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This paper is concerned with the reachable set estimation problem for neutral Markovian jump systems with bounded peak disturbances, which was rarely proposed for neutral Markovian jump systems. The main consideration is to find a proper method to obtain the no-ellipsoidal bound of the reachable set for neutral Markovian jump system as small as possible. By applying Lyapunov functional method, some derived conditions are obtained in the form of matrix inequalities. Finally, numerical examples are presented to demonstrate the effectiveness of the theoretical results.

In practice and engineering applications, many dynamical systems may cause abrupt variations in their structure, due to stochastic failures or repairs of the components, changes in the interconnections of subsystems, sudden environment changes, and so on. Markovian jump systems, modeled by a set of subsystems with transitions among the models determined by a Markov chain taking values in a finite set, have appealed to a lot of researchers in the control community. In the past few decades, the Markovian jump systems have been extensively studied, see [

The reachable set [

Recently, the reachable set estimation problems for time-delay systems have been received growing attention. Then, an increasing number of researchers have devoted their efforts to the problem of reachable set estimation [

As is well known, neutral system being a special case of time delay system exists in many dynamic systems [

Consider the following neutral Markovian jump systems with disturbances

x ˙ ( t ) − C ( t , r t ) x ˙ ( t − τ ) = A ( t , r t ) x ( t ) + B ( t , r t ) x ( t − h ( t ) ) + D ( t , r t ) w ( t ) , x ( t 0 + θ ) = φ ( θ ) , ∀ θ ∈ [ − ρ * , 0 ] , (1)

where x ( t ) ∈ ℜ n is the state vector, τ > 0 is a constant neutral delay, the discrete delay h ( t ) is a time-varying function that satisfies

0 ≤ h ( t ) ≤ h M , h ˙ ( t ) ≤ h d < 1, w T ( t ) w ( t ) ≤ w m 2 , (2)

φ ( ⋅ ) is a differentiable vector-valued initial function, ρ * = τ + h M , { r t , t ≥ 0 } is Markovian process taking values on the probability space in a finite state ℘ = { 1,2,3, ⋯ , N } with generator Λ = ( λ i j ) ( i , j ∈ ξ ) , and Λ is described as follows

P ( r t + Δ = j | r t = i ) = { λ i j Δ + o ( Δ ) , i ≠ j 1 + λ i i Δ + o ( Δ ) , i = j (3)

where Δ > 0 , l i m Δ → + 0 o ( Δ ) Δ = 0 , λ i j ≥ 0 ( i ≠ j ) is the transition probability from i to j at time t → t + Δ , λ i i = − ∑ j = 1 , j ≠ i N λ i j . A ( t , r t ) , B ( t , r t ) , C ( t , r t ) and D ( t , r t ) are known constant matrices of the Markov process.

Since the state transition probability of the Markovian jump process is considered in this paper is partially known, the transition probability matrix of Markovian jumping process Λ is defined as

Λ = [ λ 11 ? ⋯ λ 1 N ? λ 22 ⋯ λ 2 N ⋮ ⋮ ⋱ ⋮ λ 41 ? ⋯ λ N N ] , (4)

where ? represents the unknown transition rate. For notational clarity, ∀ i ∈ ℘ , the set U i denotes U i = U k i ∪ U u k i with

U k i : = { j : λ i j is known for j ∈ ℘ } , U u k i : = { j : λ i j is unknown for j ∈ ℘ } . (5)

moreover, if U k i ≠ Ø , it is further described as U k i = { k 1 i , k 2 i , ⋯ , k m i } , where m is a non-negation integer with 1 ≤ m ≤ N and k j i ∈ Z + , 1 ≤ k j i ≤ N , j = 1,2, ⋯ , m represent the known element of the ith row and jth column in the state transition probability matrix Λ .

