A Lyapunov functional for a neutral system with a distributed time delay

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Original articles

A Lyapunov functional for a neutral system with a distributed time delay Jozef Duda AGH University of Science and Technology, Department of Automatics and Engineering in Biomedicine, Krakow, Poland Received 18 February 2012; received in revised form 18 March 2014; accepted 4 August 2015

Abstract In this paper a Lyapunov functional determination method for a linear neutral system with both lumped and distributed time delay is formulated and solved. A form of a Lyapunov functional is assumed and a computing method of its coefficients is given. The Lyapunov functional is constructed for a given time derivative which is calculated on a trajectory of a neutral system with both lumped and distributed time delay. The presented method gives analytical formulas for the Lyapunov functional coefficients. An example illustrating the application of discussed theory is presented. c 2015 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights ⃝ reserved. Keywords: Lyapunov functional; Time delay system; LTI system; Neutral system

1. Introduction The Lyapunov quadratic functionals are used to test the system stability, in computation of the critical delay values for time delay systems, in computation of the exponential estimates for the solutions of the time delay systems, in calculation of the robustness bounds for the uncertain time delay systems, in calculation of a quadratic performance index of quality in a parametric optimization process for the time delay systems. One constructs the Lyapunov functionals for a time delay system with its given time derivative. For the first time such Lyapunov functional was introduced by Repin [16] for a case of a retarded time delay linear system with one delay. Repin delivered also a determination procedure for the functional coefficients. Duda [1] used a Lyapunov functional, which was proposed by Repin, for the calculation of a quadratic performance index value in the parametric optimization process for the systems with a time delay of retarded type and extended the results to a case of a neutral time delay system in [2]. In the paper [3] was presented a Lyapunov functional determination method for a linear system with two lumped retarded type time delays in a general case with no-commensurate delays and a special case with commensurate delays in which a Lyapunov functional could be determined by solving a set of ordinary differential equations. In the paper [4] was introduced a Lyapunov functional determination method for a linear system with two delays both retarded and neutral type and in [5] for a linear neutral system with k-non-commensurate time delays. In the paper [6] was proposed a Lyapunov quadratic functional determination method for a linear system with both lumped and distributed time delay

E-mail address: [email protected]. http://dx.doi.org/10.1016/j.matcom.2015.08.001 c 2015 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights 0378-4754/⃝ reserved.

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and in [7] was stated such method for a system with a time-varying delay. Infante and Castelan’s [10] construction of a Lyapunov functional is based on a solution of a matrix differential-difference equation on a finite time interval. This solution satisfies the symmetry and boundary conditions. Kharitonov and Zhabko [14] extended the Infante and Castelan’s results and proposed a construction procedure for the quadratic functionals for the linear retarded type time delay systems which could be used for the robust stability analysis of the time delay systems. This functional was expressed by means of a Lyapunov matrix which depended on a fundamental matrix of a time delay system. Kharitonov [11] extended some basic results obtained for a case of the retarded type time delay systems to a case of the neutral type time delay systems, and in [12] to the neutral type time delay systems with a discrete and distributed delay. Kharitonov and Plischke [13] formulated the necessary and sufficient conditions for the existence and uniqueness of a Lyapunov matrix for a case of a retarded system with one delay. This paper deals with a Lyapunov functional determination method for a linear neutral system with both lumped and distributed time delay. The Lyapunov functional is constructed for a given time derivative which is calculated on a trajectory of a system with both lumped and distributed neutral type time delay. The presented method gives analytical formulas for the Lyapunov functional coefficients. The novelty of the paper lies in the extension of the results presented in [6] to a neutral system with both lumped and distributed time delay. To the best of author’s knowledge, such extension has not been reported in the literature. An example illustrating this method is also presented. The organization of the paper is as follows. In Section 2, the mathematical model of a linear neutral system with both lumped and distributed time delay is presented. In Section 3 the definition of a Lyapunov functional is given. In Section 4 main result of the paper is stated. An example is included and simulation results are given in Section 5. Section 6 concludes the paper. 2. A mathematical model of a linear neutral system with both lumped and distributed time delay Let us consider a linear neutral system with both lumped and distributed time delay, in which dynamics is described by the functional-differential equation (FDE)   0 d x(t − r ) d x(t)   −C = Ax(t) + Bx(t − r ) + Gx(t + θ )dθ  dt dt −r (2.1)   x(t0 ) = x0 x(t0 + θ ) = ϕ(θ ) for t ≥ t0 , r ≥ 0, A, B, C, G ∈ Rn×n , x(t) ∈ Rn , θ ∈ [−r, 0), ϕ ∈ W 1,2 ([−r, 0), Rn ) where W 1,2 ([−r, 0), Rn ) is a space of all absolutely continuous functions [−r, 0) → Rn with derivatives in L 2 ([−r, 0), Rn ) a space of Lebesgue square integrable functions on an interval [−r, 0) with values in Rn . The norm in W 1,2 ([−r, 0), Rn ) is defined by  2   0   dϕ(t)  ∥ϕ∥2W 1,2 = ∥ϕ(t)∥2Rn +  (2.2)  dt  n dt −r

