An Impulsive Goodwin-like Model: The Case of Multiple Delays ⁎

Proceedings, 14th IFAC Workshop on Time Delay Systems Proceedings, 14th IFAC Workshop on Time Delay Systems Pesti Vigadó, Budapest, Hungary, June 28-30, 2018 Proceedings, 14th IFAC Workshop on Time Delay Systems Pesti Vigadó, Budapest, Hungary, June 28-30, 2018 Proceedings, 14th IFAC Workshop onAvailable Time Delay Systems online at www.sciencedirect.com Pesti Vigadó, Budapest, Hungary, June 28-30, 2018 Proceedings, 14th IFAC Workshop on Time Delay Systems Pesti Vigadó, Budapest, Hungary, June 28-30, 2018 Pesti Vigadó, Budapest, Hungary, June 28-30, 2018

ScienceDirect

IFAC PapersOnLine 51-14 (2018) 183–188

An Impulsive Goodwin-like Model: An Impulsive Goodwin-like Model: An Impulsive Goodwin-like Model:  AnThe Impulsive Goodwin-like Model: Case of Multiple Delays  Case of Multiple Delays AnThe Impulsive Goodwin-like Model: The The Case Case of of Multiple Multiple Delays Delays  The Case of Multiple Delays Alexander N. Churilov