For the sake of brevity, x ( t ) is used to represent the solution of the system under initial conditions x ( t , t 0 , r 0 ) , and { x ( t ) , t } satisfies the initial condition ( x 0 , r 0 ) . And its weak infinitesimal generator, acting on function V, is defined in [

L V ( x t , t , i ) = l i m Δ → 0 + 1 Δ [ E ( V ( x t + Δ , t + Δ , r t + Δ ) | ( x t , r t = i ) ) − V ( x t , t , i ) ] .

This paper aims to find a reachable set for neutral Markovian jump systems (1) based on the Lyapunov-Krasovskii functional approach. We denote the set of reachable states with w ( t ) that satisfies (2) by

ℜ x ≜ { x ( t ) ∈ ℜ n | x ( t ) , w ( t ) satisfy ( 1 ) and ( 2 ) } . (6)

We will bound ℜ x by an ellipsoid of the form

ℑ ( P ,1 ) ≜ { x ( t ) ∈ ℜ n : x T ( t ) P x ( t ) ≤ 1, P > 0 } . (7)

For simplicity, there are the following representations:

A i = A ( t , r t = i ) , B i = B ( t , r t = i ) , C i = C ( t , r t = i ) , D i = D ( t , r t = i ) , P i = P ( t , r t = i ) .

In this paper, the following Lemma and Assumption are needed:

Lemma 1 [

V ˙ ( x t ) + α V ( x t ) − β w ( t ) T w ( t ) ≤ 0 , α > 0 , β > 0 ,

then we have V ( x t ) ≤ β α w m 2 for ∀ t ≥ 0 .

Lemma 2 [

− h ∫ t − h t x ˙ T ( s ) W x ˙ ( s ) d s ≤ [ x T ( t ) x T ( t − h ) ] [ − W W W − W ] [ x ( t ) x ( t − h ) ] .

Lemma 3 [

( ∫ 0 γ ω ( s ) d s ) T Φ ( ∫ 0 γ ω ( s ) d s ) ≤ γ ∫ 0 γ ω T ( s ) Φ ω ( s ) d s

Our aim is to find an ellipsoid set as small as possible to bound the reachable set defined in (3). In this section, based on an appropriate Lyapunov functional and matrix inequality techniques (Lemma (1-3)), following Theorems are derived.

Theorem 1. Consider the Markov neutral system (1) with constraints (2), if there exist symmetric matrices P 2 i , P 3 i , P 1 i > 0 , W i > 0 ( i = 1 , 2 , ⋯ , N ), Q > 0 , R > 0 , S > 0 and a scalar α > 0 satisfying the following matrix inequalities:

[ Φ i 11 Φ i 12 Φ i 13 P 2 i T C i 0 0 P 2 i T D i * Φ i 22 Φ i 23 P 3 i T C i 0 0 P 3 i T D i * * Φ i 33 0 Φ i 35 0 0 * * * − e − α τ R 0 0 0 * * * * Φ i 55 0 0 * * * * * Φ i 66 0 * * * * * * − α w m I ] < 0 , (8)

P 1 i − W i ≤ 0, i ∈ U u k , i ≠ j , (9)

P 1 i − W i ≤ 0, i ∈ U u k , i = j , (10)

where

Φ i 11 = P 2 i T A i + A i T P 2 i + Q + α P 1 i + ∑ j ∈ U k N λ i j ( P 1 j − W j ) ,

Φ i 12 = P 1 i − P 2 i T + A i T P 3 i ,

Φ i 13 = P 2 i T B i ,

Φ i 23 = P 3 i T B i ,

Φ i 22 = R + τ h M ( h M + τ ) S − P 3 i − P 3 i T ,

Φ i 33 = − ( 1 − h d ) e − α h M Q − h M e − α ( h M + τ ) S ,

Φ i 35 = h M e − α ( h M + τ ) S ,

Φ i 55 = − h M e − α ( h M + τ ) S ,

Φ i 66 = − τ e − α ( h M + τ ) S .