R

where ∥ · ∥Rn is an arbitrary norm in Rn . Definition 1. We define a function xt ∈ W 1,2 ([−r, 0), Rn ) as shifted restrictions of x to an interval [t − r, t) by a formula xt (θ ) := x(t + θ )

for t ≥ t0 , θ ∈ [−r, 0).

(2.3)

Lemma 2. There holds the relationship ∂ xt (θ ) ∂ xt (θ ) = . ∂t ∂θ

(2.4)

Proof. xt (θ ) = x(t + θ )

for t ≥ t0 , θ ∈ [−r, 0)

J. Duda / Mathematics and Computers in Simulation (

∂ xt (θ ) ∂ x(t + θ ) ∂ x(ξ ) ∂ξ ∂ x(ξ ) = = = ∂t ∂t ∂ξ ∂t ∂ξ ∂ xt (θ ) ∂ x(t + θ ) ∂ x(ξ ) ∂ξ ∂ x(ξ ) = = = ∂θ ∂θ ∂ξ ∂θ ∂ξ

)

–

3

for ξ = t + θ for ξ = t + θ

hence ∂ xt (θ ) ∂ xt (θ ) = .  ∂t ∂θ Using the formula (2.3) we can write Eq. (2.1) in the form   0  d xt (−r ) d x(t)   − C = Ax(t) + Bx (−r ) + Gxt (θ )dθ t  dt dt −r  Rn  x(t0 ) = x0 ∈ 1,2   x t0 = ϕ ∈ W [−r, 0), Rn for t ≥ t0 . The norm of an initial value (x0 , ϕ) is given by  ∥ (x0 , ϕ) ∥ = ∥x0 ∥2Rn + ∥ϕ∥2W 1,2 .

(2.5)

(2.6)

The theorems of existence, continuous dependence and uniqueness of solutions of Eq. (2.5) are given in [8]. A solution of the functional-differential equation (2.5) with initial value (x0 , ϕ) or simply a solution through (x0 , ϕ) is an absolutely continuous function defined for t ≥ t0 − r with values in Rn . x(·; (x0 , ϕ)) ∈ W 1,2 ([t0 − r, ∞), Rn )

(2.7)

where W 1,2 ([t0 − r, ∞), Rn ) is a space of all absolutely continuous functions with derivatives in a space of Lebesgue square integrable functions on interval [t0 − r, ∞) with values in Rn . The function xt (·; (x0 , ϕ)) ∈ W 1,2 ([−r, 0), Rn ) is a shifted restriction of x(·; (x0 , ϕ)) to an interval [t − r, t) and is given by a formula xt (θ ; (x0 , ϕ)) := x(t + θ ; (x0 , ϕ)) for t ≥ t0 , θ ∈ [−r, 0)

(2.8)

xt0 (·; (x0 , ϕ)) = ϕ.