Alexander N. Churilov Alexander Alexander N. N. Churilov Churilov Faculty of Mathematics and Mechanics, St. Alexander N. Churilov Faculty of Mathematics and Mechanics, St. Petersburg Petersburg State State Faculty of Mathematics and Mechanics, St. Petersburg State University, Universitetskiy av. 28, Stary Peterhof, 198504, Faculty of Mathematics and Mechanics, St. Petersburg State University, Universitetskiy av. 28, Stary Peterhof, 198504, University, Universitetskiy 28, Peterhof, 198504, St. Russia, Faculty of Mathematics and av. Mechanics, St. Petersburg State University, Universitetskiy av. 28, Stary Staryand Peterhof, 198504, St. Petersburg, Petersburg, Russia, and St. Petersburg, Russia, and Laboratory of Control of Multi-agent, Distributed and Networked University, Universitetskiy av. 28, Stary Peterhof, 198504, St. Petersburg, Russia, and Laboratory of Control of Multi-agent, Distributed and Networked Laboratory of Control of Distributed and Networked Systems, Kronverkskiy St.University, Petersburg, Russia, andav. Laboratory Control of Multi-agent, Multi-agent, Distributed and197101, Networked Systems,ofITMO ITMO University, Kronverkskiy av. 49, 49, 197101, Systems, ITMO University, Kronverkskiy av. 49, 197101, St. Petersburg, Russia (e-mail: a [email protected]) Laboratory ofITMO Control of Multi-agent, Distributed and197101, Networked Systems, University, Kronverkskiy av. 49, St. Petersburg, Russia (e-mail: a [email protected]) St. Russia a Systems, ITMO University, Kronverkskiy av. 49, 197101, St. Petersburg, Petersburg, Russia (e-mail: (e-mail: a [email protected]) [email protected]) St. Petersburg, Russia (e-mail: a [email protected]) Abstract: An impulsive counterpart of the Goodwin biological oscillator with three discrete Abstract: An impulsive counterpart of the Goodwin biological oscillator oscillator with with three three discrete discrete Abstract: An impulsive counterpart of the Goodwin biological delays is considered. An impulse-to-impulse discrete map that captures dynamics of the Abstract: An impulsive counterpart of the Goodwin biological oscillator with three discrete delays is considered. An impulse-to-impulse discrete map that captures dynamics of the the delays is considered. An impulse-to-impulse discrete map that captures dynamics of impulsive delayed model is constructed. Abstract: An impulsive counterpart of the Goodwin biological oscillator with three discrete delays is delayed considered. Anis impulse-to-impulse discrete map that captures dynamics of the impulsive model constructed. impulsive model constructed. delays is delayed considered. Anis impulse-to-impulse discrete map that captures dynamics of the impulsive delayed model isFederation constructed. © 2018, IFAC (International of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. impulsive delayed model is constructed. Keywords: Impulsive systems, time delays, periodic solutions. Keywords: Impulsive systems, time delays, periodic solutions. Keywords: Impulsive systems, time delays, periodic solutions. Keywords: Impulsive systems, time delays, periodic solutions. Keywords: Impulsive systems, time delays, periodic solutions. 1. INTRODUCTION maps preserve most of the properties of the initial system 1. INTRODUCTION INTRODUCTION maps preserve most of the properties of the initial system 1. maps preserve most of the properties of the initial system and can be easily implemented in computer programs. 1. INTRODUCTION mapscan preserve most implemented of the properties of the initial system and be easily in computer programs. and can be easily implemented in computer programs. They are especially useful for finding periodic solutions 1. INTRODUCTION maps preserve most of the properties of the initial system A mathematical model called “Goodwin oscillator” was and can be easily implemented in computer programs. They are especially useful for finding periodic solutions A mathematical model called “Goodwin oscillator” was They are especially useful for finding periodic solutions bifurcation analysis. The basic property of a delayed A mathematical model called “Goodwin oscillator” was and can be easily implemented in computer programs. introduced in Goodwin (1965, 1966) and gained significant They are especially useful for finding periodic solutions bifurcation analysis. The basic property of a delayed A mathematical model(1965, called1966) “Goodwin oscillator” was and introduced in Goodwin and gained gained significant and bifurcation analysis. The property of a delayed that allows to construct Poincar´ ee maps introduced in 1966) and significant They are especially useful forbasic finding periodic popularity mathematical biology (see Murray (2002); A mathematical model(1965, called “Goodwin oscillator” was system and bifurcation analysis. The basic property of effectively asolutions delayed system that allows to construct Poincar´ maps effectively introduced in Goodwin Goodwin (1965, 1966) and gained significant popularity mathematical biology (see Murray (2002); system that allows to construct Poincar´ e maps effectively was called finite-dimensional reducibility. It was firstly inpopularity in mathematical biology (see Murray (2002); and bifurcation analysis. The basic property of a delayed Gonze and Abou-Jaoud´ e (2013); Mackey et al. (2012– introduced in Goodwin (1965, 1966) and gained significant system that allows to construct Poincar´ e maps effectively was called finite-dimensional reducibility. It was firstly inpopularity in mathematical biology (see Murray (2002); Gonze and and Abou-Jaoud´ Abou-Jaoud´ee (2013); (2013); Mackey Mackey et et al. al. (2012– (2012– was called finite-dimensional reducibility. It was firstly introduced in Churilov et al. (2012) and developed in a numGonze system that allows to construct Poincar´ e maps effectively 2014)). In Smith (1980, 1983) the Goodwin’s model was popularity inAbou-Jaoud´ mathematical biology (see Murray was calledinfinite-dimensional reducibility. It was in firstly introduced Churilov et al. (2012) and developed a numGonze and e1983) (2013); Mackey et model al. (2002); (2012– 2014)). In Smith (1980, the Goodwin’s was troduced in Churilov et al. (2012) and developed in a number of subsequent publications, see Churilov et al. (2013, 2014)). In Smith (1980, the Goodwin’s was was called reducibility. It was firstly inapplied to mathematical endocrinology, namely, to deGonze and Abou-Jaoud´ (2013); et model al. (2012– troduced infinite-dimensional Churilov et al. (2012) and developed in a(2013, number of subsequent publications, see Churilov et al. 2014)). In Smith (1980, e1983) 1983) the Mackey Goodwin’s model was applied to mathematical endocrinology, namely, to deber of subsequent publications, see Churilov et al. (2013, 2014a); Zhusubaliyev et al. (2014); Churilov et al. (2014b); applied to mathematical endocrinology, namely, to detroduced in Churilov et al. (2012) and developed in a numscribe oscillations of hormones’ levels in the male repro2014)). In Smith (1980, 1983) the Goodwin’s model was ber of subsequent publications, see Churilov et al. (2013, 2014a); Zhusubaliyev et al. (2014); Churilov et al. (2014b); applied to mathematical endocrinology, namely, de- 2014a); Zhusubaliyev et al. (2014); Churilov et al. (2014b); scribe oscillations oscillations of hormones’ hormones’ levels in in the the male to reproChurilov and Medvedev Zhusubaliyev (2015); scribe of levels male reproof subsequent publications, seeChurilov Churilovetet etal.al. al. (2013, ductive hormonal axis. The specific applied to mathematical endocrinology, namely, de- ber 2014a); Zhusubaliyev et (2014); al. (2014); (2014b); Churilov and Medvedev (2014); Zhusubaliyev et al. (2015); scribe oscillations of hormones’ levels of in endocrinological the male to reproductive hormonal axis. The specific of endocrinological Churilov and Medvedev (2014); Zhusubaliyev et al. (2015); Medvedev (2016); Churilov et al. (2016). ductive hormonal axis. The specific of endocrinological 2014a); Zhusubaliyev et al. (2014); Churilov et al. (2014b); regulation systems that involve the hypothalamus is that scribe oscillations of hormones’ levels in the male reproand Medvedev (2014); Zhusubaliyev et (2016). al. (2015); Medvedev (2016); Churilov et al. ductive hormonal The specific of endocrinological regulation systems axis. that involve involve the hypothalamus hypothalamus is that that Churilov Churilov and Medvedev (2016); Churilov et al. regulation systems that the is and (2014); Zhusubaliyev et (2016). al. (2015); hypothalamic hormones impulsively, governed ductive hormonal axis. Thereleased specific of endocrinological Churilov and Medvedev Medvedev (2016); Churilov etwas al. (2016). regulation systems that are involve the hypothalamus is that Churilov The Goodwin model with multiple delays studied in hypothalamic hormones are released impulsively, governed The Goodwin model with multiple delays was studied in hypothalamic hormones are released impulsively, governed Churilov and Medvedev (2016); Churilov et al. (2016). by ensembles of neurones. The conventional Goodwin regulation systems that involve the hypothalamus is that The Goodwin model with multiple delays was studied in hypothalamic hormones are released impulsively, Goodwin governed Liu and Deng (1991); Cao and Jiang (2011); Huang and by ensembles of neurones. The conventional The Goodwin model with multiple delays was studied in Liu and Deng (1991); Cao and Jiang (2011); Huang and by ensembles of neurones. The conventional Goodwin model does not suit properly to describe this impulsive hypothalamic hormones are released impulsively, governed Liu and Deng (1991); Cao and Jiang (2011); Huang and by ensembles of neurones. The conventional Goodwin Cao (2015); Zhang et al. (2015); Sun et al. (2016). In model does does not not suit suit properly properly to to describe describe this this impulsive impulsive The Goodwin model with multiple delays was studiedand in Liu and DengZhang (1991); Cao and Jiang (2011); Huang Cao (2015); et al. (2015); Sun et al. (2016). In model effect. To improve adequacy of the model Churilov et al. by ensembles of neurones. The conventional Goodwin Cao (2015); Zhang et al. (2015); Sun et al. (2016). In model does not suit properlyof to describe this impulsive endocrine systems multiple delays describe times required effect. To improve adequacy the model Churilov et al. Liu and Deng (1991); Cao and Jiang (2011); Huang and Cao (2015); Zhang et al. (2015); Sun et al. (2016). In endocrine systems multiple delays describe times required effect. To improve adequacy the model Churilov et (2009) proposed to substitute differential equamodel does not suit properlyof to this impulsive endocrine systems delays describe times required effect. improve ofordinary thedescribe model Churilov et al. al. Cao to transport hormones one organ through (2009) To proposed toadequacy substitute ordinary differential equa(2015); Zhangmultiple et from al. (2015); Sunto etanother al. (2016). In systems multiple delays describe times required to transport hormones from one organ to another through (2009) proposed to substitute ordinary differential equations presented in the Goodwin’s scheme for impulsive