}, \delta <\u03f5, if the following condition is satisfied:
a{\parallel P\parallel}^{2}{E}_{\alpha}\left(b{t}^{\alpha}\right)\le \frac{\u03f5}{\delta},\phantom{\rule{1em}{0ex}}\mathrm{\forall}t\in J=[0,T].
(3.34)
Proof From the coordinate transformation (2.13), we have
x(t)=P\overline{x}(t)=P[{x}_{1}(t)/{x}_{2}(t)],\phantom{\rule{2em}{0ex}}\phi (t)=P\overline{\phi}(t)=P[{\phi}_{1}(t)/{\phi}_{2}(t)].
(3.35)
From Theorem 3.3 we obtain
\parallel x(t)\parallel \le \parallel P\parallel \parallel \overline{x}(t)\parallel \le a{\parallel P\parallel}^{2}\parallel \phi \parallel {E}_{\alpha}\left(b{t}^{\alpha}\right).
(3.36)
Hence, using Definition 3.2 and the basic condition of Theorem 3.4, it follows that
\parallel x(t)\parallel <\u03f5,\phantom{\rule{1em}{0ex}}\mathrm{\forall}t\in J=[0,T].
(3.37)
The proof is completed. □