(2.9)

Definition 3. The zero solution of (2.5) is stable if for any ε > 0 there is a δ > 0 such that ∥ (x0 , ϕ) ∥ < δ implies ∥xt (·; (x0 , ϕ))∥ < ε for t ≥ t0 . Definition 4. The zero solution of (2.5) is asymptotically stable if ∥xt (·; (x0 , ϕ))∥ → 0

as t → ∞.

Definition 5. The zero solution of (2.5) is exponentially stable if there exist an η > 0 and a positive constant M such that ∥xt (·; (x0 , ϕ))∥ ≤ Me−ηt ∥ (x0 , ϕ) ∥

for t ≥ t0 .

(2.10)

Definition 6. The difference equation associated with (2.1) and (2.5) is given by a term x(t) = C x(t − r )

for t ≥ t0 .

(2.11)

Definition 7. The spectrum σ (C) is a set of complex numbers λ for which a matrix λI − C is not invertible. σ (C) = {λ ∈ C : det (λI − C) = 0} . The arbitrary eigenvalue of the matrix C will be denoted as λ(C).

(2.12)

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Definition 8. The spectral radius of a matrix C is given by a form γ (C) = sup {|λ| : λ ∈ σ (C)} .

(2.13)

The eigenvalues of the neutral equation (2.5) for large modulus are asymptotically equal to the eigenvalues of the difference equation (2.11). The asymptotic stability of the difference equation (2.11) is the necessary condition of the asymptotic stability of the neutral equation (2.5). According to the Theorem 9.6.1 in [9] the difference equation (2.11) is stable when the spectral radius γ (C) of the matrix C fulfills the condition γ (C) < 1.

(2.14)

We assume that the matrix C is not singular and fulfills the condition (2.14). We introduce a new function y, defined by a term y(t) = x(t) − C xt (−r )

for t ≥ t0 .

Thus Eq. (2.5) takes a form  0  dy(t)   Gxt (θ )dθ = Ay(t) + + B) x (−r ) + (AC t    dt −r y(t) = x(t) − C xt (−r )   1,2 n  x   t0 = ϕ ∈ W ([−r, 0), R ) y(t0 ) = y0 for t ≥ t0 where y0 = x0 − Cϕ(−r ) A state of a system (2.16) is a vector   y(t) S(t) = for t ≥ t0 . xt

(2.15)

(2.16)

(2.17)

A state space is defined by a formula X = Rn × W 1,2 ([−r, 0), Rn ).

(2.18)

The norm in the state space X is defined by ∥S(t)∥2X = ∥y(t)∥2Rn + ∥xt ∥2W 1,2

for t ≥ t0 .

(2.19)

The controllability of the systems with time delay is presented in [15]. 3. A Lyapunov functional Definition 9. A functional V : X → R is positive definite if and only if it is continuous and V (x) > 0 for x ̸= 0 and V (0) = 0. A functional V : X → R is negative definite if and only if it is continuous and V (x) < 0 for x ̸= 0 and V (0) = 0. Definition 10. We define a time derivative of the functional V (y(t), xt ) at (y(t0 ), ϕ) on a trajectory of a system (2.16) by the formula   d V (y(t0 ), ϕ) 1  := lim sup V y (t0 + h) , xt0 +h − V (y(t0 ), ϕ) . dt h h→0 Definition 11. We say that V : X → R is a Lyapunov functional if 1. V is positive definite. 2. V is differentiable. 3. A time derivative of V on a trajectory of the system (2.16) is negative definite.

(3.1)

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The existence of the Lyapunov functional for the system (2.16) is a sufficient condition for asymptotic stability of its zero solution. It means that y(t) → 0 f or t → ∞. The condition (2.14) implies the stability of the difference equation (2.11) and therefore x(t) → 0 for t → ∞. When the system (2.16) is asymptotically stable  ∞ d V (y(t), xt ) dt = lim V (y(t), xt ) − lim V (y(t), xt ) t→∞ t→t0 dt t0 = V ( lim (y(t), xt )) − V ( lim (y(t), xt )) t→∞

t→t0

= V (0) − V (y(t0 ), ϕ) = −V (y(t0 ), ϕ).

