1 Introduction

The bounding of a reachable set is of practical importance in the design of the controller for ellipsoid dynamic systems with disturbance. It plays an important role for state estimation and parameter estimation in control theory [1–4]. Therefore, the reachable set estimation and its related fields have been investigated by many researchers. For a dynamic system, the reachable set is a set which bounds all the states starting from the origin by inputs with peak values. The problem of reachable set bounding for time-delay systems has received considerable attention in recent years, for instance, [1–11] and the references therein. However, time delays cannot be avoided during practice, and they cause undesirable dynamic network behaviors such as oscillation and instability [12–30]. Then, it is natural to ask what about the reachable set of systems with time delays.

Boyd researched linear systems without time-delay, and an LMI condition for an ellipsoid that bounds the reachable set was given in [31]. In [9], Fridman and Shaked studied the uncertain linear systems with time-varying delays and bounded peak input, and LMIs criteria of an ellipsoid that bounds the reachable set based on the Razumikhin theory were firstly obtained. In [3], Kim improved the condition by using the modified Lyapunov-Razumikhin functional. More recently, Nam and Pathirana obtained a smaller reachable set bound by the delay decomposition technique [8]. The maximal Lyapunov functional, combined with the Razumikhin methodology, was utilized to give a non-ellipsoidal description of the reachable set in [10]. Moreover, the authors focused on reachable set bounding for linear systems with discrete and distributed delays in [1, 2]. However, up to now, there have been only few literature works available about linear systems with mixed delays. Hence, it is necessary to make further study on linear systems with mixed delays.

The paper [8] considered non-differentiable time-varying delays, and differentiable time-varying delays were considered in [3–7, 9–11]. In [3], the derivative of time delays was assumed to be less than 1. As is well known, a large value of the derivative of time delays may yield bigger reachable set bounding. In fact, the constraint for time-varying delays may be relaxed, that is, a value of the derivative of delays may not necessarily be less than 1.

Motivated by the above discussions, in this paper we aim to study the reachable set bounding for linear systems with discrete and distributed delays. The constraint of delay is relaxed. The value of derivatives of time delay is not necessary to be less than 1. We construct Lyapunov functionals, combined with the delay decomposition technique, reciprocally convex approach and free-weighting matrix method to derive a more accurate description of the reachable set bound. To the best of our knowledge, it is the first time to introduce triple integrations for reachable set bounds of linear systems with discrete and distributed delays. Numerical examples are given to illustrate the effectiveness of the obtained results, and reachable set bound is tighter than the ones in [1–4, 6–8].

Notations: The following notations are used in our paper except where otherwise specified. \(R^{n}\) is the n-dimension Euclidean space, \(R^{n\times m}\) denotes the set of \(n\times m\)-dimension real matrices; real matrix \(P>0\) (≥0) means that P is a symmetric positive definite (positive semi-definite) matrix. Superscript ‘T’ denotes the transposition of a vector or a matrix; ⋆ represents the elements below the main diagonal of a symmetric block matrix; I denotes an identity matrix; ‘-’ in tables represents no feasible solution for matrix inequality.

2 Preliminaries

Consider the following delayed linear systems with disturbances:

$$ \dot{z}(t)=Az(t)+Bz\bigl(t-\tau(t)\bigr)+D\int_{t-\sigma}^{t}z(s)\,ds+Ew(t), \qquad z(t)=0,\quad t\in[-d,0], $$
(1)

where \(z(t)\in R^{n}\) is the state vector, \(w(t)\in R^{m}\) is the disturbance, \(\tau(t)\) is discrete time delay and σ is distributed time delay, \(A\in R^{n\times n}\), \(B\in R^{n\times n}\), \(D\in R^{n\times n}\) and \(E\in R^{n\times m}\), A, D, B and E are known constant matrices.

\(\tau(t)\) is time-varying discrete delay satisfying

$$0\leq\tau(t)\leq\tau, \qquad\dot{\tau}(t)\leq\mu, $$

where τ and μ are constants. Moreover, \(d=\max\{\tau, \sigma\}\).

The disturbance \(w(t)\in R^{m}\) is the input with bounded peak value, that is,

$$ w^{T}(t)w(t)\leq w_{m}^{2}, $$
(2)

where \(w_{m}\) is a constant.

The following lemmas are useful in deriving the criteria.

Lemma 2.1

The following relation is known as the Leibniz rule:

$$\frac{d}{dt}\int_{a(t)}^{b(t)}f(t,s)\,ds= \dot{b}(t)f\bigl[t,b(t)\bigr]-\dot{a}(t)f\bigl[t,a(t)\bigr] +\int _{a(t)}^{b(t)}\frac{\partial}{\partial t}f(t,s)\,ds. $$

Lemma 2.2

[29]

Given constant symmetric matrices \(R_{1}\), \(R_{2}\), \(R_{3}\), where \(R_{1}=R_{1}^{T}\), \(R_{2}=R_{2}^{T}>0\), then \(R_{1}+R_{3}^{T}R_{2}^{-1}R_{3}<0\) if and only if

$$ \begin{pmatrix} R_{1} & R_{3}^{T} \\ R_{3} & -R_{2} \end{pmatrix} < 0, $$

or

$$ \begin{pmatrix} -R_{2} & R_{3} \\ R_{3} ^{T} & R_{1} \end{pmatrix} < 0. $$

Lemma 2.3

[13]

For any constant matrix \(P=P^{T}>0\) and \(h_{2}>h_{1}\geq0\) such that the following integrations are well defined, then

$$-(h_{2}-h_{1})\int_{t-h_{2}}^{t-h_{1}}z^{T}(s)Pz(s)\,ds \leq-\biggl(\int_{t-h_{2}}^{t-h_{1}}z(s)\,ds \biggr)^{T}P\biggl(\int_{t-h_{2}}^{t-h_{1}}z(s)\,ds \biggr). $$

Lemma 2.4

[30]