Then, the reachable set of the system (1) having the constraints (2) is bounded by a non-ellipsoid boundary ∩ i ∈ ℘ ℑ ( P 1 i ,1 ) , which ℑ ( P 1 i ) ( i ∈ ℘ ) is defined in (7).

Proof. We choose the following Lyapunov-Krasovskii functional candidate as follows:

V ( x t , t , r t ) = ∑ i = 1 4 V i ( x t , t , r t ) (11)

where

V 1 ( x t , t , r t ) = x T ( t ) P 1 r t x ( t ) = [ x T ( t ) x ˙ T ( t ) ] [ I 0 0 0 ] [ P 1 r t 0 P 2 r t P 3 r t ] [ x ( t ) x ˙ ( t ) ] ,

V 2 ( x t , t , r t ) = ∫ t − h ( t ) t e α ( s − t ) x T ( s ) Q x ( s ) d s ,

V 3 ( x t , t , r t ) = ∫ t − τ t e α ( s − t ) x ˙ T ( s ) R x ˙ ( s ) d s ,

V 4 ( x t , t , r t ) = τ h M ∫ − h M − τ 0 ∫ t + θ t e α ( s − t ) x ˙ T ( s ) S x ˙ ( s ) d s d θ ,

where r t ∈ ℘ , P 2 i , P 3 i , P 1 i > 0 ( i = 1,2, ⋯ , N ), Q > 0 , R > 0 , S > 0 and α > 0 are solutions of (10).

First, we show that V ( x t ) in (11) is a good L-K functional candidate. For t − h M ≤ s ≤ t , we have 0 < e − α h M ≤ e α ( s − t ) ≤ 1 . Furthermore, for t − τ ≤ s ≤ t , we have ∑ i = 2 4 V i ( t , x t , r t ) ≥ 0 .

Therefore, we get

{ V ( x t , t , r t ) = ∑ j = 1 4 V j ≥ V 1 ( x t , t , r t ) = x T ( t ) P 1 r t x ( t ) , V ( x t ) = 0 , when x ( θ ) = 0 , θ ∈ [ t − ρ * , t ] . (12)

Then, for given r t = i ∈ ℘ , P 1 r t = P 1 i , P 2 r t = P 2 i , P 3 r t = P 3 i and the weak infinitesimal operator L of the stochastic process x ( t ) along the evolution of V k ( x t , t , r t ) ( k = 1 , 2 , ⋯ , N ) are given as

L V 1 ( x t , t , i ) = 2 [ x T ( t ) x ˙ T ( t ) ] [ P 1 i P 2 i T 0 P 3 i T ] [ x ( t ) 0 ] + x T ( t ) [ ∑ j = 1 N λ i j P 1 j ] x ( t ) = 2 [ x T ( t ) x ˙ T ( t ) ] [ P 1 i P 2 i T 0 P 3 i T ] [ x ( t ) A i x ( t ) − x ˙ ( t ) + C x ˙ ( t − τ ) + B i x ( t − h ( t ) ) + D w ( t ) ] + x T ( t ) [ ∑ j = 1 N λ i j P 1 j ] x (t)

= x T ( t ) [ P 2 i T A i + A i T P 2 i ] x ( t ) + 2 x T ( t ) [ P 1 i − P 2 i T + A i T P 3 i ] x ˙ ( t ) + 2 x T ( t ) P 2 i T B i x ( t − h ( t ) ) + 2 x T ( t ) P 2 i T C i x ˙ ( t − τ ) + 2 x T ( t ) P 2 i T D i w ( t ) − x ˙ T ( t ) [ P 3 i T + P 3 i ] x ˙ ( t ) + 2 x ˙ T ( t ) [ P 3 i T B i x ( t − h ( t ) ) + 2 x ˙ T ( t ) P 3 i T C i x ( t − τ ) + 2 x ˙ T ( t ) P 3 i T D i w ( t ) + x T ( t ) ∑ j = 1 N λ i j P 1 j ] x ( t ) .

