The paper is devoted to the formalization of a concept of impulse-sliding regimes generated by positional impulse controls for systems with delay. We define the notion of impulse-sliding trajectory as a limit of a sequence of Euler polygonal lines generated by a discrete approximation of the impulse position control. The equations describing the trajectory of impulse-sliding regime are received.The research was supported by Russian Science Foundation (RSF) (project No. 16-11-10146)

Sesekin, A. N.

Zhelonkina, N. I.

Ural mathematical journal

The paper is devoted to the formalization of a concept of impulse-sliding regimes generated by positional impulse controls for systems with delay. We define the notion of impulse-sliding trajectory as a limit of a sequence of Euler polygonal lines generated by a discrete approximation of the impulse position control. The equations describing the trajectory of impulse-sliding regime are received

Sesekin, Alexander N.

Zhelonkina, Natalya I.

Ural Mathematical Journal (UMJ)

URAL MATHEMATICAL JOURNAL, Vol. 2, No. 2, 2016IMPULSE{SLIDING REGIMES IN SYSTEMS WITH DELAY1Alexander N. SesekinKrasovskii Institute of Mathematics and Mechanics,Ural Branch of Russian Academy of Sciences, Ekaterinburg, Russia,sesekin@imm.uran.ruNatalya I. ZhelonkinaKrasovskii Institute of Mathematics and Mechanics,Ural Branch of Russian Academy of Sciences, Ekaterinburg, Russia,312115@mail.ruAbstract: The paper is devoted to the formalization of a concept of impulse-sliding regimes generated bypositional impulse controls for systems with delay. We dene the notion of impulse-sliding trajectory as a limitof a sequence of Euler polygonal lines generated by a discrete approximation of the impulse position control.The equations describing the trajectory of impulse-sliding regime are received.Key words: Impulse position control, Systems with delay, Impulse-sliding regime, Euler polygonal lines.IntroductionUsually, the positional control algorithms are introduced by substitution in the program controlmodels the initial time and the initial model position to an arbitrary time moment and to anarbitrary state. Such replacement may result in that we will need to realize an impulse controlaction at each time moment. This fact leads to the appearance of a moving or a so called slidingimpulse. Such phenomenon from the point of view of the theory of dierential equations requiresan appropriate formalization. In addition, this motion type in the space of positions creates themotion sliding on some functional manifold. Impulse-sliding regimes in systems without delay wereconsidered in [1{3]. Impulse-sliding regimes for linear systems with delay were studied in [4]. Thereaction of nonlinear systems with delay to impulse actions is understood here as in the paper [5].The denition of a solution of nonlinear systems with delay given in [5] is a generalization for thenotion of solution for systems without delay in [6, 7].1. Formalization of impulse-sliding regimeConsider a dynamic system with impulse control_x(t) = f t; x(t); x(t  )+B(t; x(t))u; t 2 [t0; #]; (1.1)with the initial conditionx(t) = '(t); t 2 [t0   ; t0]; (1.2)where f(; ; ) is a function with value in Rn, B(; ) is a m  n-matrix function. Elements off and B are continuous functions and satisfy the conditions, which guarantee the existence and1The research was supported by Russian Science Foundation (RSF) (project No.16{11{10146).142 Alexander N. Sesekin and Natalya I. Zhelonkinauniqueness of a solution for any summable function u(t). Let xt() be a function-prehistory xt() =fx(t+s);    s < 0g. The function '(t) here is a function of bounded variation for t 2 [t0 ; t0].We will assume that the function B(t; x) satises the well-known Frobenius condition [8],nX=1@bij(t; x)@xbl(t; x) =nX=1@bil(t; x)@xbj(t; x): (1.3)According to [5, 6] this condition ensures the uniqueness of the system response to the impactof a generalized control u(t) (the generalized derivative of a bounded variation function). We