For any constant matrix \(R>0\), scalars \(h_{2}>h_{1}\geq0\) such that the following integrations are well defined, then

$$\begin{aligned} -\frac{1}{2}(h_{2}-h_{1})^{2}\int _{-h_{2}}^{-h_{1}}\int_{t+\theta }^{t-h_{1}}z^{T}(s)Rz(s)\,ds\,d\theta \leq{}&{-}\biggl(\int_{-h_{2}}^{-h_{1}}\int _{t+\theta}^{t-h_{1}}z^{T}(s)\,ds\,d\theta\biggr) \\ &{}\cdot R \biggl(\int_{-h_{2}}^{-h_{1}}\int _{t+\theta}^{t-h_{1}}z(s)\,ds\,d\theta\biggr). \end{aligned}$$

Lemma 2.5

[18]

Let \(f_{1},f_{2},\ldots,f_{N}:R^{m}\mapsto R\) have positive values in an open subset D of \(R^{m}\). Then the reciprocally convex combination of \(f_{i}\) over D satisfies

$$\min_{\{\alpha_{i}|\alpha_{i}>0,\sum_{i=1}^{N}\alpha_{i}=1\}}\sum_{i=1}^{N} \frac{1}{\alpha_{i}}f_{i}(t) +\max_{g_{i,j}(t)}\sum _{i\neq j}g_{i,j}(t), $$

subject to

$$\left\{g_{i,j}(t):R^{m}\mapsto R, g_{j,i}(t)=g_{i,j}(t), \begin{bmatrix} f_{i}(t) & g_{i,j}(t) \\ g_{i,j}(t) & f_{j}(t) \end{bmatrix}\geq0 \right\}. $$

Lemma 2.6

[26]

For any vectors \(x_{1}\), \(x_{2}\), constant matrices \(T_{i}\) (\(i=1,2,3,4\)), S and scalars \(\alpha>0\), \(\beta>0\) satisfying \(\alpha+\beta=1\), then the following inequality holds:

$$-\frac{1}{\alpha}x_{1}^{T}T_{1}x_{1}- \frac{1}{\beta }x_{2}^{T}T_{2}x_{2}- \frac{\beta}{\alpha}x_{1}^{T}T_{3}x_{1} - \frac{\alpha}{\beta}x_{2}^{T}T_{4}x_{2} \leq - \left.\begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix} \right.^{T} \begin{bmatrix} T_{1} & S \\ S^{T} & T_{2} \end{bmatrix} \begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix}, $$

subject to

$$ \begin{bmatrix} T_{1}+T_{3} & S \\ S^{T} & T_{2}+T_{4} \end{bmatrix}\geq0. $$

Lemma 2.7

[5]

Let V be a Lyapunov function for system (1)-(2). If

$$\dot{V}+\alpha V-\frac{\alpha}{w_{m}^{2}}w^{T}(t)w(t)\leq0, $$

then \(V\leq1\).

3 Main results

Theorem 3.1

If there exist matrices \(P>0\), \(Q>0\), \(R>0\), \(K_{1}>0\), \(K_{2}>0\), \(M_{1}>0\), \(M_{2}>0\), S, N with appropriate dimensions, and a scalar \(\alpha>0\) such that the following inequalities holds:

$$\begin{aligned}& \begin{bmatrix} R+K_{1} & S \\ S^{T} & R+K_{2} \end{bmatrix}\geq0, \end{aligned}$$
(3)
$$\begin{aligned}& \Phi= \begin{bmatrix} \Phi_{11} & \Phi_{12} & \Phi_{13} &\Phi_{14} &0 & \Phi_{16} & \Phi _{17} & \Phi_{18} \\ \star& \Phi_{22} & \Phi_{23} & \Phi_{24} & \Phi_{25} & \Phi_{26} & 0 & 0 \\ \star& \star& \Phi_{33} & 0 & \Phi_{35} & 0 & 0 & 0 \\ \star& \star& \star& \Phi_{44} & 0 & 0 & 0 & 0 \\ \star& \star& \star& \star& \Phi_{55} & 0 & 0 & 0 \\ \star& \star& \star& \star& \star& \Phi_{66} & \Phi_{67} & \Phi _{68} \\ \star& \star& \star& \star& \star& \star& \Phi_{77} & 0 \\ \star& \star& \star& \star& \star& \star& \star& -\frac{\alpha }{w^{2}_{m}}I \end{bmatrix} \leq0, \end{aligned}$$
(4)

where

$$\begin{aligned}& \Phi_{11}=\alpha P+ PA+ A^{T}P+ Q+\sigma M_{1}+ M_{2}-2e^{-\alpha\tau }K_{1}-e^{-\alpha\tau}R, \\& \Phi_{12}=PB+e^{-\alpha\tau}R-e^{-\alpha\tau}S,\qquad \Phi_{13}=e^{-\alpha \tau}S,\qquad \Phi_{14}= 2e^{-\alpha\tau}K_{1}, \\& \Phi_{16}=A^{T}N^{T}, \qquad\Phi_{17}= \sigma PD, \qquad\Phi_{18}=PE, \\& \Phi_{22}=-(1-\mu) e^{-\alpha\tau}M_{2}-2 e^{-\alpha\tau}K_{1}-2 e^{-\alpha\tau}K_{2}-2 e^{-\alpha\tau}R + e^{-\alpha\tau}\bigl(S^{T}+S\bigr), \\& \Phi_{23}= e^{-\alpha\tau}R- e^{-\alpha\tau}S, \qquad \Phi_{24}= 2e^{-\alpha \tau}K_{2}, \\& \Phi_{25}= 2e^{-\alpha\tau}K_{1},\qquad \Phi_{26}=B^{T}N^{T}, \\& \Phi_{33}=- e^{-\alpha\tau}Q-2 e^{-\alpha\tau}K_{2}- e^{-\alpha\tau}R, \qquad\Phi_{35}=2 e^{-\alpha\tau}K_{2}, \\& \Phi_{44}=-2 e^{-\alpha\tau}K_{1}-2 e^{-\alpha\tau}K_{2},\qquad \Phi_{55}=-2 e^{-\alpha\tau}K_{1}-2 e^{-\alpha\tau}K_{2}, \\& \Phi_{66}=\tau^{2}R +\frac{1}{2}\tau^{2}K_{1}+ \frac{1}{2}\tau^{2}K_{2}- N- N^{T},\qquad \Phi _{67}=\sigma ND,\qquad \Phi_{68}=NE, \\& \Phi_{77}=-\sigma e^{-\alpha\sigma}M_{1}. \end{aligned}$$