Taking into account the situation that the information of transition probabilities is not accessible completely, due to ∑ j = 1 N λ i j = 0 ( i ∈ ℘ ), the following equations hold for arbitrary appropriate matrices W i = W i T are satisfied

− x T ( t ) ∑ j = 1 N λ i j W i x ( t ) = 0 , ∀ i ∈ ℘ .

It is trivial to obtain the following equality:

L V 1 ( x t , t , i ) = x T ( t ) [ P 2 i T A i + A i T P 2 i ] x ( t ) + 2 x T ( t ) [ P 1 i − P 2 i T + A i T P 3 i ] x ˙ ( t ) + 2 x T ( t ) P 2 i T B i x ( t − h ( t ) ) + 2 x T ( t ) P 2 i T C i x ˙ ( t − τ ) + 2 x T ( t ) P 2 i T D i w ( t ) − x ˙ T ( t ) [ P 3 i T + P 3 i ] x ˙ ( t ) + 2 x ˙ T ( t ) P 3 i T B i x ( t − h ( t ) ) + 2 x ˙ T ( t ) P 3 i T C i x ˙ ( t − τ ) + 2 x ˙ T ( t ) P 3 i T D i w ( t ) + x T ( t ) [ ∑ j ∈ U k N λ i j ( P 1 j − W j ) ] x ( t ) + x T ( t ) [ ∑ j ∈ U u k N λ i j ( P 1 j − W j ) ] x ( t ) , (13)

L V 2 ( x t , t , i ) = x T ( t ) Q x ( t ) − ( 1 − h ˙ ( t ) ) e − α h ( t ) x T ( t − h ( t ) ) Q x ( t − h ( t ) ) − α ∫ t − h ( t ) t e α ( s − t ) x T ( s ) Q x ( s ) d s ≤ x T ( t ) Q x ( t ) − ( 1 − h ˙ M ) e − α h M x T ( t − h ( t ) ) Q x ( t − h ( t ) ) − α V 2 , (14)

L V 3 ( x t , t , i ) = x ˙ T ( t ) R x ˙ ( t ) − e − α τ x ˙ T ( t − τ ) R x ˙ ( t − τ ) − α ∫ t − τ t e α ( s − t ) x ˙ T ( s ) R x ˙ ( s ) d s = x ˙ T ( t ) R x ˙ ( t ) − e − α τ x ˙ T ( t − τ ) R x ˙ ( t − τ ) − α V 3 , (15)

L V 4 ( x t , t , i ) = τ h M ( h M + τ ) x ˙ T ( t ) S x ˙ ( t ) − τ h M ∫ t − h M − τ t e α ( s − t ) x ˙ T ( s ) S x ˙ ( s ) d s − α V 4 ≤ τ h M ( h M + τ ) x ˙ T ( t ) S x ˙ ( t ) − h M e − α ( h M + τ ) [ ∫ t − h M − τ t − h ( t ) x ˙ T ( s ) S x ˙ ( s ) d s + ∫ t − h ( t ) t x ˙ T ( s ) S x ˙ ( s ) d s ] − α V 4 ≤ τ h M ( h M + τ ) x ˙ T ( t ) S x ˙ ( t ) + h M e − α ( h M + τ ) [ x ( t − h M − τ ) x ( t − h ( t ) ) ] T [ − S S S − S ] ⋅ [ x ( t − h M − τ ) x ( t − h ( t ) ) ] − τ e − α ( h M + τ ) ( ∫ t − h ( t ) t x ˙ T ( s ) d s ) S ∫ t − h ( t ) t x ˙ ( s ) d s − α V 4 . (16)