effect. To improve adequacy of the model Churilov et al. endocrine to transport hormones from one organ to another through (2009) proposed in to the substitute ordinary differential equathe bloodstream, and also times for hormones’ synthesis. tions presented Goodwin’s scheme for impulsive endocrine systems multiple delays describe times required transport hormones fromtimes one organ to anothersynthesis. through the bloodstream, and also for hormones’ tions presented Goodwin’s scheme for (functional-differential) equations used in applied math(2009) proposed in to the substitute ordinary differential equa- to the bloodstream, and also times for tions presented in the Goodwin’s scheme for impulsive impulsive In Churilov et al. (2017) an impulsive counterpart of the (functional-differential) equations used in applied mathto transport hormones from one organ to anothersynthesis. through the bloodstream, and also times for hormones’ hormones’ synthesis. In Churilov et al. (2017) an impulsive counterpart of the (functional-differential) equations used in applied mathematics (see Lakshmikantham et al. (1989); Bainov and tions presented in the Goodwin’s scheme for impulsive In Churilov et al. (2017) an impulsive counterpart of the (functional-differential) equations used in applied maththree-dimensional Goodwin oscillator was considered with ematics (see Lakshmikantham et al. (1989); Bainov and the bloodstream, and also times for hormones’ synthesis. In Churilov et al. Goodwin (2017) anoscillator impulsivewas counterpart ofwith the three-dimensional considered ematics (see Lakshmikantham et al. (1989); Bainov and Simeonov (1993); Samoilenko and Perestyuk (1995); Sta(functional-differential) equations used in applied maththree-dimensional Goodwin oscillator was considered with ematics (see Lakshmikantham et al. (1989); (1995); Bainov Staand In the help of an impulse-to-impulse map. In this paper Simeonov (1993); Samoilenko and Perestyuk Churilov et al. (2017) an impulsive counterpart of the three-dimensional Goodwin oscillator was considered with the help of an impulse-to-impulse map. In this paper the Simeonov (1993); Samoilenko and Perestyuk (1995); Stamova and Stamov (2016)). In electrical engineering ematics (see Lakshmikantham et al. (1989); Bainov such and the help of an impulse-to-impulse map. In this paper the Simeonov (1993); Samoilenko and Perestyuk (1995); Stasame approach is extended to a Goodwin-like impulsive mova and Stamov (2016)). In electrical engineering such three-dimensional Goodwin oscillator was considered with the help of an impulse-to-impulse map. In this paper the same approach is extended to a Goodwin-like impulsive mova and Stamov (2016)). In electrical engineering such systems are known as pulse modulated (see e. g. Jones Simeonov (1993); Samoilenko and Perestyuk (1995); Sta- system same approach is extended to a Goodwin-like impulsive mova andare Stamov (2016)). Inmodulated electrical engineering such of a higher dimension that necessitates reworking systems known as pulse (see e. g. Jones the help ofa an impulse-to-impulse map. In this reworking paper the same approach is extended tothat a Goodwin-like impulsive system of higher dimension necessitates systems are known as pulse modulated (see e. g. Jones et al. (1961); Gelig and Churilov (1998)). mova and Stamov (2016)). In electrical engineering such system of aa higher dimension necessitates reworking systems are known as pulse modulated the proof of the main statement 4). et al. al. (1961); (1961); Gelig and and Churilov (1998)). (see e. g. Jones same approach is extended tothat a(Theorem Goodwin-like impulsive system of higher dimension that necessitates reworking the proof of the main statement (Theorem 4). et Gelig Churilov (1998)). systems are known as with pulse modulated (see e.studied g. Jones the the statement (Theorem 4). et al. Goodwin (1961); Gelig and Churilov (1998)). system of of a higher dimension that necessitates The model time delays was in the proof proof of the main main statement (Theorem 4). reworking The Goodwin model with time delays was studied in et al. (1961); Gelig and Churilov (1998)). The Goodwin model with time delays was studied in the proof of the main statement (Theorem 4). a number of works Smith (1983); Cartwright and HuThe Goodwin modelSmith with (1983); time delays was studied in 2. FD-REDUCIBILITY OF A LINEAR SYSTEM a number number of works works Cartwright and HuHu2. FD-REDUCIBILITY OF A LINEAR SYSTEM aThe of Smith (1983); Cartwright and sain (1986); Das et al. (1994); Mukhopadhyay and BhatGoodwin model with time delays was studied in 2. FD-REDUCIBILITY OF A LINEAR SYSTEM asain number ofDas works Smith (1983); Cartwrightand andBhatHuWITH TWO DISCRETE DELAYS (1986); et al. (1994); Mukhopadhyay 2. FD-REDUCIBILITY OF A LINEAR SYSTEM WITH TWO DISCRETE DELAYS sain (1986); Das et al. (1994); Mukhopadhyay and Bhattacharyya (2004); Ren (2004); Enciso and Sontag (2004); atacharyya number of works Smith (1983); Cartwright and HuWITH TWO DISCRETE DELAYS sain (1986); Das et al. (1994); Mukhopadhyay and Bhat(2004); Ren Ren (2004); (2004); Enciso Enciso and and Sontag Sontag (2004); (2004); 2. FD-REDUCIBILITY OF A LINEAR SYSTEM WITH TWO DISCRETE DELAYS tacharyya (2004); Efimov and Fradkov (2007); Greenhalg and Khan (2009); sain (1986); Das etRen al. (1994); Mukhopadhyay and(2004); Bhat- Let us consider tacharyya (2004); (2004); Enciso and Sontag Efimov and Fradkov (2007); Greenhalg and Khan (2009); WITH TWO DISCRETE DELAYS a linear system with two discrete Efimov and Fradkov (2007); Greenhalg and Khan (2009); Li (2015). Beginning with Churilov al. (2012) aa time us consider aa linear system with two discrete delays delays tacharyya Ren (2004); Encisoet Efimov and(2004); Fradkov (2007); Greenhalg and Khan (2004); (2009); Li (2015). (2015). Beginning with Churilov etand al. Sontag (2012) time Let Let us consider  linear system with two discrete delays u = U u(t), (1) Li Beginning with Churilov et al. (2012) a time Let us consider a linear system with two discrete delays  delay was introduced into the impulsive Goodwin model. 1 Efimov and Fradkov (2007); Greenhalg and Khan (2009); = U (1) Li (2015). Beginning into with the Churilov et al. (2012) model. a time Let us consider u  linear delay was introduced introduced impulsive Goodwin 1 u(t), a system with two discrete delays u = U u(t), (1)  delay was into the impulsive Goodwin model. 1 The main mathematical tool that was employed was a Li (2015). Beginning withtool Churilov et al. (2012) model. awas time v(t) + G − ττ1 )) (2) vv  = u U (1) 2 u(t), 1 u(t delay was introduced into the impulsive Goodwin 1 The main mathematical that was employed a = U v(t) + G u(t − (2)  2 v(t) 1 u(t − τ1 ) The main mathematical tool that was employed was a construction of a discrete impulse-to-impulse map (called u u(t), (1) = U + G (2) v  delay was introduced into the impulsive Goodwin model. 1 2 1 1 The main mathematical tool that was employed was a construction of a discrete impulse-to-impulse map (called w(t) + G v(t − τ ) (3) w v(t) + G u(t − τ ) (2) v  = U2 3 2 2 1 1 construction of a discrete impulse-to-impulse map (called (3) w  = U3 w(t) + G2 v(t − τ2 ) the Poincar´ e map in the theory of hybrid systems, see The main mathematical tool that was employed was a v(t) + G − ττ12)) (2) v  = U23 w(t) construction a discrete map (called + G v(t − (3) the Poincar´ Poincar´ee ofmap map in the the impulse-to-impulse theory of of hybrid hybrid systems, systems, see for t  t . Herew 12u(t U , U , U , G , G are constant matrix = U w(t) + G v(t − τ ) (3) w the in theory see 0 . Here U 32 , U3 , G1 ,2 G2 are2 constant matrix Haddad et al. (2006)). This map can also be considered construction a discrete impulse-to-impulse map (called  1 t  t , U the Poincar´ e ofmap in the theory of hybrid systems, see for 0 1 2 3 1 2 Haddad et al. (2006)). This map can also be considered = U w(t) + G v(t − τ ) (3) w for tt0 .. Here U ,, U ,,1U ,,22G constant matrix blocks sizes × × p × p 3 ,,pG 2 pare Haddad et (2006)). This map can also be considered 1 ,, 1p 2 ,, 2p 3 × 3 ,, p 2 1 ,, as special operator the trajectories of the etranslation map in the theory of see for tt  whose U11are U322p constant matrix whose sizes are p × × p × p 0 Here 3 pG 2 pare Haddad et al. al. (2006)). This mapalong canhybrid also besystems, considered 1U 1 , 1p 2G 2, p 3 × 3, p 2 1 as aa a Poincar´ special translation operator along the trajectories trajectories of blocks blocks whose sizes are p × p p × p p × p p × p p × p , respectively, and u(·), v(·), w(·) are functions with 1 1 2 2 3 3 2 as special translation operator along the of for t  t . Here U , U , U , G , G are constant matrix 3 2 a continuous-time system Krasnoselskii (1968). Poincar´ e 0 1 2 3 1 2 Haddad et al. (2006)). This map can also be considered sizes are p1u(·), × p1v(·), , p2 × p2 ,are p3 functions × p3 , p2 ×with p11 ,, p p respectively, and w(·) as a special translation operator along the trajectories ofe blocks 3× 2 ,,whose a continuous-time system Krasnoselskii (1968). Poincar´ p × p respectively, and u(·), v(·), w(·) are functions with vector values of dimensions p , p , p . 3 × p2 ,whose aas system Krasnoselskii (1968). Poincar´ sizes are p1u(·), × pp11v(·), , p22, × 3p 2 , p3 × p3 , p2 × p1 , a special translation operator along the trajectories ofee blocks p respectively, and vector of dimensions p 3 2 values a continuous-time continuous-time system Krasnoselskii (1968). Poincar´ 1 ,, p 2 , w(·) 3 .. are functions with  vector values of dimensions p p p 1v(·), 2 , w(·) 3 . are functions with work was supported byKrasnoselskii the Russian Foundation for Basic p × p , respectively, and u(·), 3 2 a The continuous-time system (1968). Poincar´ e  vector values of dimensions p , p p 1 2vector 3 Let us introduce aa p-dimensional The work was supported by the Russian Foundation for Basic  Research (Grant Government of thefor Russian Let us introduce The work was 17-01-00102-a) supported by and the the Russian Foundation Basic vector values of dimensions p1 , p2vector , p3 .  Research (Grant Government of thefor Russian Let us introduce aa p-dimensional p-dimensional vector The work was 17-01-00102-a) supported by and the the Russian Foundation Basic x(·) = col{u(·), v(·), w(·)} Federation (Grant 08-08). Research (Grant 17-01-00102-a) and the Government of the Russian Let us introduce p-dimensional vector  The work was supported by the Russian Foundation for Basic x(·) = col{u(·), v(·), w(·)} Federation(Grant (Grant 08-08). Research 17-01-00102-a) and the Government of the Russian Let us introduce a p-dimensional vector x(·) = col{u(·), v(·), Federation (Grant 08-08). Research 17-01-00102-a) and the Government of the Russian x(·) = col{u(·), v(·), w(·)} w(·)} Federation(Grant (Grant 08-08). v(·), w(·)} Federation 08-08).(International Federation of Automatic Control) Copyright 2018 183 Hosting by Elsevierx(·) 2405-8963 © ©(Grant 2018, IFAC IFAC Ltd. = Allcol{u(·), rights reserved. Copyright © 2018 IFAC 183 Copyright © under 2018 IFAC 183 Control. Peer review responsibility of International Federation of Automatic Copyright © 2018 IFAC 183 10.1016/j.ifacol.2018.07.220 Copyright © 2018 IFAC 183