Then the reachable sets of system (1) are bounded by a ball \(B(0,r)=\{ z\in R^{n}|\|z\|\leq r\}\) with

$$ r=\frac{1}{\sqrt{\lambda_{\min}(P)}}. $$
(5)

Proof

Construct the following Lyapunov-Krasovskii functional:

$$V(z_{t})=\sum_{i=1}^{6}V_{i}(z_{t}), $$

where

$$\begin{aligned}& V_{1}(z_{t})=z^{T}(t)Pz(t), \\& V_{2}(z_{t})=\int_{t-\tau}^{t}e^{\alpha(s-t)}z^{T}(s)Qz(s)\,ds, \\& V_{3}(z_{t})=\tau\int_{-\tau}^{0} \int_{t+\theta}^{t}e^{\alpha(s-t)}\dot {z}^{T}(s)R\dot{z}(s)\,ds, \\& V_{4}(z_{t})=\int_{-\tau}^{0} \int_{\eta}^{0}\int_{t+\theta}^{t}e^{\alpha (s-t)} \dot{z}^{T}(s)K_{1}\dot{z}(s)\,ds\,d\theta\,d\eta, \\& V_{5}(z_{t})=\int_{-\tau}^{0} \int_{-\tau}^{\eta}\int_{t+\theta }^{t}e^{\alpha(s-t)} \dot{z}^{T}(s)K_{2}\dot{z}(s)\,ds\,d\theta\,d\eta, \\& V_{6}(z_{t})=\int_{t-\sigma}^{t}e^{\alpha(s-t)}( \sigma-t+s)z^{T}(s)M_{1}z(s)\,ds +\int_{t-\tau(t)}^{t}e^{\alpha(s-t)}z^{T}(s)M_{2}z(s)\,ds. \end{aligned}$$

Taking the time derivative of \(V(z_{t})\) along the trajectory of system (1), we obtain

$$\begin{aligned}& \begin{aligned}[b] \dot{V}_{1}(z_{t})={}&2z^{T}(t)P \dot{z}(t)=-\alpha V_{1}(z_{t})+2z^{T}(t)P \dot{z}(t)+\alpha z^{T}(t)Pz(t) \\ ={}&{-}\alpha V_{1}(z_{t})+\alpha z^{T}(t)Pz(t)+2z^{T}(t)P \biggl(Az(t)+Bz\bigl(t-\tau (t)\bigr) \\ &{}+D\int_{t-\sigma}^{t}z(s)\,ds+Ew(t)\biggr), \end{aligned} \end{aligned}$$
(6)
$$\begin{aligned}& \dot{V}_{2}(z_{t})=-\alpha V_{2}(z_{t})+z^{T}(t)Qz(t)-e^{-\alpha\tau }z^{T}(t- \tau)Qz(t-\tau), \end{aligned}$$
(7)
$$\begin{aligned}& \begin{aligned}[b] \dot{V}_{3}(z_{t})={}&{-}\alpha V_{3}(z_{t})+\tau^{2}\dot{z}^{T}(t)R \dot{z}(t) -\tau\int_{t-\tau}^{t}e^{\alpha(s-t)} \dot{z}^{T}(s)R\dot{z}(s)\,ds \\ \leq{}&{-}\alpha V_{3}(z_{t})+\tau^{2} \dot{z}^{T}(t)R\dot{z}(t) -e^{-\alpha\tau}\tau\int_{t-\tau(t)}^{t} \dot{z}^{T}(s)R\dot{z}(s)\,ds \\ &{}-e^{-\alpha\tau}\tau\int_{t-\tau}^{t-\tau(t)} \dot{z}^{T}(s)R\dot{z}(s)\,ds, \end{aligned} \end{aligned}$$
(8)
$$\begin{aligned}& \begin{aligned}[b] \dot{V}_{4}(z_{t})={}&{-}\alpha V_{4}(z_{t})+\frac{1}{2}\tau^{2}\dot {z}(t)K_{1}\dot{z}(t) -\int_{-\tau}^{0} \int_{t+\theta}^{t}e^{\alpha(s-t)}\dot {z}^{T}(s)K_{1}\dot{z}(s)\,ds\,d\theta \\ \leq{}&{-}\alpha V_{4}(z_{t})+\frac{1}{2} \tau^{2}\dot{z}(t)K_{1}\dot{z}(t) -e^{-\alpha\tau}\int _{-\tau}^{0}\int_{t+\theta}^{t} \dot {z}^{T}(s)K_{1}\dot{z}(s)\,ds\,d\theta \\ ={}&{-}\alpha V_{4}(z_{t})+\frac{1}{2} \tau^{2}\dot{z}(t)K_{1}\dot{z}(t) -e^{-\alpha\tau}\int _{-\tau(t)}^{0}\int_{t+\theta}^{t} \dot {z}^{T}(s)K_{1}\dot{z}(s)\,ds\,d\theta \\ &{}-e^{-\alpha\tau}\int_{-\tau}^{-\tau(t)}\int _{t+\theta}^{t-\tau(t)}\dot {z}^{T}(s)K_{1} \dot{z}(s)\,ds\,d\theta \\ &{}-e^{-\alpha\tau}\int_{-\tau}^{-\tau(t)}\int _{t-\tau(t)}^{t}\dot {z}^{T}(s)K_{1} \dot{z}(s)\,ds\,d\theta, \end{aligned} \end{aligned}$$
(9)
$$\begin{aligned}& \begin{aligned}[b] \dot{V}_{5}(z_{t})={}&{-} \alpha V_{5}(z_{t})+\frac{1}{2}\tau^{2}\dot {z}(t)K_{2}\dot{z}(t) -\int_{-\tau}^{0}\int _{t-\tau}^{t+\theta}e^{\alpha(s-t)}\dot {z}^{T}(s)K_{2} \dot{z}(s)\,ds\,d\theta \\ \leq{}&{-}\alpha V_{5}(z_{t})+\frac{1}{2} \tau^{2}\dot{z}(t)K_{2}\dot{z}(t) -e^{-\alpha\tau}\int _{-\tau}^{0}\int_{t-\tau}^{t+\theta} \dot {z}^{T}(s)K_{2}\dot{z}(s)\,ds\,d\theta \\ ={}&{-}\alpha V_{5}(z_{t})+\frac{1}{2} \tau^{2}\dot{z}(t)K_{2}\dot{z}(t) -e^{-\alpha\tau}\int _{-\tau(t)}^{0}\int_{t-\tau(t)}^{t+\theta} \dot {z}^{T}(s)K_{2}\dot{z}(s)\,ds\,d\theta \\ &{}-e^{-\alpha\tau}\int_{-\tau(t)}^{0}\int _{t-\tau}^{t-\tau(t)}\dot {z}^{T}(s)K_{2} \dot{z}(s)\,ds\,d\theta \\ &{}-e^{-\alpha\tau}\int_{-\tau}^{-\tau(t)}\int _{t-\tau}^{t+\theta}\dot {z}^{T}(s)K_{2} \dot{z}(s)\,ds\,d\theta, \end{aligned} \end{aligned}$$
(10)
$$\begin{aligned}& \begin{aligned}[b] \dot{V}_{6}(z_{t})={}&{-} \alpha V_{6}(z_{t})-\int_{t-\sigma}^{t}e^{\alpha (s-t)}z^{T}(s)M_{1}z(s)\,ds+ \sigma z^{T}(t)M_{1}z(t) \\ &{}+z^{T}(t)M_{2}z(t)-\bigl(1-\dot{\tau}(t) \bigr)z^{T}\bigl(t-\tau(t)\bigr)e^{-\alpha\tau (t)}M_{2}z\bigl(t- \tau(t)\bigr) \\ \leq{}& {-}\alpha V_{6}(z_{t})-e^{-\alpha\sigma}\int _{t-\sigma }^{t}z^{T}(s)M_{1}z(s)\,ds+ \sigma z^{T}(t)M_{1}z(t) \\ &{}+z^{T}(t)M_{2}z(t)-(1-\mu)e^{-\alpha\tau}z^{T} \bigl(t-\tau(t)\bigr)M_{2}z\bigl(t-\tau(t)\bigr). \end{aligned} \end{aligned}$$
(11)