Then substituting (13-16) into (12), we further have

L ( x t , t , i ) + α V ( x t , t , i ) − α w m w T ( t ) w ( t ) ≤ x T ( t ) [ P 2 i T A i + A i T P 2 i + Q + α P 1 i + ∑ j ∈ U k N λ i j ( P 1 j − W j ) ] x ( t ) + 2 x T ( t ) [ P 1 i − P 2 i T + A i T P 3 i ] x ˙ ( t ) + 2 x T ( t ) P 2 i T B i x ( t − h ( t ) ) + 2 x T ( t ) P 2 i T C i x ˙ ( t − τ ) + 2 x T ( t ) P 2 i T D i w ( t ) + 2 x ˙ T ( t ) [ R + τ h M ( h M + τ ) S − P 3 i − P 3 i T ] x ˙ (t)

+ 2 x ˙ T ( t ) P 3 i T B i x ( t − h ( t ) ) + 2 x ˙ T ( t ) P 3 i T C i x ˙ ( t − τ ) + 2 x ˙ T ( t ) P 3 i T D i w ( t ) − x T ( t − h ( t ) ) [ − ( 1 − h d ) e − α h M Q − h M e − α ( h M + τ ) S ] x ( t − h ( t ) ) + 2 h M x T ( t − h ( t ) ) e − α ( h M + τ ) S x ( t − h M − τ ) − e − α τ x ˙ T ( t − τ ) R x ˙ ( t − τ ) − h M e − α ( h M + τ ) x T ( t − h M − τ ) S x ( t − h M − τ )

− τ e − α ( h M + τ ) ( ∫ t − h ( t ) t x ˙ T ( s ) d s ) S ( ∫ t − h ( t ) t x ˙ ( s ) d s ) − α w m w T ( t ) w ( t ) + x T ( t ) [ ∑ j ∈ U u k N λ i j P 1 j ] x ( t ) = ξ T ( t ) Φ i ξ ( t ) + x T ( t ) [ ∑ j ∈ U u k N λ i j ( P 1 j − W j ) ] x ( t ) .

where Φ i is the same as defined in the Theorem 1 and ξ T ( t ) = [ x T ( t ) x ˙ T ( t ) x T ( t − h ( t ) ) x ˙ T ( t − τ ) x T ( t − h M − τ ) ∫ t − h ( t ) t x ˙ ( s ) d s T w T ( t ) ] . Thus, from matrix inequalities (10)-(11), we get

L V ( x t ) + α V ( x t ) − α w m 2 w ( t ) T w ( t ) ≤ 0 (17)

which means, by Lemma 1, that V ( x t , t , i ) = V 1 ( x t , t , i ) + V 2 ( x t , t , i ) + V 3 ( x t , t , i ) + V 4 ( x t , t , i ) ≤ 1 , and this results in V 1 ( x t , t , i ) = x T ( t ) P 1 i x ( t ) ≤ 1 for ∀ i ∈ ℘ , since V 2 ( x t , t , i ) + V 3 ( x t , t , i ) + V 4 ( x t , t , i ) ≥ 0 from (11).

So the reachable set of the system (1) is bounded by ellipsoid ℑ ( P 1 i ,1 ) ( i ∈ ℘ ) defined in (7), which implies that the reachable sets of the system (1) having the constraints (2) is bounded by a non-ellipsoid boundary ∩ i ∈ ℘ ℑ ( P 1 i ,1 ) . This completes the proof. £

Remark 1. It should be noted that the more the unknown elements there are in (4), the lower the maximum of time delay will be in Theorem 1. Actually, if all transition probabilities are unknown, the corresponding system can be viewed as a switched neutral Markovian jump system under arbitrary switching. Thus, the conditions obtained in Theorem 1 will cover the results for arbitrary switched neutral Markovian jump systems with disturbances. In that case, one can see the bounds of reachable sets in Theorem 1 become seriously conservative, for many constraints. Fortunately, we can use the Lyapunov functional method to analyze the bound of reachable sets for the neutral Markovian jump system under the assumption that all transition probabilities are not known.