notethat there are various ways of dening a solution for the equation (1.1) which lead generally tovarious implementations of the trajectories [6]. We will use the denition that is based on theclosure of the set of smooth trajectories in the space of functions of bounded variation [6]. Thisdenition is the most natural from the point of view of control theory. This is due to the fact thatimpulse controls are often some control idealizations operating in short time intervals and withgreat intensity.By an impulse positional control we will mean an operator t; xt()  ! U(t; x(t)) mapping thespace of extended states ft; x(t)g into the space of m-vector-valued distributionsU(t; x(t)) = r(t; x(t)) t: (1.4)In this paper we assume that a delay is only in f(t; x(t); x(t  )) and a control function doesnot contain a delay.Here r(t; x(t)) ism-dimensional vector function, t is the Dirac impulse function concentrated att. The system reaction to the impulse position control U(t; x(t)) (which we call an impulse-slidingregime) is dened as the set of Euler polygonal functions xh(); h = max(tk+1   tk) correspondingto all decompositions t0 < t1 < ::: < tp = # of the interval [t0; #]. The Euler polygonal function(Euler line) xh() is constructed as a left continuous function of bounded variation such that theequation holds_xh(t) = f t; xh(t); xh(t  )+ pXi=1B(t; xh(t))r(ti; x(ti))ti (1.5)with the initial condition x(t) = '(t); t 2 [t0   ; t0]:The Euler line satises the equationxh(t) = '(t0) +Z tt0f ; xh(); xh(   ) d +Xti<tS ti; xh(ti); r(ti; x(ti))(1.6)and the jump functions are dened by the equationsS ti; xh(ti); r(ti; xh(ti))= z(1)  z(0); (1.7)_z() = B(t; z()) r(ti; xh(ti)); z(0) = xh(ti): (1.8)Here the jump function S(t; x; ) is the solution of the equation@y@= B(t; y): (1.9)We will assume that the equalityrt; x(t) + S t; x(t); r(t; x(t))= 0: (1.10)is true.This equality means that after the action of impulse at the system at time t, the state ft; x(t)gwill belong to the manifold r(t; x(t)) = 0:Impulse-sliding regimes in systems with delay 1432. Properties of the impulse-sliding regimeLemma 1. Assume that for all admissible values t1; t2; x1; x2; y1 and y2 the following inequal-ities are truekf(t; x; y)k  C 1 + sup[t0  ]kx()k; (2.1)kS(t1; x1; r(t1; x1))  S(t2; x2; r(t2; x2))k  L jt1   t2j+ kx1   x2k: (2.2)Then for all decompositions h and all t 2 [t0; #] the set of Euler polygonal functions xh() is bounded,what means that there exists a constant M such thatkxh(t)k M: (2.3)P r o o f. From (1.6) and (2.1) the following inequality followskxh(t)k  k'(t0)k+ CZ tt0 1 + sup[t0 ;]kxh()kd +Xti<tS ti; xh(ti); r(ti; xh(ti): (2.4)Due to the fact thatS ti 1; xh(ti 1 + 0); r(ti 1; xh(ti 1 + 0)= 0;in view of (2.2), we have the inequalitieskS(ti; xh(ti); r(ti; xh(ti))k = kS(ti; xh(ti); r(ti; xh(ti))k  S ti 1; xh(ti 1 + 0); r(ti 1; xh(ti 1 + 0)  L ti   ti 1 + kxh(ti)  xh(ti 1 + 0)k: (2.5)At the same time, in view of (2.1),kxh(ti)  xh(ti 1 + 0)k Z titi 1kf(; xh(); xh(   ))kd Cti   ti 1 + LZ titi 1(1 + sup[t0 ;]kxh()k)d:(2.6)In consequence, from (2.4) in view of (2.5) and (2.6) we get the following inequalitykxh(t)k  k'(t0)k+ (L+ C)(t  t0) + L(1 + C)Z tt0sup[t0 ;]kxh()kd: (2.7)As in [9], from the last inequality we getsup[t0 ;t]kxh()k  R+ (L+ C)(t  t0) + L(1 + C)Z tt0sup[t0 ;]kxh()kd; (2.8)whereR = sup[t0 ;t0]k'()k:Applying the result of [10] we get from (2.8) the estimatesup kxh()k   R+ (L+ C)(#  t0)eL(1+C)(# t0);which completes the proof of the lemma. 