2018 IFAC TDS 184 Budapest, Hungary, June 28-30, 2018

Alexander Churilov et al. / IFAC PapersOnLine 51-14 (2018) 183–188

with p = p1 +p2 +p3 . Then system (1)–(3) can be rewritten in a matrix form as (4) x = A0 x(t) + A1 x(t − τ1 ) + A2 x(t − τ2 ), where A0 , A1 , A2 are constant p × p matrices,       U1 0 0 0 00 0 0 0 A0 = 0 U2 0 , A1 = G1 0 0 , A2 = 0 0 0 . 0 0 U3 0 00 0 G2 0 (5) The initial problem for (4) can be defined as u(t) = ϕu (t), t0 − τ1  t  t0 , v(t) = ϕv (t), t0 − τ2  t  t0 , w(t0 ) = w0 , where ϕu (·), ϕv (·) are initial functions and w0 is a constant vector. Definition 1. A time-delay linear system (4) is called finite-dimensional reducible (FD-reducible) if for every t0 there exist a constant p×p matrix D and a positive number τ¯ such that any solution x(t) of (4) defined for t  t0 satisfies the ordinary differential equation (6) x = Dx(t) for t  t0 + τ¯. The matrix D and the number τ¯ are independent of the initial data. Certainly, the property of FD-reducibility is rather restrictive, but it holds for linear systems with matrices A0 , A1 , A2 having a special “cyclic” structure (5). Lemma 1. System (4) with coefficients (5) is FD-reducible with τ¯ = τ1 + τ2 and the matrix D can be defined as D = D0 + A2 e−D0 τ2 with D0 = A0 + A1 e−A0 τ1 . (7) Proof. Since (1) is independent of (2), (3), we have u(t) = eU1 (t−t0 ) u(t0 ),

t  t0 ,

and hence u(t − τ1 ) = e−U1 τ1 u(t), Thus (1), (2) can be rewritten as

t  t 0 + τ1 .

It is easily seen that   Dz 0 , D0 = 0 U3

which implies (7). 

e−D0 τ2 =



 0 e−Dz τ2 , 0 e−U3 τ2

Notice that det(sIp − A0 − A1 e−A0 τ1 − A2 e−A0 τ2 ) = det(sI − A0 ) for all complex s. Thus the spectrum of the linear system (4) is finite and independent of τ1 , τ2 . 3. IMPULSIVE MODEL WITH THREE DELAYS In this section we consider an impulsive system whose continuous part has the form of (4) including two discrete delays τ1 and τ2 . The third discrete delay τ3 is incorporated into the discrete part of the system. We shall deal with an impulsive system x = A0 x(t) + A1 x(t − τ1 ) + A2 x(t − τ2 ), (10) tn < t < tn+1 , σ(t) = Cx(t), (11) − x(t+ ) − x(t ) = λ B, n = 0, 1, . . . , (12) n n n tn+1 = tn + Tn , Tn = Φ(σ(tn − τ3 )), (13) λn = F (σ(tn − τ3 )). Assume that its matrix coefficients have specific block forms (5) and B = col{Bu , 0, 0}, C = [0 0 Cw ] . (14) Thus Bu is p1 × 1 and Cw is 1 × p3 .