Using Lemma 2.2, we have

$$\begin{aligned} \dot{V}_{3}(z_{t})\leq{}&{-}\alpha V_{3}(z_{t})+\tau^{2}\dot{z}^{T}(t)R\dot {z}(t) \\ &{}-e^{-\alpha\tau}\frac{\tau}{\tau(t)}\bigl(z^{T}(t)-z^{T} \bigl(t-\tau (t)\bigr)\bigr)R\bigl(z(t)-z\bigl(t-\tau(t)\bigr)\bigr) \\ &{}-e^{-\alpha\tau}\frac{\tau}{\tau-\tau(t)}\bigl(z^{T}\bigl(t-\tau(t) \bigr)-z^{T}(t-\tau )\bigr)R\bigl(z\bigl(t-\tau(t)\bigr)-z(t-\tau)\bigr) \\ ={}&{-}\alpha V_{3}(z_{t})+\tau^{2} \dot{z}^{T}(t)R\dot{z}(t) \\ &{}-e^{-\alpha\tau}\frac{1}{\beta_{1}}\bigl(z^{T}(t)-z^{T} \bigl(t-\tau (t)\bigr)\bigr)R\bigl(z(t)-z\bigl(t-\tau(t)\bigr)\bigr) \\ &{}-e^{-\alpha\tau}\frac{1}{\beta_{2}}\bigl(z^{T}\bigl(t-\tau(t) \bigr)-z^{T}(t-\tau )\bigr)R\bigl(z\bigl(t-\tau(t)\bigr)-z(t-\tau)\bigr), \end{aligned}$$
(12)

where \(\beta_{1}=\frac{\tau(t)}{\tau}\), \(\beta_{2}=\frac{\tau-\tau (t)}{\tau}\) and \(\beta_{1}+\beta_{2}=1\).