(3.2)

We assume that the time derivative of the Lyapunov functional V is given as a quadratic form d V (y(t), xt ) ≡ −y T (t)W y(t) for t ≥ t0 dt

(3.3)

where W ∈ Rn×n is a positive definite matrix. Taking (3.2) and (3.3) into account we obtain a relationship  ∞ J= y T (t)W y(t)dt = V (y0 , ϕ).

(3.4)

t0

Corollary 12. If we construct a Lyapunov functional such that its time derivative computed on the trajectory of the system (2.16) will be given as a quadratic form (3.3) we can not only investigate the system (2.16) stability but also we can calculate a value of a square indicator of quality (3.4) of the parametric optimization problem. To calculate the value of the performance index (3.4), which is equal to the value of the Lyapunov functional at the initial state of the system (2.16), we need a mathematical formula of that functional. 4. Main result. Determination of the Lyapunov functional Let us consider a quadratic functional on X , given by a formula  0  0 T T V (y(t), xt ) = y (t)M y(t) + y (t)N (θ )xt (θ )dθ + −r

−r

0

−r

xtT (θ )L(θ, σ )xt (σ )dσ dθ.

(4.1)

M = M T ∈ Rn×n ; N ∈ C 1 ([−r, 0], Rn×n ); L ∈ C 1 (Ω , Rn×n ); Ω = {(θ, σ ) : θ ∈ [−r, 0], σ ∈ [−r, 0]}. C 1 is a space of continuous functions with a continuous derivative. Conjecture 13. We introduce a procedure of determination of the functional (4.1) coefficients M, N , L to obtain the Lyapunov functional. We compute the time derivative of the functional (4.1) on the trajectory of the system (2.16). This time derivative is defined by the formula (3.1). We take the following procedure. We compute the time derivative of each term of the ∂ xt (θ) right-side of the formula (4.1) and we substitute in place of dy(t) dt and ∂t the following terms dy(t) = Ay(t) + (AC + B) xt (−r ) + dt



0

Gxt (θ )dθ

(4.2)

−r

∂ xt (θ ) ∂ xt (θ ) = . ∂t ∂θ After some calculations we obtain   d V (y(t), xt ) N (0) + N T (0) = y T (t) A T M + M A + y(t) dt 2 + y T (t) [2M (AC + B) + N (0)C − N (−r )] xt (−r )

(4.3)

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 +

−r 0

 +

−r

)

–

 d N (θ ) T T T y (t) 2M G + A N (θ ) − + L(0, θ ) + L (θ, 0) xt (θ )dθ dθ 

xtT (−r )[(AC + B)T N (θ ) + C T L(0, θ ) − L(−r, θ ) + C T L T (θ, 0) +

− L (θ, −r )]xt (θ )dθ + T

0



−r



0

−r

xtT (θ )



 ∂ L(θ, σ ) ∂ L(θ, σ ) xt (σ )dσ dθ. G N (σ ) − − ∂θ ∂σ T

(4.4)

The time derivative of the Lyapunov functional should be negative definite, therefore we identify the coefficients of the functional (4.1) assuming that the time derivative (4.4) satisfies the relationship (3.3). From relations (4.4) and (3.3) we attain the set of equations N (0) + N T (0) = −W 2 2M (AC + B) + N (0)C − N (−r ) = 0 d N (θ ) + L(0, θ ) + L T (θ, 0) = 0 2M G + A T N (θ ) − dθ AT M + M A +

(4.5) (4.6) (4.7)

(AC + B)T N (θ ) + C T L(0, θ ) + C T L T (θ, 0) − L(−r, θ ) − L T (θ, −r ) = 0 ∂ L(θ, σ ) ∂ L(θ, σ ) + = G T N (σ ) ∂θ ∂σ

(4.8) (4.9)

for θ, σ ∈ [−r, 0]. Let us consider a solution of Eq. (4.9) as below  σ L(θ, σ ) = Φ(θ − σ ) + Φ T (σ − θ ) + G T N (ξ )dξ

(4.10)

0

where Φ ∈ C 1 ([−r, r ]). From Eq. (4.10) it attains L(0, θ ) + L (θ, 0) = 2Φ (θ ) + 2Φ(−θ ) + T

T



θ

G T N (ξ )dξ

(4.11)

0

and L(−r, θ ) + L (θ, −r ) = 2Φ(−r − θ ) + 2Φ (θ + r ) + T

T

θ



G N (ξ )dξ + T

0



−r

N T (ξ )Gdξ.