Here Φ(·), F (·) are continuous R → R functions with bounds (15) 0 < Φ1  Φ(·)  Φ2 , 0 < F1  F (·)  F2 , where Φi , Fi , i = 1, 2, are positive constants. In particular, inequality Φ1 > 0 implies that the sequence {tn }∞ n=0 has no accumulation points and tn → +∞ as n → ∞.

u(t), t  t0 + τ1 . u = U1 u(t), v = U2 v(t) + G1 e In a matrix notation we have z  = Dz z(t), t  t0 + τ1 , where     U1 0 u(t) , z(t) = . (8) Dz = v(t) G1 e−U1 τ1 U2

As previously, x(·) = col{u(·), v(·), w(·)}. Thus equation (10) can be rewritten as (1)–(3). Then σ(t) = Cw w(t). Evidently, only the function u(t) has jumps − (16) u(t+ n ) − u(tn ) = λn Bu , while the functions v(t), w(t), σ(t) are continuous. Formulas (11)–(13) describe an amplitude-frequency pulse modulation with a modulating signal σ(t) (see Gelig and Churilov (1998)). The right-hand side of system (10) has jumps at the points tn , tn + τ1 , tn + τ2 , n  0, while in the time intervals between these points system (10) is linear.

Gz = [0 G2 ]e−Dz τ2 .

System (10)–(13) is subject to initial conditions given by the continuous initial functions ϕu (·), ϕv (·), ϕw (·). More precisely, u(t) = ϕu (t), t0 − τ1  t < t0 , v(t) = ϕv (t), t0 − τ2  t  t0 , w(t) = ϕw (t), t0 − τ3  t  t0 , T0 = Φ(Cw ϕw (t0 − τ3 )), λ0 = F (Cw ϕw (t0 − τ3 )) − and u(t+ 0 ) − u(t0 ) = λ0 Bu . Notice that the values of ϕw (t) inside the interval t0 − τ3 < t < t0 do not influence the solution and can be chosen arbitrarily.





−U1 τ1

This implies z(t − τ2 ) = e−Dz τ2 z(t), t  t0 + τ1 + τ2 . From (3) it follows w = U3 w(t) + [0 G2 ]z(t − τ2 ) Introduce a p3 × (p1 + p2 ) matrix

(9)

Hence z  = Dz z(t), w = U3 w(t) + Gz z(t),

t  t0 + τ1 + τ2 .

Because x(·) = col{z(·), w(·)}, (6) is satisfied with   0 Dz . D= Gz U3

We will use notation (7), (8) for z(·), Dz , Gz , D0 , D introduced in the previous section. 184

2018 IFAC TDS Budapest, Hungary, June 28-30, 2018

Alexander Churilov et al. / IFAC PapersOnLine 51-14 (2018) 183–188

The following two assumptions are made with respect to the time delay values: (17) τ2 > τ1 , Φ1 > τ1 + τ2 + τ3 . The second inequality in (17) implies that the lower bound for the length of a sampling interval (18) Tn > τ1 + τ2 + τ3 for all solutions and all n  0. For D given in (7) introduce a row vector C˜ = Ce−Dτ3 .

(19)

x(t− n ).

Introduce the shorthand notation x ¯n = Lemma 2. Under assumption (17) the formulas for Tn , λn in (13) can be rewritten for n  1 as ¯n ), λn = F (C˜ x ¯n ), (20) Tn = Φ(C˜ x where C˜ is defined by (19). Proof. As proved in Lemma 1, the continuous part of the system given by (10) is FD-reducible for t  τ1 +τ2 . Hence x(t) satisfies x = Dx(t) for tn +τ1 +τ2 < t < tn+1 , n = 0, 1, 2, . . . . (21) It follows from (18) that tn − τ3 > tn−1 + τ1 + τ2 for all n  1. Then (21) implies x(tn − τ3 ) = e−Dτ3 x ¯n . Thus −Dτ3 ˜ x ¯n = C x ¯n for all n = 1, 2, . . . . Cx(tn − τ3 ) = Ce Then for n  1 the expressions in (13) can be rewritten as (20). 

Finally, notice that the next statement can be obtained immediately. Theorem 3. Under supposition (15) the following properties of system (10)–(13) are valid. (i) System (10)–(13) has no equilibria. (ii) Let the matrices U1 , U2 , U3 be Hurwitz stable. Then for any solution of (10)–(13) the Euclidean norm x(t) is bounded for t  t0 . (iii) Assume that the matrices U1 , U2 , U3 are Metzler, i. e. all their nondiagonal elements are nonnegative. Suppose that all the elements of the matrices G1 , G2 , Bu are also nonnegative. Then system (10)–(13) is positive (see Luenberger (1979)). This means that if the initial functions ϕu (·), ϕv (·), ϕw (·) are elementwise nonnegative, then all the components of x(t) are also nonnegative for t  t0 . 4. REDUCTION TO A DISCRETE-TIME SYSTEM In this section we will obtain explicit formulas for the solutions of (10)–(13). For brevity introduce notation t∗n = tn + τ1 , tˆn = tn + τ2 , t∗∗ n = tn + τ1 + τ2 . From (17), (18) it follows that tn < t∗n < tˆn < t∗∗ n < tn+1 . Theorem 4. For t > t1 any solution of (10)–(13) satisfies a delay-free equation (22) x = Dx(t) − (D − A0 )α(t) − (D − D0 )β(t). Here on every interval tn < t < tn+1 , n  1, the functions α(t), β(t) are defined as 185

185



λn eA0 (t−tn ) B, tn < t < t∗n , 0, t∗n < t < tn+1 ,  tn < t < t∗n , 0, ∗ β(t) = λn eD0 (t−tn ) eA0 τ1 B, t∗n < t < t∗∗ ,  0, t∗∗ n < t < tn+1 .