Using Lemma 2.3, we get

$$\begin{aligned} \dot{V}_{4}(z_{t})\leq{}&{-}\alpha V_{4}(z_{t})+\frac{1}{2}\tau^{2}\dot {z}(t)K_{1}\dot{z}(t) \\ &{}-2e^{-\alpha\tau}\frac{1}{(\tau(t))^{2}}\biggl(\int_{-\tau(t)}^{0} \int_{t+\theta}^{t}\dot{z}^{T}(s)\,ds\,d\theta \biggr) K_{1}\biggl(\int_{-\tau(t)}^{0}\int _{t+\theta}^{t}\dot{z}(s)\,ds\,d\theta\biggr) \\ &{}-2e^{-\alpha\tau}\frac{1}{(\tau-\tau(t))^{2}}\biggl(\int_{-\tau}^{-\tau (t)} \int_{t+\theta}^{t-\tau(t)}\dot{z}^{T}(s)\,ds\,d\theta \biggr) \\ &{}\cdot K_{1}\biggl(\int_{-\tau}^{-\tau(t)}\int _{t+\theta}^{t-\tau(t)}\dot {z}(s)\,ds\,d\theta\biggr) \\ &{}-e^{-\alpha\tau}\frac{\tau-\tau(t)}{\tau(t)}\biggl(\int_{t-\tau(t)}^{t} \dot {z}^{T}(s)\,ds\,d\theta\biggr) K_{1}\biggl(\int _{t-\tau(t)}^{t}\dot{z}^{T}(s)\,ds\,d\theta\biggr) \\ ={}&{-}\alpha V_{4}(z_{t})+\frac{1}{2} \tau^{2}\dot{z}(t)K_{1}\dot{z}(t) \\ &{}-2e^{-\alpha\tau}\biggl(z^{T}(t)-\frac{1}{\tau(t)}\int _{t-\tau(t)}^{t}z^{T}(s)\,ds\biggr) K_{1}\biggl(z(t)-\frac{1}{\tau(t)}\int_{t-\tau(t)}^{t}z(s)\,ds \biggr) \\ &{}-2e^{-\alpha\tau}\biggl(z\bigl(t-\tau(t)\bigr)-\frac{1}{\tau-\tau(t)}\int _{t-\tau }^{t-\tau(t)}z(s)\,ds\biggr)^{T} K_{1}\biggl(z\bigl(t-\tau(t)\bigr) \\ &{}-\frac{1}{\tau-\tau(t)}\int_{t-\tau}^{t-\tau(t)}z(s)\,ds\biggr) \\ &{}-e^{-\alpha\tau}\frac{\beta_{2}}{\beta_{1}}\bigl(z^{T}(t)-z^{T} \bigl(t-\tau(t)\bigr)\bigr) K_{1}\bigl(z(t)-z\bigl(t-\tau(t)\bigr) \bigr), \end{aligned}$$
(13)
$$\begin{aligned} \dot{V}_{5}(z_{t})\leq{}&{-}\alpha V_{5}(z_{t})+\frac{1}{2}\tau^{2}\dot {z}(t)K_{2}\dot{z}(t) \\ &{}-2e^{-\alpha\tau}\frac{1}{(\tau(t))^{2}}\biggl(\int_{-\tau(t)}^{0} \int_{t-\tau(t)}^{t+\theta}\dot{z}^{T}(s)\,ds\,d\theta \biggr) K_{2}\biggl(\int_{-\tau(t)}^{0}\int _{t-\tau(t)}^{t+\theta}\dot {z}^{T}(s)\,ds\,d\theta\biggr) \\ &{}-e^{-\alpha\tau}\frac{\tau(t)}{\tau-\tau(t)}\biggl(\int_{t-\tau}^{t-\tau (t)} \dot{z}^{T}(s)\,ds\,d\theta\biggr) K_{2}\biggl(\int _{t-\tau}^{t-\tau(t)}\dot{z}(s)\,ds\,d\theta\biggr) \\ &{}-2e^{-\alpha\tau}\frac{1}{(\tau-\tau(t))^{2}}\biggl(\int_{-\tau}^{-\tau (t)} \int_{t-\tau}^{t+\theta}\dot{z}^{T}(s)\,ds\,d\theta \biggr) K_{2}\biggl(\int_{-\tau}^{-\tau(t)}\int _{t-\tau}^{t+\theta}\dot {z}(s)\,ds\,d\theta\biggr) \\ ={}&{-}\alpha V_{5}(z_{t})+\frac{1}{2} \tau^{2}\dot{z}(t)K_{2}\dot{z}(t) \\ &{}-2e^{-\alpha\tau}\biggl(\frac{1}{\tau(t)}\int_{t-\tau (t)}^{t}z^{T}(s)\,ds-z \bigl(t-\tau(t)\bigr)\biggr) \\ &{}\cdot K_{2}\biggl(\frac{1}{\tau(t)}\int_{t-\tau(t)}^{t}z(s)\,ds-z \bigl(t-\tau(t)\bigr)\biggr) \\ &{}-e^{-\alpha\tau}\frac{\beta_{1}}{\beta_{2}}\bigl(z^{T}\bigl(t-\tau (t) \bigr)-z^{T}(t-\tau)\bigr) K_{2}\bigl(z\bigl(t-\tau(t) \bigr)-z(t-\tau)\bigr) \\ &{}-2e^{-\alpha\tau}\biggl(\frac{1}{\tau-\tau(t)}\int_{t-\tau}^{t-\tau (t)}z^{T}(s)\,ds-z^{T}(t- \tau)\biggr) \\ &{}\cdot K_{2}\biggl(\frac{1}{\tau-\tau(t)}\int_{t-\tau}^{t-\tau (t)}z(s)\,ds-z(t- \tau)\biggr). \end{aligned}$$
(14)

From Lemma 2.5 and inequality (6), one can obtain

$$\begin{aligned} &{-}e^{-\alpha\tau}\frac{1}{\beta_{1}}\bigl(z^{T}(t)-z^{T} \bigl(t-\tau (t)\bigr)\bigr)R\bigl(z(t)-z\bigl(t-\tau(t)\bigr)\bigr) \\ &\qquad{}-e^{-\alpha\tau}\frac{1}{\beta_{2}}\bigl(z^{T}\bigl(t-\tau(t) \bigr)-z^{T}(t-\tau )\bigr)R\bigl(z\bigl(t-\tau(t)\bigr)-z(t-\tau)\bigr) \\ &\qquad{}-e^{-\alpha\tau}\frac{\beta_{2}}{\beta_{1}}\bigl(z^{T}(t)-z^{T} \bigl(t-\tau(t)\bigr)\bigr) K_{1}\bigl(z(t)-z\bigl(t-\tau(t)\bigr) \bigr) \\ &\qquad{}-e^{-\alpha\tau}\frac{\beta_{1}}{\beta_{2}}\bigl(z^{T}\bigl(t-\tau (t)\bigr)-z^{T}(t-\tau)\bigr) K_{2}\bigl(z\bigl(t-\tau(t) \bigr)-z(t-\tau)\bigr) \\ &\quad\leq-e^{-\alpha\tau} \left.\begin{bmatrix} z(t)-z(t-\tau(t)) \\ z(t-\tau(t))-z(t-\tau) \end{bmatrix} \right.^{T} \begin{bmatrix} R & S \\ S^{T} & R \end{bmatrix} \begin{bmatrix} z(t)-z(t-\tau(t)) \\ z(t-\tau(t))-z(t-\tau) \end{bmatrix}. \end{aligned}$$
(15)

