In the paper using algebraic methods and method of steps new sufficient conditions for positivity of linear
continuous-time dynamical systems with variable coefficients and distributed delay and several constant point delays are established. Neutral linear dynamical systems with several constant point delays are also discussed. As a special case linear dynamical systems without delays are also considered. Finally, illustrative example is given

Kalinowski, Józef

Klamka, Jerzy

Fasciculi Mathematici

English

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             Title: Positive dynamical systems with delays  Author: Józef Kalinowski, Jerzy Klamka  Citation style: Kalinowski Józef, Klamka Jerzy. (2005). Positive dynamical systems with delays. "Fasciculi Mathematici" (Nr 36 (2005), s. 65-71). F A S C I C U L I M A T H E M A T I C INr 36 2005Jo´zef Kalinowski and Jerzy KlamkaPOSITIVE DYNAMICAL SYSTEMS WITH DELAYSAbstract: In the paper using algebraic methods and methodof steps new sufficient conditions for positivity of linearcontinuous-time dynamical systems with variable coefficients anddistributed delay and several constant point delays are estab-lished. Neutral linear dynamical systems with several constantpoint delays are also discussed. As a special case linear dynamicalsystems without delays are also considered. Finally, illustrativeexample is given.Key words: linear systems, continuous time systems, positivesystems with delays.1. IntroductionIn the paper using algebraic methods and method of steps and resultsgiven in monograph [2] and paper [4] sufficient conditions for positivity oflinear continuous-time systems with distributed delay and several constantpoint delays and variable coefficients are formulated and proved. Roughlyspeaking in the positive dynamical systems trajectory of the system andthe initial conditions are non-negative. The results obtained in the presentpaper generalize to the case of linear dynamical systems with time-variablecoefficients and different types of delays, positivity conditions given in themonograph [2] for linear dynamical systems without delays and constantcoefficients, and in the paper [4] for linear dynamical systems with variablecoefficients and without delays.Proof of the main result is based on integral formula of the solution andmethod of steps for the differential state equation. Generally it is shown,that there are important relationships between Metzler matrix and posi-tivity conditions for linear dynamical systems with delays. Moreover, itshould be pointed out that the presented results can be extended to covermore general class of dynamical systems with delays e.g., dynamical systemswith Lebesque-Stieltjes integral. Finally, simple numerical example, whichillustrates theoretical considerations is presented.66 J. Kalinowski, J. Klamka2. PreliminariesLet Rn×m be the set of n×m real matrices. The set of n×m real matriceswith non-negative entries will be denoted by Rn×m+ and Rn+ := Rn×1+ .Consider the continuous-time linear dynamical system with variable co-efficients and distributed delay and several constant point delays in the statevariables described by the following vector functional differential nonhomo-geneous equation(1) x′(t) =M∑k=0Ak(t)x(t− hk) +∫ 0−hMA(s)x(t+ s)ds+ f(t),defined for t ∈ [t0,∞) a.e., where x(t) ∈ Rn, Ak(t) ∈ Rn×n, k = 0, 1, . . . ,M ,the entries akij ∈ L2loc([t0,∞),R), i, j = 1, 2, . . . , n and the function f ∈L1loc([to,∞),Rn), 0 = h0 < h1 < · · · < hM are constant point delays, A(s)for s ∈ [−hM , 0] is n × n dimensional matrix with continuous entries aij ∈C([−hM , 0],R) for i, j = 1, 2, . . . , n.It is well known (see e.g. [1]), that for the dynamical system with delays(1) it is convenient to introduce the following notation: xt(s) = x(t+ s) fors ∈ [−hM , 0].In order to solve vector functional differential equation (1) it is necessaryto give initial condition (x(t0), xt0) ∈ Rn × L2([−hM , 0], Rn).From the form of the vector functional differential equation (1) it immedi-ately follows, that for a given initial condition (x(t0), xt0) and given functionf(t) the unique solution x(t) of the equation (1) is absolutely continuousvector function for t ∈ [t0,∞).In the sequel we shall use the following basic definitions of positive dy-namical systems, taken from monograph [2].Definition 1. ([2]). The dynamical system (1) is called positive if for allinitial conditions (x(t0), xt0) ∈ Rn+×L2([−hM , 0],Rn+) and arbitrary functionf ∈ L1loc([t0,∞),Rn+), we have x(t) ∈ Rn+ for t ≥ t0.Definition 2. ([2]). A matrix M ∈ Rn×n is called the Metzler matrix ifits all off-diagonal entries are non-negative.It is easy to show (see e.g. monograph [2]) that exp(Mt) ∈ Rn×n+ , if andonly if M is a Metzler matrix.In the sequel we shall also consider a special case of the dynamical system(1), i.e., dynamical system without delays described by the following linearvector differential equation with variable coefficients(2) x′(t) = A0(t)x(t) + f(t),defined for t ≥ t0.Positive dynamical systems with delays 67It should be pointed out, that for linear dynamical system (2) with con-stant coefficients (i.e., A0(t) = A0) without loss of generality we can takethe initial time t0 = 0 (see e.g. [3]).It is well known that the solution x(t) of vector differential equation (2)with a given initial condition x(t0) ∈ Rn and given function f ∈ L1loc([t0,∞),Rn) has the following form (see e.g. [3])(3) x(t) = F (t, t0)x(t0) +∫ tt0F (t, s)f(s)dsfor t ≥ t0, where F (t, s) is so called state transition matrix [3] given byF (t, t0) = In +∫ tt0A0(τ)dτ +∫ tt0A0(τ)∫ τt0A0(s)dsdτ + . . .In the special case, when the matrix A0(t) satisfies the following Lappo-Danilewski additional assumptionA0(t) ·∫ tt0A0(τ)dτ =∫ tt0A0(τ)dτ ·A0(t) for t ∈ [t0,∞)the transition matrix F (t, t0) is given byF (t, t0) = exp(∫ tt0A0(τ)dτ).It should be mentioned, that state transition matrix F (t, τ) is nonsingularfor every t, τ ∈ R.Moreover, it should be also pointed out, that for the case whenA0(t) = A0we have (see [2])F (t, s) = exp(A0(t− s)).Now, we shall recall two lemmas concerning the relationships betweenMetzler matrix and positive dynamical systems without delays.Lemma 1. ([4]). Let us assume that A0(t) is a Metzler matrix for everyt ≥ t0. Then the system (2) has the state transition matrix F (t, t0) ∈ Rn×n+for every t ≥ t0.Lemma 2. ([4]). The system (2) with f ∈ L1loc([t0,∞),Rn+) is positive,if A0(t) is a Metzler matrix for every t ≥ t0.In the next sections integral formula (3) will be used in the proof of themain results. Let us observe, that for positive linear dynamical systemswithout delays we have x(t) ∈ Rn+ for t ≥ t0.68 J. Kalinowski, J. Klamka3. Main ResultsIn this section we shall formulate and prove sufficient condition for posi-tivity of the dynamical systems with delays (1).Theorem 1. The dynamical system (1) is positive if1. A0(t) is a Metzler matrix for every t ≥ t0,2. entries akij ∈ L2loc([t0,∞),R+), k = 1, 2, . . . ,M , i, j = 1, 2, . . . , n,3. f ∈ L1loc([t0,∞),Rn+),4. entries aij ∈ C([−hM , 0],R+) for all i, j = 1, 2, . . . , n.Proof. Proof of the theorem is based on the method of steps. It is wellknown that the solution x(t) of the functional differential equation (1) for agiven initial condition (x(t0), xt0) ∈ Rn+×L1([−hM , 0],Rn+) has the followingform [1], for t ∈ (t0, t0 + h1]x(t) = F (t, t0)x(t0)+∫ tt0F (t, s)[M∑k=0Ak(s)xt0(s− hk) +∫ 0−hMA(τ)x(τ + s)dτ + f(s)]ds.Let the symbol W 1,2([t0, t0 + h1],Rn+) denotes the Sobolev space of non-negative vector functions defined in the interval [t0, t0 + h1], which havesquare integrable first order derivative [3]. Since A0(t) is a Metzler matrixfor t ∈ [t0, t0 + h1] andM∑k=0Ak(s)x0(s− hk) +∫ 0−hMA(τ)x(τ + s)dτ + f(s) ∈W 1,2([t0, t0 + h1],Rn+)then x(t) ∈ Rn+ for t ∈ [t0, t0 + h1]. Hence x(t0 + h1) ∈ Rn+.Similarly, using the method of steps, it can be shown that for any k =2, 3, . . . ,M , we have x(t0+hk) ∈ Rn+. It directly follows from the equalities:a) for t ∈ (t0 + h1, t0 + h2]x(t) = F (t, t0 + h1)x(t0 + h1)+∫ tt0+h1F (t, s)[M∑k=0Ak(s)xt0+h1(s− hk)+∫ 0−hMA(τ)x(τ + s)dτ + f(s)]ds,Positive dynamical systems with delays 69b) for t ∈ (t0 + hk, t0 + hk+1],x(t) = F (t, t0 + hk)x(t0 + hk)+∫ tt0+hkF (t, s)[M∑k=0Ak(s)xt0+hk(s− hk)+∫ 0−hMA(τ)x(τ + s)dτ + f(s)]ds,c) for t ∈ (t0 + hM ,∞),x(t) = F (t, t0 + hM )x(t0 + hM )+∫ tt0+hMF (t, s)[M∑k=0Ak(s)xt0+hM (s− hk)+∫ 0−hMA(τ)x(τ + s)dτ + f(s)]ds.Therefore, x(t) ∈ Rn+ for t ≥ t0 and Theorem 1 is proved. Now, let us discuss special case of dynamical system (1) i.e., linear dy-namical system without distributed delay but with only several point delaysdescribed by the following linear vector differential equation(4) x′(t) =M∑k=0Ak(t)x(t− hk) + f(t).Corollary 1. The dynamical system (4) is positive if1. A0(t) is a Metzler matrix for every t ≥ t0,2. akij ∈ L2loc([t0,∞),R+), k = 1, 