α(t) =

(23)

(24)

Proof. The proof of Theorem 4 is lengthy and is given in Appendix. Theorem 4 is the key result of this paper. Equations (22), (23), (24) present a reduction of the initial impulsive system with delays to delay-free equations, however only for t  t1 . By integrating (22) we can get explicit formulas for x(t). Corollary 1. Any solution (tn , x(t)) of (10)–(13) obeys for all tn < t < tn+1 , n  1, the following relationships x(t) = eD(t−tn ) x ¯n + λn θ(t)B,

(25)

where

 A (t−tn )  , e 0 D0 (t−t∗ ) A0 τ 1 n θ(t) = e e ,  eD(t−t∗∗ n ) e D0 τ 2 e A0 τ 1 ,

tn < t  t∗n , t∗n < t  t∗∗ n , t∗∗ < t < t n+1 . n

(26)

Proof. From Theorem 4 it follows that any solution (tn , x(t)) of (10)–(13) obeys the delay-free equation (22). Let us demonstrate that by integrating (22) we will come to (25). Case (i). Let tn < t < t∗n . Theorem 4 yields (22) with α(t) = λn eA0 (t−tn ) B, β(t) ≡ 0. Obviously, α = A0 α(t). Introduce a difference y(t) = x(t) − α(t). From (22) it follows y  = Dy(t), tn < t < t∗n , (27) + y(t+ ¯n . n ) = x(tn ) − λn B = x Integrating (27), we obtain ¯n , tn < t < t∗n . y(t) = eD(t−tn ) x Since x(t) = y(t) + α(t), we come to the first case in (25), (26). Notice that by setting t = t∗n from the first formula of (26) we have x(t∗n ) = eDτ1 x ¯n + λn eA0 τ1 B.

(28)

Case (ii). Let t∗n < t < t∗∗ 4 we obtain n . From Theorem ∗ (22) with α(t) = 0, β(t) = λn eD0 (t−tn ) eA0 τ1 B. Hence β  = D0 β(t), β(t∗n ) = λn eA0 τ1 B. (29) From (22), (28) and (29) we conclude that the difference y(t) = x(t) − β(t) satisfies y  = Dy(t),

t∗n < t < t∗∗ n ,

y(t∗n ) = eDτ1 x ¯n .

(30)

Integrating (30) and using x(t) = y(t) + β(t), we get the second case in (25), (26). Moreover, we have D(τ1 +τ2 ) x ¯n + λn eD0 τ2 eA0 τ1 B. x(t∗∗ n )=e

(31)

 Case (iii). Let t∗∗ n < t < tn+1 . Then x = Dx(t) and ∗∗

x(t) = eD(t−tn ) x(t∗∗ t∗∗ (32) n ), n < t < tn+1 . From (31) and (32) we come to the third case in (25), (26). 

2018 IFAC TDS 186 Budapest, Hungary, June 28-30, 2018

Alexander Churilov et al. / IFAC PapersOnLine 51-14 (2018) 183–188

It can be concluded from Corollary 1 that the dynamics of (10)–(13) at points tn , n = 1, 2, . . ., obey the following discrete (Poincar’e) map. Corollary 2. For any solution (tn , x(t)) of (10)–(13) it holds for n = 1, 2, . . . that x ¯n+1 = Q(¯ xn ), (33) where ˜ ˜ Q(¯ x) = eDΦ(C x¯) x ¯ + F (C˜ x ¯)eD(Φ(C x¯)−τ1 −τ2 ) eD0 τ2 eA0 τ1 B. Proof. With t = tn+1 formulas (31) and (32) imply x ¯n+1 = eDTn x ¯n + λn eD(Tn −τ1 −τ2 ) eD0 τ2 eA0 τ1 B. Thus (33) follows.  Thus given two initial points x ¯0 , x ¯1 , the subsequent points x ¯2 , x ¯3 , . . . can be found by recurrence (33). The initial functions ϕu (t), ϕv (t), ϕw (t), are necessary to calculate the values of x ¯0 , x ¯1 , but they do not influence further values x ¯n . Obviously, the function Q(·) is continuous as a composition of continuous functions. If the functions Φ(·), F (·) are smooth, then Q(·) is also smooth. 5. CONCLUSION An impulsive delayed system whose continuous part has a specific “cyclic” structure is considered. It is shown that solutions of this system calculated at sampling times (beginning with the third sample) obey a discrete-time equation (an impulse-to-impulse map) (33) that is independent on the initial data (Corollary 2). Additionally, explicit formulas for solutions are provided for times taken inside a sampling period (Corollary 1). Thus the impulseto-impulse map thoroughly characterizes solutions of the continuous-time equation with time delays. The discretetime equation (33) can be readily explored by means of a computer modelling (examples of such analysis are given in Churilov et al. (2017)). The results of this paper generalize those of Churilov et al. (2017) for a multidimensional case. ACKNOWLEDGEMENTS The author acknowledges Alexander Medvedev, Uppsala University, Sweden, and Zhanybai Zhusubaliyev, Southwest State University, Russia, for cooperation and discussion. REFERENCES Bainov, D. and Simeonov, P. (1993). Impulsive Differential Equations: Periodic Solutions and Applications. Longman, Harlow, UK. Cao, J. and Jiang, H. (2011). Stability and Hopf bifurcation analysis on Goodwin model with three delays. Chaos Solitons Fractals, 44, 613–618. Cartwright, M. and Husain, M. (1986). A model for the control of testosterone secretion. J. Theor. Biol., 123, 239–250. Churilov, A.N. and Medvedev, A. (2014). An impulse-toimpulse discrete-time mapping for a time-delay impulsive system. Automatica, 50(8), 2187–2190. Churilov, A.N. and Medvedev, A. (2016). Discrete-time map for an impulsive Goodwin oscillator with a distributed delay. Math. Control Signals Syst., 28(1), 1–22. 186

Churilov, A.N., Medvedev, A., and Mattsson, P. (2012). Periodical solutions in a time-delay model of endocrine regulation by pulse-modulated feedback. In Proc. 51st IEEE Conf. Decis. Control, 362–367. Maui, Hawaii. Churilov, A.N., Medvedev, A., and Mattsson, P. (2013). Finite-dimensional reducibility of time-delay systems under pulse-modulated feedback. In Proc. 52nd IEEE Conf. Decis. Control, 2078–2083. Florence, Italy. Churilov, A.N., Medvedev, A., and Mattsson, P. (2014a). Periodical solutions in a pulse-modulated model of endocrine regulation with time-delay. IEEE Trans. Automat. Control, 59(3), 728–733. Churilov, A.N., Medvedev, A., and Shepeljavyi, A.I. (2009). Mathematical model of non-basal testosterone regulation in the male by pulse modulated feedback. Automatica, 45(1), 78–85. Churilov, A.N., Medvedev, A., and Zhusubaliyev, Zh.T. (2014b). Periodic modes and bistability in an impulsive Goodwin oscillator with large delay. In 19th World Congress IFAC Proc., 3340–3345. Cape Town, South Africa. Churilov, A.N., Medvedev, A., and Zhusubaliyev, Zh.T. (2016). Impulsive Goodwin oscillator with large delay: periodic oscillations, bistability, and attractors. Nonlinear Anal. Hybrid Syst., 21, 171–183. Churilov, A., Medvedev, A., and Zhusubaliyev, Z. (2017). Discrete-time mapping for an impulsive Goodwin oscillator with three delays. Int. J. Bifurcation Chaos, 27(12), 1750182 [17 pages]. Das, P., Roy, A.B., and Das, A. (1994). Stability and oscillations of a negative feedback delay model for the control of testosterone secretion. BioSystems, 32(1), 61– 69. Efimov, D. and Fradkov, A. (2007). Oscillatority conditions for nonlinear systems with delay. J. Appl. Math., 2007. Article ID 72561. Enciso, G. and Sontag, E.D. (2004). On the stability of a model of testosterone dynamics. J. Math. Biol., 49, 627–634. Gelig, A.Kh. and Churilov, A.N. (1998). Stability and Oscillations of Nonlinear Pulse-modulated Systems. Birkh¨auser, Boston. Gonze, D. and Abou-Jaoud´e, W. (2013). The Goodwin model: Behind the Hill function. PLoS ONE, 8(8), e69573. Goodwin, B.C. (1965). Oscillatory behavior in enzymatic control processes. In G. Weber (ed.), Adv. Enzyme Regul., volume 3, 425–438. Pergamon, Oxford. Goodwin, B.C. (1966). An intrainment model for timed enzyme synthesis in bacteria. Nature, 209(5022), 479– 481. Greenhalg, D. and Khan, Q.J.A. (2009). A delay differential equation mathematical model for the control of the hormonal system of the hypothalamus, the pituitary and the testis in man. Nonlin. Anal., 71, e925–e935. Haddad, W.M., Chellaboina, V., and Nersesov, S.G. (2006). Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control. Princeton Univ. Press, Princeton and Oxford. Huang, C. and Cao, J. (2015). Hopf bifurcation in an n-dimensional Goodwin model via multiple delays feedback. Nonlinear Dynam., 79(4), 2541–2552.