It is clear that the following equation holds:

$$ 2\dot{z}^{T}(t)N\biggl(-\dot{z}(t)+Az(t)+Bz\bigl(t- \tau(t)\bigr)+D\int_{t-\sigma }^{t}z(s)\,ds+Ew(t)\biggr)=0. $$
(16)

Combining (9)-(19), one gets

$$ \dot{V}(z_{t})+\alpha V(z_{t})- \frac{\alpha}{w_{m}^{2}}w^{T}(t)w(t) \leq\frac{1}{\sigma}\int _{t-\sigma}^{t}\xi^{T}(t)\Phi\xi(t)\,ds, $$
(17)

where

$$\begin{aligned} \xi^{T}(t)={}& \biggl[z^{T}(t),\qquad z^{T} \bigl(t-\tau(t)\bigr),\qquad z^{T}(t-\tau),\qquad \frac{1}{\tau(t)}\int _{t-\tau(t)}^{t}z^{T}(s)\,ds, \\ &{}\frac{1}{\tau-\tau(t)}\int_{t-\tau}^{t-\tau(t)}z^{T}(s)\,ds, \qquad \dot {z}^{T}(t), \qquad z^{T}(s),\qquad w^{T}(t)\biggr]. \end{aligned}$$

Since (6) and (7) hold, we can conclude that \(\dot{V}+\alpha V-\frac{\alpha}{w_{m}^{2}}w^{T}(t)w(t)\leq0\).

Therefore, one can obtain \(V(z_{t})\leq1\) by Lemma 2.7.

Using the spectral properties of a symmetric positive definite matrix P, the following inequality holds:

$$ \lambda_{\min}(P)\bigl\| z(t)\bigr\| ^{2}\leq V(z_{t}). $$
(18)

This further implies that \(\|z(t)\|\leq r=\frac{1}{\sqrt{\lambda _{\min}(P)}}\) due to (19). This completes the proof. □

If \(D=0\) in system (1), the following corollary is true.

Corollary 3.1

If there exist matrices \(P>0\), \(Q>0\), \(R>0\), \(K_{1}>0\), \(K_{2}>0\), \(M>0\), S, N with appropriate dimensions, and a scalar \(\alpha>0\) such that the following inequalities holds:

$$\begin{aligned}& \begin{bmatrix} R+K_{1} & S \\ S^{T} & R+K_{2} \end{bmatrix}\geq0, \end{aligned}$$
(19)
$$\begin{aligned}& \Phi= \begin{bmatrix} \Phi_{11} & \Phi_{12} & \Phi_{13} &\Phi_{14} &0 & \Phi_{16} & \Phi _{17} \\ \star& \Phi_{22} & \Phi_{23} & \Phi_{24} & \Phi_{25} & \Phi_{26} & 0 \\ \star& \star& \Phi_{33} & 0 & \Phi_{35} & 0 & 0 \\ \star& \star& \star& \Phi_{44} & 0 & 0 & 0 \\ \star& \star& \star& \star& \Phi_{55} & 0 & 0 \\ \star& \star& \star& \star& \star& \Phi_{66} & 0 \\ \star& \star& \star& \star& \star& \star & -\frac{\alpha }{w^{2}_{m}}I \end{bmatrix} \leq0, \end{aligned}$$
(20)

where

$$\begin{aligned}& \Phi_{11}=\alpha P+ PA+ A^{T}P+ Q+ M-2e^{-\alpha\tau}K_{1}-e^{-\alpha \tau}R, \\& \Phi_{12}=PB+e^{-\alpha\tau}R-e^{-\alpha\tau}S, \qquad \Phi_{13}=e^{-\alpha \tau}S, \\& \Phi_{14}= 2e^{-\alpha\tau}K_{1}, \qquad \Phi_{16}=A^{T}N^{T}, \qquad\Phi_{17}=PE, \\& \Phi_{22}=-(1-\mu) e^{-\alpha\tau}M-2 e^{-\alpha\tau}K_{1}-2 e^{-\alpha \tau}K_{2}-2 e^{-\alpha\tau}R + e^{-\alpha\tau} \bigl(S^{T}+S\bigr), \\& \Phi_{23}= e^{-\alpha\tau}R- e^{-\alpha\tau}S, \qquad \Phi_{24}= 2e^{-\alpha \tau}K_{2}, \\& \Phi_{25}= 2e^{-\alpha\tau}K_{1}, \qquad \Phi_{26}= B^{T}N^{T}, \\& \Phi_{33}=- e^{-\alpha\tau}Q-2 e^{-\alpha\tau}K_{2}- e^{-\alpha\tau}R, \qquad\Phi_{35}=2 e^{-\alpha\tau}K_{2}, \\& \Phi_{44}=-2 e^{-\alpha\tau}K_{1}-2 e^{-\alpha\tau}K_{2},\qquad \Phi_{55}=-2 e^{-\alpha\tau}K_{1}-2 e^{-\alpha\tau}K_{2}, \\& \Phi_{66}=\tau^{2}R +\frac{1}{2}\tau^{2}K_{1}+ \frac{1}{2}\tau^{2}K_{2}- N- N^{T}. \end{aligned}$$

Then the reachable sets of system (1) are bounded by a ball \(B(0,r)=\{ z\in R^{n}|\|z\|\leq r\}\) with

$$ r=\frac{1}{\sqrt{\lambda_{\min}(P)}}. $$
(21)

Proof

By setting \(D=0\), \(M_{1}=0\) in the proof of Theorem 3.1, one can easily get the conclusion in the corollary. □

Furthermore, consider the following uncertain polytopic time-delayed linear systems with disturbances:

$$ \begin{aligned} &\dot{z}(t)=(A+\Delta A)z(t)+(B+\Delta B)z\bigl(t- \tau(t)\bigr)+(E+\Delta E)w(t), \\ & z(t)=0,\quad t\in[-\tau_{M},0]. \end{aligned} $$
(22)

The uncertainties are expressed as a linear convex-hull of known matrices \(A_{i}\), \(B_{i}\) and \(E_{i}\),

$$\Delta A=\sum_{i=1}^{N} \theta_{i}(t)A_{i},\qquad \Delta B=\sum _{i=1}^{N}\theta _{i}(t)B_{i}, \qquad \Delta E=\sum_{i=1}^{N} \theta_{i}(t)E_{i} $$

with \(\theta_{i}(t)\in[0,1]\) and \(\sum_{i=1}^{N}\theta_{i}(t)=1\), \(\forall t>0\).