(4.12)

0

We put a term (4.11) into Eq. (4.7) and we obtain a formula d N (θ ) + 2Φ T (θ ) + 2Φ(−θ ) + 2M G + A N (θ ) − dθ T



θ

G T N (ξ )dξ = 0.

Now we put the terms (4.11) and (4.12) into Eq. (4.8) and we get a relationship     C T A T + B T N (θ ) + C T 2Φ T (θ ) + 2Φ(−θ ) − 2Φ(−r − θ ) − 2Φ T (θ + r )  θ  −r T T + (C − I ) · G N (ξ )dξ − N T (ξ )Gdξ = 0. 0

(4.13)

0

(4.14)

0

From Eq. (4.13) we attain terms d N T (θ ) 2Φ(θ ) + 2Φ (−θ ) = − N T (θ )A − 2G T M − dθ T

 0

θ

N T (ξ )Gdξ

(4.15)

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and 2Φ(−θ − r ) + 2Φ T (θ + r ) = −

d N T (−θ − r ) − N T (−θ − r ) − 2G T M − dθ



−θ −r

N T (ξ )Gdξ.

(4.16)

0

We put the terms (4.15) and (4.16) into Eq. (4.14) and after some computations we obtain a formula  θ d N (θ ) d N T (−θ − r ) CT + = −B T N (θ ) − N T (−θ − r )A + G T N (ξ )dξ dθ dθ 0  θ + N T (−ξ − r )Gdξ + 2C T M G − 2G T M.

(4.17)

0

In computations we used a relationship  −θ −r  θ T − N (ξ )Gdξ = N T (−ξ − r )Gdξ. 0

−r

We introduce a substitution d N (θ ) = Q(θ ) for θ ∈ [−r, 0]. dθ

(4.18)

We compute a derivative of a term N T (−θ − r ) d N T (−θ − r ) = −Q T (−θ − r ) dθ We can write Eq. (4.17) in a form

for θ ∈ [−r, 0].

C T Q(θ ) − Q T (−θ − r ) = −B T N (θ ) − N T (−θ − r )A +

(4.19)

θ



G T N (ξ )dξ

0 θ

 +

N T (−ξ − r )Gdξ + 2C T M G − 2G T M.

(4.20)

0

Taking into account the formulas (4.18) and (4.19) we calculate a derivative of both sides of Eq. (4.20) d Q(θ ) d Q T (−θ − r ) − = −B T Q(θ ) + Q T (−θ − r )A + G T N (θ ) + N T (−θ − r )G (4.21) dθ dθ for θ ∈ [−r, 0]. We transpose both sides of Eq. (4.21) and then we change a variable putting θ = −ξ − r and dθ = −dξ . In this way we obtain CT

d Q(ξ ) d Q T (−ξ − r ) − C = −Q T (−ξ − r )B + A T T (ξ ) + N T (−ξ − r )G + G T N (ξ ) dξ dξ

(4.22)

for ξ ∈ [−r, 0]. The sense of a formula (4.22) does not depend on the notation of the variable, so we can use symbol θ instead of ξ . We introduce new functions K (θ ) = N T (−θ − r ) for θ ∈ [−r, 0]

(4.23)

P(θ ) = Q T (−θ − r )

(4.24)

and for θ ∈ [−r, 0].

The formula (4.20) takes a form C T T (θ ) − P(θ ) = −B T N (θ ) − K (θ )A +

 0

for θ ∈ [−r, 0].