2018 IFAC TDS Budapest, Hungary, June 28-30, 2018

Alexander Churilov et al. / IFAC PapersOnLine 51-14 (2018) 183–188

Jones, R., Li, C., Meyer, A., and Pinter, R. (1961). Pulse modulation in physiological systems, phenomenological aspects. IRE Trans. Biomed. Electron., 8(1), 59–67. Krasnoselskii, M.A. (1968). The Operator of Translation along the Trajectories of Differential Equations. Amer. Math. Soc., Providence, RI. Lakshmikantham, V., Bainov, D.D., and Simeonov, P.S. (1989). Theory of Impulsive Differential Equations. World Scientific, Singapore. Li, X. (2015). Global existence of periodic solutions in a physiological model with delay. Acta Math. Appl. Sinica, English Ser., 31(4), 1043–1048. Liu, B.Z. and Deng, G.M. (1991). An improved mathematical model of hormone secretion in the hypothalamopituitary-gonadal axis in man. J. Theor. Biol., 150, 51– 58. Luenberger, D.G. (1979). Introduction to Dynamic Systems: Theory, Models, and Applications. Wiley, New York. Mackey, M.C., Santill´ an, M., Tyran-Kami´ nska, M., and Zeron, E.S. (2012–2014). The utility of simple mathematical models in understanding gene regulatory dynamics. In Silico Biol., 12(1–2), 23–53. Mukhopadhyay, B. and Bhattacharyya, R. (2004). A delayed mathematical model for testosterone secretion with feedback control mechanism. Int. J. Math. Math. Sci., 2004(3), 105–115. Murray, J.D. (2002). Mathematical Biology, I: An Introduction (3rd ed.). Springer, New York. Ren, H. (2004). Stability analysis of a simplified model for the control of testosterone secretion. Discrete Contin. Dynam. Syst., Ser. B, 4(3), 729–738. Samoilenko, A.M. and Perestyuk, N.A. (1995). Impulsive Differential Equations. World Scientific, Singapore. Smith, W.R. (1980). Hypothalamic regulation of pituitary secretion of lutheinizing hormone—II Feedback control of gonadotropin secretion. Bull. Math. Biol., 42, 57–78. Smith, W.R. (1983). Qualitative mathematical models of endocrine systems. Am. J. Physiol., 245(4), R473–R477. Stamova, I. and Stamov, G. (2016). Applied Impulsive Mathematical Models. Springer, Berlin. Sun, X., Yuan, R., and Cao, J. (2016). Bifurcations for Goodwin model with three delays. Nonlinear Dynam., 84(2), 1093–1105. Zhang, Y., Cao, J., and Xu, W. (2015). Stability and Hopf bifurcation of a Goodwin model with four different delays. Neurocomputing, 165(C), 144–151. Zhusubaliyev, Zh.T., Mosekilde, E., Churilov, A.N., and Medvedev, A. (2015). Multistability and hidden attractors in an impulsive Goodwin oscillator with time delay. Eur. Phys. J. Special Topics, 224, 1519–1539. Zhusubaliyev, Z., Churilov, A., and Medvedev, A. (2014). Time delay induced multistability and complex dynamics in an impulsive model of endocrine regulation. In Proc. 13th European Control Conf. (ECC), 2304–2309. Strasbourg, France.

Formulas (7) yield so (10) can be rewritten as

x = A0 x(t) + (D0 − A0 )eA0 τ1 x(t − τ1 )

+ (D − D0 )eD0 τ2 x(t − τ2 ). With D = A0 + (D0 − A0 ) + (D − D0 ), the last equality is equivalent to   x = Dx(t) − (D0 − A0 ) x(t) − eA0 τ1 x(t − τ1 )   (A.1) − (D − D0 ) x(t) − eD0 τ2 x(t − τ2 ) . From the previously deduced formulas   0 0 0 D0 − A0 = G1 e−U1 τ1 0 0 , 0 0 0   0 0 D − D0 = , Gz 0 i. e. the last columns of these matrices are zero. Using (8), equation (A.1) can be expressed as (A.2) x = Dx(t) − (D0 − A0 )η(t) − (D − D0 )ξ(t), where     ξz (t) ηu (t) (A.3) , ξ(t) = η(t) = 0 0 and (A.4) ηu (t) = u(t) − eU1 τ1 u(t − τ1 ),

ξz (t) = z(t) − eDz τ2 z(t − τ2 ). (A.5) Lemma 5. Let n  1. Then from the first formula (A.3) we obtain (A.6) η(t) = α(t), t > t1 . Moreover, η(t) = 0 for t∗0 < t < t1 .

Proof. Since u = U1 u(t), it follows that U1 (t−tn )

u(t) = e

×

Then



u(t− n ), u(t+ n ),

t = tn , tn−1 < t < tn , tn < t < tn+1 .