As for system (22), the following corollary is true.

Corollary 3.2

If there exist matrices \(P>0\), \(Q>0\), \(R>0\), \(K_{1}>0\), \(K_{2}>0\), \(M>0\), S, N with appropriate dimensions, and a scalar \(\alpha>0\) satisfying the following inequalities for all \(i=1,2,\ldots,N\):

$$\begin{aligned}& \begin{bmatrix} R+K_{1} & S \\ S^{T} & R+K_{2} \end{bmatrix}\geq0, \end{aligned}$$
(23)
$$\begin{aligned}& \Phi_{i}= \begin{bmatrix} \Phi_{11i} & \Phi_{12i} & \Phi_{13i} &\Phi_{14i} &0 & \Phi_{16i} & \Phi _{17i} \\ \star& \Phi_{22i} & \Phi_{23i} & \Phi_{24i} & \Phi_{25i} & \Phi_{26i} & 0 \\ \star& \star& \Phi_{33i} & 0 & \Phi_{35i} & 0 & 0 \\ \star& \star& \star& \Phi_{44i} & 0 & 0 & 0 \\ \star& \star& \star& \star& \Phi_{55i} & 0 & 0 \\ \star& \star& \star& \star& \star& \Phi_{66i} & 0 \\ \star& \star& \star& \star& \star& \star & -\frac{\alpha }{w^{2}_{m}}I \end{bmatrix} \leq0, \end{aligned}$$
(24)

where

$$\begin{aligned}& \Phi_{11i}=\alpha P+ P(A+A_{i})+ (A+A_{i})^{T}P+ Q+ M-2e^{-\alpha\tau }K_{1}-e^{-\alpha\tau}R, \\& \Phi_{12i}=P(B+B_{i})+e^{-\alpha\tau}R-e^{-\alpha\tau}S, \qquad \Phi _{13i}=e^{-\alpha\tau}S, \\& \Phi_{14i}= 2e^{-\alpha\tau}K_{1}, \Phi_{16i}=(A+A_{i})^{T}N^{T}, \qquad \Phi _{17i}=P(E+E_{i}), \\& \Phi_{22i}=-(1-\mu) e^{-\alpha\tau}M-2 e^{-\alpha\tau}K_{1}-2 e^{-\alpha\tau}K_{2}-2 e^{-\alpha\tau}R + e^{-\alpha\tau} \bigl(S^{T}+S\bigr), \\& \Phi_{23i}= e^{-\alpha\tau}R- e^{-\alpha\tau}S, \qquad \Phi_{24i}= 2e^{-\alpha \tau}K_{2}, \\& \Phi_{25i}= 2e^{-\alpha\tau}K_{1}, \qquad \Phi_{26i}= (B+B_{i})^{T}N^{T}, \\& \Phi_{33i}=- e^{-\alpha\tau}Q-2 e^{-\alpha\tau}K_{2}- e^{-\alpha\tau}R,\qquad \Phi_{35i}=2 e^{-\alpha\tau}K_{2}, \\& \Phi_{44i}=-2 e^{-\alpha\tau}K_{1}-2 e^{-\alpha\tau}K_{2}, \qquad \Phi_{55i}=-2 e^{-\alpha\tau}K_{1}-2 e^{-\alpha\tau}K_{2}, \\& \Phi_{66i}=\tau^{2}R +\frac{1}{2}\tau^{2}K_{1}+ \frac{1}{2}\tau^{2}K_{2}- N- N^{T}. \end{aligned}$$

Then the reachable sets of system (25) are bounded by a ball \(B(0,r)=\{ z\in R^{n}|\|z\|\leq r\}\) with

$$ r=\frac{1}{\sqrt{\lambda_{\min}(P)}}. $$
(25)

Proof

Replacing A, B, E with \(\sum_{i=1}^{N}\theta_{i}(t)(A+A_{i})\), \(\sum_{i=1}^{N}\theta_{i}(t)(B+B_{i})\), \(\sum_{i=1}^{N}\theta_{i}(t)(E+E_{i})\) in the proof of Corollary 3.1, respectively, one can easily obtain the conclusion. □

Remark 1

In this paper, the delay decomposition technique, free-weighting matrix approach and reciprocally convex method are used to construct a Lyapunov functional. Triple integrals are introduced in the Lyapunov functional for the first time to investigate bounding of a reachable set for linear systems with discrete and distributed delays, which may lead to tighter bounding. It will be verified by the following numerical examples.

Remark 2

In order to guarantee negative definite, μ is required to be less than 1 in [3]. It should be noted that the values of derivatives of time delays are not necessary to be less than 1 in Theorem 3.1 because the term \(\Phi_{22}=-(1-\mu) e^{-\alpha\tau}M_{2}-2 e^{-\alpha\tau}K_{1}-2 e^{-\alpha\tau}K_{2}-2 e^{-\alpha\tau}R + e^{-\alpha\tau}(S^{T}+S)\) can be negative definite by choosing appropriate \(K_{1}\), \(K_{2}\), R, S, \(M_{2}\) when \(\mu>1\). Obviously, the results in this paper have a wider scope of application than the ones in [3].