θ

G T N (ξ )dξ +

 0

θ

K (ξ )Gdξ + 2C T M G − 2G T M

(4.25)

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From definition (4.23) and Eqs. (4.19) and (4.24) it results that d K (θ ) = −P(θ ) for θ ∈ [−r, 0]. dθ Using the definitions (4.23) and (4.24) we can rewrite the relationships (4.21) and (4.22) in a form

(4.26)

d Q(θ ) d P(θ ) − = −B T Q(θ ) + P(θ )A + G T N (θ ) + K (θ )G (4.27) dθ dθ d Q(θ ) d P(θ ) − C = −P(θ )B + A T Q(θ ) + K (θ )G + G T N (θ ) (4.28) dθ dθ for θ ∈ [−r, 0]. We reshape a set of Eqs. (4.27) and (4.28) then we add to them Eqs. (4.18) and (4.26). In this way we obtain a differential equations set  d N (θ )  = Q(θ )    dθ    d K (θ )   = −P(θ )    dθ    d Q(θ ) d Q(θ ) − CT C = G T N (θ ) − G T N (θ )C + K (θ )G (I − C) + A T Q(θ ) (4.29) dθ dθ   T  + B Q(θ )C − P(θ ) (B + AC)       d P(θ ) d P(θ )    − CT C = − I − C T G T N (θ ) − K (θ )G + C T K (θ )G   dθ  dθ     + B T + C T A T Q(θ ) − P(θ )A − C T P(θ )B CT

for θ ∈ [−r, 0] with initial conditions N (−r ), K (−r ), Q(−r ), P(−r ). From formulas (4.23) and (4.24) it implies that a solution of Eq. (4.29) satisfies the relationships K (θ )|θ =− r2 = N T (θ )|θ =− r2

(4.30)

P(θ )|θ =− r2 = Q T (θ )|θ=− r2 .

(4.31)

We determine a value of the initial conditions of a system (4.29) to obtain a solution of a set of the differential equations (4.29) on an interval [−r, 0]. From formulas (4.23) and (4.24) it implies that there exist the connections between initial conditions N (0) = K T (−r )

K (0) = N T (−r )

Q(0) = P T (−r )

P(0) = Q T (−r ).

(4.32)

We calculate a value of a formula (4.25) for θ = 0. Taking into account the relationships (4.32) after transposition we obtain P(−r )C − Q(−r ) + K (−r )B + A T N (−r ) − 2G T MC + 2M G = 0.

(4.33)

Taking into consideration the conditions (4.32) we rewrite Eqs. (4.5) and (4.6) K (−r ) + K T (−r ) = −W 2

(4.34)

2M (AC + B) + K T (−r )C − N (−r ) = 0.

(4.35)

AT M + M A +

A set of Eqs. (4.33)–(4.35) and the terms (4.30) and (4.31) enable us to compute an initial condition value of the differential equations (4.29) and the matrix M. This equations composition constitutes an algebraic equation set with unknown N (−r ), K (−r ), Q(−r ), P(−r ), M. Taking into account a term (4.18) we can write formula (4.15) in a form  1 T 1 T 1 θ T T T Φ(θ ) + Φ (−θ) = Q (θ ) − N (θ )A − G M − N (ξ )Gdξ. (4.36) 2 2 2 0

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According to formulas (4.10) and (4.36) we attain a term   σ 1 1 1 θ−σ T N (ξ )Gdξ + G T N (ξ )dξ − G T M. L(θ, σ ) = Q T (θ − σ ) − N T (θ − σ )A − 2 2 2 0 0

9

(4.37)

In this way we obtained all coefficients of the functional (4.1). These coefficients depend on the matrices A, B, C, G of the system (2.16). The time derivative of the functional (4.1) is negative definite. When the matrices M, N (θ ) and L (θ, σ ) for θ ∈ [−r, 0]; σ ∈ [θ, 0] are positive definite the functional (4.1) becomes the Lyapunov functional. 5. The example Let us consider a system described by an equation   0 d x(t) d x(t − r )   − c = ax(t) + bx(t − r ) + gx(t + θ )dθ  dt dt −r   x(t0 ) = x0 x(t0 + θ ) = ϕ(θ )

(5.1)

for t ≥ t0 , θ ∈ [−r, 0), x(t) ∈ R, ϕ ∈ W 1,2 ([−r, 0), R), a, b, c, g ∈ R, c ̸= 0, |c| < 1, r ≥ 0. We introduce a new variable y(t) = x(t) − cx(t − r )

for t ≥ t0 .