2, . . . ,M , i, j = 1, 2, . . . , n,3. f ∈ L1loc([t0,∞),Rn+).From Theorem 1 it directly follows sufficient condition for positivity ofthe linear dynamical system with variable coefficients and without delays.Corollary 2. ([3]). The dynamical system without delays (2) is positiveif1. A0(t) is a Metzler matrix for every t ≥ t0,2. f ∈ L1loc([t0,∞),Rn+).The next corollary proved in the monograph [2] gives necessary and suffi-cient condition for positivity of the linear dynamical systems without delaysand with constant coefficients.70 J. Kalinowski, J. KlamkaCorollary 3. ([2]). The dynamical system without delays (2) and withconstant coefficients (i.e., A0(t) = A0) is positive if and only if1. A0 is a Metzler matrix,2. f ∈ L1loc([0,∞),Rn+).Finally, let us consider linear vector neutral differential equation withvariable coefficients and several constant point delays of the following form(5) x′(t) =M∑k=0Ak(t)x(t− hk) +M∑k=1Ck(t)x′(t− hk) + f(t),where Ck(t), k = 1, 2, . . . ,M are n × n dimensional matrices with entriesckij ∈ C1([−hM , 0],R). For dynamical system (5) the initial condition xt0 ∈C1([−hM , 0],Rn+).Since x(t), t > 0, depends on the past values of the derivatives x′(t−hk),then simple observation shows, that in this case even nonnegativity of allmatrices Ck(t), k = 1, 2, . . . ,M does not ensure positivity of the solutionx(t) for t > 0.The next simple numerical example illustrates the essential role of theassumption 2 in Theorem 1.Example 1. For n = 2, M = 1, t0 = 0 and h1 = 1 consider thehomogeneous linear delay system (1) with Metzler matrices A0 and A1 ofthe following formx′(t) =[ −1 0a −1]x(t) +[ −5 0b c]x(t− 1) for t ≥ 0,where a > 0, b > 0 and c ∈ R are constants.The initial function x0(t) and initial vector x0 are given byx0(t) =[11]for t ∈ [−1, 0]; x0 =[11].Let us observe, that variable x1(t) = e−t − 5 for t ≥ 0 and x1(1) < 0.Then, dynamical system (1) is not positive. Hence, the assumption, that A1is a Metzler matrix does not ensure positivity of the system (1). However,it should be pointed out, that for positivity of the dynamical system (1) itis necessary to assume A1 ∈ R2×2+ .Positive dynamical systems with delays 714. ConclusionsIn the paper using algebraic methods and method of steps sufficient condi-tions for positivity of linear continuous-time systems with distributed delayand several constant point delays and variable coefficients have been formu-lated and proved. The results obtained in the present paper generalize to thecase of dynamical systems with time-variable coefficients and different typesof delays, positivity conditions given in the monograph [2] and in the paper[4]. Monograph [2] contains positivity results for linear dynamical systemswithout delays and constant coefficients and paper [4] discusses positivityproperty for linear dynamical system with variable coefficients and withoutdelays.Finally, it should be pointed out that using quite similar algebraic meth-ods based on Metzler matrix and generalized permutation matrix it is pos-sible to formulate and to prove controllability and reachability results (seee.g., [3] for more details) for linear positive dynamical systems with severalpoint delays in control or in the state variables.Acknowledgements: This work of second author was partially sup-ported by Komitet Badan´ Naukowych under Grant 8T11A 019 018.References[1] Hale J., Theory of Functional Differential Equations, Springer-Verlag, NewYork Heidelberg Berlin 1997.[2] Kaczorek T., Positive 1D and 2D Systems, Springer-Verlag, Berlin 2001.[3] Klamka J., Kalinowski J., Reachability and Controllability of PositiveContinuous Time Linear Systems with Time Variable Coefficients, Bull. Pol.Acad. Sci. Techn. Sci., 49(4)(2001), 623–632.[4] Ratajczak Z., Onmonotonic solutions of system of n linear homogeneous dif-ferential equations, Zeszyty Naukowe Politechniki Poznan´skiej, seria Matem-atyka, 42(3)(1967), 59–65.Jo´zef KalinowskiDepartment of Mathematics, Silesian UniversityBankowa 14, 40–007 Katowice, Polande-mail: kalinows@ux2.math.us.edu.plJerzy KlamkaDepartment of Automation, Silesian Technical UniversityAkademicka 16, 44-100 Gliwice, Polande-mail: jklamka@ia.polsl.gliwice.plReceived on 28.12.2001 and, in revised from, on 03.11.2004.

Positive dynamical systems with delays

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Positive dynamical systems with delays

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