Remark 3

The reachable set of system (1) can be minimized by solving the following optimization problem for a scalar \(\delta>0\):

$$ \begin{aligned} &\min \bar{\delta}\quad \biggl(\bar{\delta}= \frac{1}{\delta}\biggr) \\ &\mbox{s.t.} \begin{cases} (\mathrm{a}) &P\geq\delta I,\\ (\mathrm{b}) &(6)\mbox{-}(7)\mbox{ or }(22)\mbox{-}(23) \mbox{ or }(26)\mbox{-}(27). \end{cases} \end{aligned} $$
(26)

Remark 4

It should be noted that the matrix inequalities in Theorem 3.1 and Corollaries 3.1-3.2 cannot be simplified to LMIs. However, when α is fixed, the matrix inequalities reduced to LMIs, and Matlab’s Toolbox is employed to solve the matrix inequalities in Theorem 3.1 and Corollaries 3.1-3.2.

4 Examples

In this section, two numerical examples are presented to show the validity of the main results derived in this paper.

Example 1

Consider the following time-delayed system with parameters:

$$\begin{aligned} \dot{z}(t)={}& \begin{bmatrix} 0 & 1 \\ -2 & -3 \end{bmatrix} z(t) + \begin{bmatrix} 0 & -0.1 \\ 0.2 & 0.3 \end{bmatrix} z\bigl(t- \tau(t)\bigr) \\ &{}+ \begin{bmatrix} 0.2 & 0.2 \\ 0 & -0.2 \end{bmatrix} \int_{t-\sigma}^{t}z(s)\,ds + \begin{bmatrix} 1 \\ 1 \end{bmatrix} w(t), \end{aligned}$$
(27)

and \(w^{T}(t)w(t)\leq1\).

In order to compare with previous results, r’s for different values of σ with \(\tau=0.2\), \(\mu=0.5\) are listed in Table 1. r’s for different values of τ with \(\sigma=0.1\), \(\mu=0.5\) are listed in Table 2. r’s for different values of μ with \(\tau=0.2\), \(\sigma=0.1\) are listed in Table 3. It is shown that tighter bounds are obtained than the ones in [1, 2] by the proposed method in this paper.

Table 1 Computed r ’s of Example 1 for different values of σ with \(\pmb{\tau=0.2}\) , \(\pmb{\mu=0.5}\)
Full size table
Table 2 Computed r ’s of Example 1 for different values of τ with \(\pmb{\sigma=0.1}\) , \(\pmb{\mu=0.5}\)
Full size table
Table 3 Computed r ’s of Example 1 for different values of μ with \(\pmb{\sigma=0.1}\) , \(\pmb{\tau=0.2}\)
Full size table

Solving system (1) with parameters \(\sigma=0.1\), \(\mu=0.5\), \(\tau=0.2\) by Theorem 3.1, we have

$$ P= \begin{bmatrix} 0.8599 & 0.7400 \\ 0.7400 & 2.1083 \end{bmatrix},\qquad r=\sqrt{1.9380}. $$
(28)

To give a direct comparison, we plotted the ellipsoid of the reachable set in Figure 1. The solid line is the result computed by the proposed method in this paper and the dotted line is the one obtained by [2].

Figure 1
figure 1

Bounding ellipsoids computed by different methods.

Full size image

Example 2

Consider the following uncertain time-delayed system with parameters:

$$ \dot{z}(t)= \begin{bmatrix} -2 & 0 \\ 0 & -0.7 \end{bmatrix} z(t) + \begin{bmatrix} -1 & 0 \\ -1 & -0.9 \end{bmatrix} z\bigl(t- \tau(t)\bigr) + \begin{bmatrix} -0.5 \\ 1 \end{bmatrix} w(t), $$
(29)

and \(w^{T}(t)w(t)\leq1\).

By employing the method of Corollary 3.1 in this paper, r’s for different values of τ with \(\mu=0\) are listed in Table 4. It is clear that the bounds obtained in this paper are better than the ones of [1, 3, 4, 6–8].

Table 4 Computed r ’s of Example 2 for different values of τ with \(\pmb{\mu=0}\)
Full size table

Example 3

Consider the following uncertain time-delayed system with parameters:

$$ \begin{aligned} &A+A_{1}= \begin{bmatrix} -2 & 0 \\ 0 & -0.7 \end{bmatrix},\qquad A+A2= \begin{bmatrix} -2 & 0 \\ 0 & -1.1 \end{bmatrix}, \\ &B+B_{1}= \begin{bmatrix} -1 & 0 \\ -1 & -0.9 \end{bmatrix},\qquad B+B_{2}= \begin{bmatrix} -1 & 0 \\ -1 & -1.1 \end{bmatrix}, \\ &E+E_{1}= \begin{bmatrix} -0.5 \\ 1 \end{bmatrix} =E+E_{2},\qquad w^{T}(t)w(t)\leq1. \end{aligned} $$
(30)

By solving optimization problems (26), computed \(\bar{\delta}\)’s for the case \(0\leq\tau(t)\leq0.75\) with different values of μ are listed in Table 5. It is clear that the proposed method in this paper yields tighter bounds than the one in [3, 6].

Table 5 Computed \(\pmb{\bar{\delta}}\) ’s of Example 3 for the case \(\pmb{0 \leq\tau (t)\leq0.75}\) , \(\pmb{\dot{\tau}(t)\leq\mu}\)
Full size table

5 Conclusions

In this paper, the problem of reachable set bounding for linear systems with both discrete and distributed delays has been investigated. By using Lyapunov-Krasovskii functional theory, delay decomposition technique, reciprocally convex method and free-weighting matrix approach, new reachable set bounds are obtained. Triple integrations are introduced for the first time in Lyapunov functionals to study reachable set bounding for linear systems with mixed delays. Meanwhile, delay decomposition technique is employed, which leads to tighter reachable set bounds. The results have been given to illustrate the advantages over the ones in [1–4, 6–8]. The foregoing results have great potential to be useful for further study in this area, such as linear neutral systems and nonlinear systems. Meanwhile, it is expected that the approach will be used for practical applications in the future, for instance, finite-time boundedness of state estimation for neural networks.