(5.2)

Formula (5.1) takes a form   0 dy(t)   = ay(t) + + b) x(t − r ) + gx(t + θ )dθ (ac    dt −r y(t) = x(t) − cx(t − r )      y(t0 ) = x0 − cϕ(−r ) x(t0 + θ ) = ϕ(θ ) for t ≥ t0 , θ ∈ [−r, 0), y(t) ∈ R, ϕ ∈ W 1,2 ([−r, 0), R), a, b, c, g ∈ R, c ̸= 0, |c| < 1, r ≥ 0. A Lyapunov functional is defined by a formula  0  0 0 V (y(t), xt ) = my(t) + y(t)n(θ )xt (θ )dθ + l(θ, σ )xt (θ )xt (σ )dσ dθ. −r

−r

(5.3)

(5.4)

−r

We write a set of the differential equations (4.29) for a system (5.3)  dn(θ )    0 0 1 0  dθ      0 0 0 −1  n(θ )  dk(θ )         g a + bc b + ac  g  dθ     k(θ )   = −     2 2 dq(θ ) q(θ )   1+c  1+c 1−c 1−c      p(θ )  dθ  g g b + ac a + bc   − − − d p(θ ) 1+c 1 + c 1 − c2 1 − c2 dθ for θ ∈ [−r, 0]. A solution of a differential equations (5.5) system is given by a term     n(θ ) n(−r )  k(θ )   k(−r )       q(θ )  = Ψ (θ + r )  q(−r )  p(θ ) p(−r ) for θ ∈ [−r, 0] where Ψ (θ ) is a fundamental matrix of a system (5.5).

(5.5)

(5.6)

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Fig. 5.1. Parameters n(θ ), k(θ ), q(θ ), p(θ ).

A coefficient m and an initial condition value of a system (5.5) one obtains by solving of an algebraic equation set  2am + k(−r ) = −w      2 (ac + b) m − n(−r ) + ck(−r ) = 0 2g (1 − c) m + an(−r ) + bk(−r ) − q(−r ) + cp(−r ) = 0 (5.7)   r = k(θ )| r n(θ )|  θ=− θ=−  2 2   q(θ )|θ=− r2 = p(θ )|θ=− r2 where w is a positive real number. Having solution of Eq. (5.5) we can obtain a coefficient l(θ, σ ) l(θ, σ ) =

1 1 1 q(θ − σ ) − an(θ − σ ) − gm − 2 2 2

 0

θ −σ

gn(ξ )dξ +



σ

gn(ξ )dξ.

(5.8)

0

Fig. 5.1 shows the functions n(θ ), k(θ ), q(θ ), p(θ ) graphs and an m value attained by means of the matlab code for a given parameters a, b, c, g, w value of a system (5.1). 6. Conclusions The paper presents a Lyapunov functional determination procedure for a linear neutral system with both lumped and distributed time delay. A neutral system is described by Eq. (2.5). In the paper a form of a Lyapunov functional is assumed and a computing method of its coefficients is given. Presented method allows achieving the analytical formulas on the Lyapunov functional coefficients. This article extends the results presented in [6] to a neutral system with both lumped and distributed time delay. To find the Lyapunov functional coefficients we need to solve a set of differential equations (4.29). The dimension of that set is equal to 4n 2 because y(t) ∈ Rn and therefore we often use numerical methods to obtain the fundamental matrix of that system. To this end can be used Vandermonde matrices [17,18]. Acknowledgments The author wishes to thank the editors and the reviewers for their suggestions, which have improved the quality of the paper.

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