For finding the bound of reachable sets for neutral Markovian jump systems with all transition probabilities are not known, one can construct the following Lyapunov functional

V ( x t , t , r t ) = [ x T ( t ) x ˙ T ( t ) ] [ I 0 0 0 ] [ P 1 0 P 2 P 3 ] [ x ( t ) x ˙ ( t ) ] + ∫ t − h ( t ) t e α ( s − t ) x T ( s ) Q x ( s ) d s + τ h M ∫ − h M − τ 0 ∫ t + θ t e α ( s − t ) x ˙ T ( s ) S x ˙ ( s ) d s d θ + ∫ t − τ t e α ( s − t ) x ˙ T ( s ) R x ˙ ( s ) d s .

Following a similar line as in proof of Theorem 1, we can obtain the following Theorem.

Theorem 2. Consider the Markov neutral system (1) with all elements unknown in transition rate matrix (4), if there exist symmetric matrices P 2 , P 3 , P 1 > 0 , Q > 0 , R > 0 , S > 0 and a scalar α > 0 satisfying the following matrix inequalities for i = 1,2, ⋯ , N

[ Φ i 11 Φ i 12 Φ i 13 P 2 i T C i 0 0 P 2 i T D i * Φ i 22 Φ i 23 P 3 i T C i 0 0 P 3 i T D i * * Φ i 33 0 Φ i 35 0 0 * * * − e − α τ R 0 0 0 * * * * Φ i 55 0 0 * * * * * Φ i 66 0 * * * * * * − α w m I ] < 0 , (18)

P 1 i − W i ≤ 0, i ∈ U u k , i ≠ j , (19)

P 1 i − W i ≤ 0 , i ∈ U u k , i = j , (20)

where

Φ i 11 = P 2 i T A i + A i T P 2 i + Q + α P 1 i ,

Φ i 12 = P 1 i − P 2 i T + A i T P 3 i ,

Φ i 13 = P 2 i T B i ,

Φ i 23 = P 3 i T B i ,

Φ i 22 = R + τ h M ( h M + τ ) S − P 3 i − P 3 i T ,

Φ i 33 = − ( 1 − h d ) e − α h M Q − h M e − α ( h M + τ ) S ,

Φ i 35 = h M e − α ( h M + τ ) S ,

Φ i 55 = − h M e − α ( h M + τ ) S ,

Φ i 66 = − τ e − α ( h M + τ ) S .

Then, the reachable sets of the system (1) having the constraints (2) is bounded by a non-ellipsoid boundary ∩ i ∈ ℘ ℑ ( P 1 i ,1 ) , which ℑ ( P 1 i ) ( i ∈ ℘ ) is defined in (7).

Remark 2. The solution for (8-10) or (18-20), if it exists, need not be unique. It is well-known [

minimize δ ¯ ( δ ¯ = 1 δ ) s .t . { ( a ) [ δ ¯ I I I P 1 i ] ≥ 0, i ∈ ℘ , ( b ) ( 8 - 10 ) or ( 18 - 20 ) . (21)

Remark 3. The matrix inequalities in Theorem 1 and Theorem 2 contain only one non-convex scalar α > 0 (for given h M and h d ), and these become LMI by fixing the scalar α . The feasibility check of a matrix inequality having only one non-convex scalar parameter is numerically tractable, and a local optimum value of α can be found by fminsearch.m.