(A.7)



∗ ∗ u(t− n ), tn−1 < t < tn , (A.8) ∗ ∗ + u(tn ), tn < t < tn+1 . From (A.4), (A.7), (A.8) we get  t∗n−1 < t < tn , 0, U1 (t−tn ) − + × u(tn ) − u(tn ), tn < t < t∗n , ηu (t) = e  0, t∗n < t < tn+1 . (A.9) Then (16), (A.9) imply  t∗n−1 < t < tn , 0, U1 (t−tn ) (A.10) ηu (t) = λn e Bu , tn < t < t∗n ,  ∗ 0, tn < t < tn+1 . Since  Ut  e 1 Bu e A0 t B = 0 for all t, the statement of Lemma 5 follows from (A.10) and (23). 

u(t − τ1 ) = e

U1 (t−t∗ n)

×

Deduce explicit formulas for the function z(t). Consider a matrix Az and a vector Bz defined as     U 0 Bu , Bz = . Az = 1 0 U2 0

Appendix A. PROOF OF THEOREM 4

A1 = (D0 − A0 )eA0 τ1 ,

187

A2 = (D − D0 )eD0 τ2 ,

(The number of elements in Bz is p1 + p2 .) Recall that the matrix Dz is defined by (8).

187

2018 IFAC TDS 188 Budapest, Hungary, June 28-30, 2018

Alexander Churilov et al. / IFAC PapersOnLine 51-14 (2018) 183–188

Proof. From (A.11) it follows that

Lemma 6. The function z(t) can be calculated as z(t) = e

Dz (t−tn )

z(t− n)

+ λn θz (t)Bz ,

(A.11)

where

  0, θz (t) = eAz (t−tn ) ,  eDz (t−t∗n ) eAz τ1 ,

t∗n−1 < t < tn , tn < t < t∗n , t∗n < t < tn+1 .

(A.12)

Proof. Notice that w(t), ξ(t) are not involved in the first two block rows of (A.2). We have   0 0 Dz − Az = . G1 e−U1 τ1 0 Then the first p1 +p2 rows of equation (A.2) can be written as   η (t) z  = Dz z(t) − (Dz − Az )ηz (t), ηz (t) = u , (A.13) 0 where 0 stands for the zero vector of dimension p2 . Since ηu (t) is defined by (A.10) and  Ut  e 1 Bu for all t, e Az t B z = 0 the function ηz (t) in (A.13) takes the form of  t∗n−1 < t < tn , 0, Az (t−tn ) ηz (t) = λn e (A.14) Bz , tn < t < t∗n ,  ∗ 0, tn < t < tn+1 . t∗n−1



If < t < tn then (A.13), (A.14) imply z = Dz z(t). Then the first case in (A.11) follows.

Let us consider the case tn < t < tn+1 . In this interval the function z(t) is continuous, while ηz (t) has a jump at t = t∗n . Evidently, ηz = Az ηz (t), tn < t < tn+1 , t = t∗n . Consider a difference yz (t) = z(t) − ηz (t). Then yz = Dz yz (t), tn < t < tn+1 , t = t∗n , + − yz (tn ) = z(t+ n ) − λn Bz = z(tn ), − − + yz (t+ n + τ1 ) − yz (tn + τ1 ) = ηz (tn + τ1 ) − ηz (tn + τ1 ) = λn eAz τ1 Bz .

(A.15) Relationships (A.15) imply yz (t) = eDz (t−tn ) z(t− n)  (A.16) 0, tn < t < t∗n , + Dz (t−t∗ ) A τ ∗ n e z 1B , λn e t < t < t . z n+1 n Since z(t) = yz (t) + ηz (t), from (A.14), (A.16) we come to (A.11) for tn < t < tn+1 .  Consider now the calculation of ξ(t). Recall that the functions ξ(t), ξz (t) are defined by (A.3), (A.5). Lemma 7. Let n  1. The function ξz (t) defined by (A.5) is calculated as ξz (t) = λn θ˜z (t)Bz , tn < t < tn+1 , (A.17) where  eAz (t−tn ) , tn < t < t∗n ,  ∗   eDz (t−tn ) eAz τ1 , t∗n < t < tˆn ,  ∗ θ˜z (t) = eDz (t−tn ) eAz τ1 − eDz τ2 eAz (t−tˆn ) , (A.18)   ∗∗ ˆ  t n < t < tn ,    0, t∗∗ < t < tn+1 . n

188

ˆ

z(t − τ2 ) = eDz (t−tn ) z(t− n ) + λn θz (t − τ2 )Bz , tn < t < tn+1 . Hence from (A.5) we have   ξz (t) = λn θz (t) − eDz τ2 θz (t − τ2 ) Bz , tn < t < tn+1 . Thus (A.20) can be represented as with θ˜z (t) = θz (t) − eDz τ2 θz (t − τ2 ). Formula (A.12) implies   tn < t < tˆn , 0, θz (t − τ2 ) = eAz (t−tˆn ) , tˆn < t < t∗∗ n ,  eDz (t−t∗∗ n ) e Az τ 1 , t∗∗ < t < t

(A.19) (A.20) (A.17)

n+1 .

n

Thus we obtain (A.18). Lemma 8. Let n  1. Then Gz ξz (t) = Gz ξ˜z (t), tn < t < tn+1 , (A.21) where  λn eAz (t−tn ) Bz , tn < t < t∗n ,  ∗ D (t−t ) A τ n e z 1B , (A.22) ξ˜z (t) = λn e z t∗n < t < t∗∗ z n ,  ∗∗ 0, tn < t < tn+1 . Proof. From (A.18) and (A.22) we have ξz (t) − ξ˜z (t)  ˆ −λn eDz τ2 eAz (t−tn ) Bz , tˆn < t < t∗∗ n , = 0, otherwise.

Since



e U1 t 0 Gz e e Bz = [0 G2 ] 0 e U2 t for all t, (A.23) implies (A.21).  D z τ 2 Az t

(A.23)



 Bu =0 0

Now we can complete the proof of Theorem 4. From (A.21) we have       0 0 ξ˜z (t) 0 0 ξz (t) (D − D0 )ξ(t) = . = 0 Gz 0 Gz 0 0 At the same time    D t A τ e z e z 2 Bz e Az t B z A0 t D0 t A0 τ 2 e B= , e e B= 0 0

for all t. Thus from (A.22) we have (D − D0 )ξ(t) = (D − D0 )(α(t) + β(t)) and for n  1, t > t1 any solution (tn , x(t)) satisfies x = Dx(t)−(D0 −A0 )α(t)−(D−D0 )(α(t)+β(t)). (A.24) Let tn < t < t∗n . Then β(t) ≡ 0. Since (D0 − A0 ) + (D − D0 ) = D − A0 , from (A.24) we conclude that x = Dx(t) − (D − A0 )α(t), tn < t < t∗n . Let t∗n < t < t∗∗ n then α(t) ≡ 0, so (A.24) can be rewritten as x = Dx(t) − (D − D0 )β(t), t∗n < t < t∗∗ n . Finally, let t∗∗ n < t < tn+1 . Then the functions α(t), β(t) are equal to zero, so we come to (22) again. The proof of Theorem 4 is complete.