In this section, a numerical example demonstrates the effectiveness of the approaches presented in this paper. Consider the neutral Markov jump systems with three operation modes whose state matrices are listed as follow:

x ˙ ( t ) − C ( t , r t ) x ˙ ( t − 0.1 ) = A ( t , r t ) x ( t ) + B ( t , r t ) x ( t − 0.5 ) + D ( t , r t ) w ( t ) , x ( t 0 + θ ) = φ ( θ ) , ∀ θ ∈ [ − ρ * , 0 ] , (22)

where

A 1 = [ − 2 − 1 0 − 2 ] , A 2 = [ − 3 0 0 − 2 ] , A 3 = [ − 2 0 − 1 − 1.5 ] , B 1 = [ − 1 0 − 1 − 2 ] , B 2 = [ − 2 0 − 1 − 1 ] , B 3 = [ − 2 0 0 − 2 ] ,

C 1 = [ − 1 0 − 1 − 2 ] , C 2 = [ − 2 0 − 1 − 1 ] , C 3 = [ − 2 0 0 − 2 ] , D 1 = [ − 0.15 0.15 ] , D 2 = [ − 0.14 0.35 ] , D 3 = [ − 0.2 0.3 ] .

The transition rate matrix Λ is considered as the following three cases.

Case 1. The transition rate matrix Λ is completely known, which is considered as

Λ = [ − 0.6 0.2 0.4 0.6 − 1 0.4 0.3 0.5 − 0.8 ] .

Case 2. The transition rate matrix Λ is partly known, which is considered as

Λ = [ − 0.6 0.2 0.4 ? − 1 ? ? ? ? ] .

Case 3. The transition rate matrix Λ is completely unknown, which is considered as

Λ = [ ? ? ? ? ? ? ? ? ? ] .

Firstly, by giving the transition probabilities Λ , a possible mode evolution of the neutral Markov jump system (22) is derived as shown in

By using theorem 2 and solving the optimization problem (20) in case 1, we can obtain the maximize δ = 0.61 when α = 0.1 , and the corresponding feasible matrices are given as P 11 = [ 7.2962 3.0026 3.0026 2.0132 ] , P 12 = [ 6.7864 2.6325 2.6325 1.7363 ] , P 13 = [ 6.4850 2.4905 2.4905 1.6669 ] . The reachable sets of the system (22) in case 1 is bounded by a intersection of three ellipsoids: ∩ i = 1 3 ℑ ( P 1 i , 1 ) , which is depicted in

By using theorem 1 and solving the optimization problem (22) in case 2, we can obtain the maximize δ = 0.61 when α = 0.1 , and the corresponding feasible matrices are given as P 11 = [ 6.5942 2.5361 2.5361 1.6849 ] , P 12 = [ 6.5941 2.5362 2.5362 1.6849 ] , P 13 = [ 6.5954 2.5366 2.5366 1.6851 ] . The reachable sets of the system (22) in case 2 is bounded by a intersection of three ellipsoids: ∩ i = 1 3 ℑ ( P 1 i , 1 ) , which is depicted in

By using theorem 3 and solving the optimization problem (22) in case 3, we can obtain the maximize δ = 0.65 when α = 0.1 , and the corresponding feasible matrices are given as P 11 = [ 8.1386 4.3002 4.3002 4.8168 ] , P 12 = [ 9.0931 2.6563 2.6563 2.5718 ] , P 13 = [ 6.8607 2.6735 2.6735 1.8013 ] . The reachable sets of the system (22) in case 3 is bounded by a intersection of three ellipsoids: ∩ i = 1 3 ℑ ( P 1 i , 1 ) , which is depicted in

In this paper, the problem of robust stability for a class of uncertain neutral systems with time-varying delays is investigated. Sufficient conditions are given in terms of linear matrix inequalities which can be easily solved by LMI Toolbox in Matlab. Numerical examples are given to indicate significant improvements over some existing results.

This research was supported by Science and Technology Foundation of Guizhou Province (No. LKM[

The authors declare no conflicts of interest regarding the publication of this paper.

Shen, C.C., Zhou, S.W. and Deng, H.Y. (2020) The No-Ellipsoidal Bound of Reachable Sets for Neutral Markovian Jump Systems with Disturbances. Journal of Applied Mathematics and Physics, 8, 799-813. https://doi.org/10.4236/jamp.2020.85062