144 Alexander N. Sesekin and Natalya I. ZhelonkinaNote that as a constant M we can take the following numberM = R+ (L+ C)(#  t0)eL(1+C)(# t0):Let D be a bounded closed set which contains all xh(). By continuity we may assume that allfunctions f(t; x; y), B(t; x) and r(t; x) are bounded.DenoteM1 = max[t0;#]DDkf(t; x; y)k; M2 = max[t0;#]DkB(t; x)k; M3 = max[t0;#]Dkr(t; x)k: (2.9)Lemma 2. Under the above assumptions from each connal sequence of Euler functions fxh()gwe can select a subsequence fxhp()g uniformly at (t0; #] converging to absolutely continuous func-tion x(). Moreover for all t 2 (t0; #] we have r(t; x(t)) = 0 x(t) = '(t) for t 2 [t0 ; t0], in otherwords the limit element of the impulse-sliding regime moves over the manifold which is describedby the equation r(t; x(t)) = 0.P r o o f. The proof of convergence of xh() uses the generalization of Arzela's lemma from [11].Let xhi() be a connal sequence. Then according to (1.6) we havekxhi(t00)  xhi(t0)k Z t00t0kf(t; xh(t); xh(t  )kds+m(t00)Xk=m(t0)+1kS(tk; xhi(tk); r(tk; xhi(tk)))k; (2.10)where m(t) is the nearest point on the left in the decomposition which generates the polygonal linexhi(): In accordance with (1.6) we havekS(tk; xhi(tk); r(tk; xh(tk)))k = kS(tk; xh(tk); r(tk; xh(tk)))k  S tk 1; xhi(tk 1 + 0); r(tk 1; xhi(tk 1 + 0)):Considering (2.2) we getkS(tk; xhi(tk); r(tk; xh(tk)))k  L tk   tk 1 + kxhi(tk)  xhi(tk 1 + 0)k:At the same timexhi(tk)  xhi(tk 1 + 0) =Z tktk 1f(; xh())d:By taking into account (2.8), we obtainS tk; xhi(tk); r(tk; xh(tk))  L tk   tk 1 +M1(tk   tk 1) = L(1 +M1)(tk   tk 1): (2.11)From (2.10) and (2.11) it follows thatkxhi(t00)  xhi(t0)k   M1 + L(1 +M1)(t00   t0) + L(2 +M)(t0   ttihi); (2.12)where ttihi is the nearest point at the left in partition hi to the point t0: The last inequality allowsto apply the generalization of Arzela's lemma from [11] and ensures the existence of a subsequencexhi() which uniformly converges to the function x():Now we pass to the limit in the inequality (2.12) as i!1: As a result we havekx(t00)  x(t0)k  (M1 + L(1 +M1))(t00   t0):This means that x(t) is an absolutely continuous function at (t0; #]:Impulse-sliding regimes in systems with delay 145Now let us show that the limit element xh() belongs to the manifold r(t; x) = 0: Let tmthi bethe nearest point from the left in partition hi by the time t. The following inequality holdskr(t; x(t)k  kr(t; x(t))  r(t; xhi(t)) + r(t; hi(t))k kr(t; x(t))  r(t; xhi(t))k+ kr(tmthixhi(tmthi + 0))  r(t; xhi(t))k Lkx(t)  xhi(t)k+ (t  tmthi) + kxhi(tmthi + 0)  xhi(t)k Lkx(t)  xhi(t)k+ (L+M)(t  tmihi):By the uniform convergence of the sequence xhi() the rst term at the right hand part at thelast inequality tends to zero. The second one tends to zero because hi ! 0 when i!1. Thereforer(t; x(t))  0 when t 2 (t0; #], this completes the proof of lemma. Lemma 3. Let r(t; x) be a vector function continuously dierentiable in all variables. Thenthe following equality holdsS(tk; xh(tk); r(tk; xh(tk)))  S(tk 1; xh(tk 1 + 0); r(tk 1; xh(tk 1 + 0)))=Z tktk 1@S(; xh(); r(t; xh()))@+@S(; xh(); r(; xh()))@xf(; xh(); xh(   ))+ (2.13)+@S(; xh(); r(; xh()))@r@r(; xh())@+@r(; xh())@x_f(; xh(); xh(   ))d:The lemma follows from the the formula for dierentiating a composite function.Theorem 1. Let all assumptions given above hold. Then an impulse-sliding regime on (t0; #]is described by the equation_x(t) =@S(t; x(t); r(t; x(t))))@t+@S(t; x(t); r(t; x(t)))@r@r(t; x(t))@t++E +@S(t; x(t); r(t; x(t)))@x+@S(t; x(t); r(t; x(t)))@r @r(t; x(t)))@xf(t; x(t); x(t  ));(2.14)x(t0 + 0) = x(t0) + S t0; x(t0); r(t0; x(t0)):P r o o f. According to (1.6) and Lemma 3 xh(t) satises the equationxhi(t) = '(t0) +Z tt0f(; xhi(); xhi(   ))d +Z tmhit0@S(; xhi(); r(t; xhi()))@++@S(; xhi(); r(; xhi()))@x+@S(; xhi(); r(; xhi()))@r @r(; xhi())@xf(; xhi(); xhi(   ))++@S(; xhi(); r(; xhi()))@r @r(; xhi())@d:Passing to the limit at the last equation and bearing in mind that x(t) is an absolutely continuousfunction, we can see that the theorem is true. 146 Alexander N. Sesekin and Natalya I. Zhelonkina3. ConclusionThe formalization of the impulse-sliding regime for a nonlinear system with time delay is made.The equation to describe the limiting element of impulse-sliding regime is obtained.REFERENCES1. Zavalishchin S.T., Sesekin A.N. Impulse-sliding regimes of nonlinear dynamic systems // Dier.Equations, 1983. Vol. 19, no. 5. P. 562{571.2. Finogenko I.A., Ponomarev D.V. On dierential inclusions with positional discontinuous and pulsecontrols // Trudy Inst. Mat. i Mekh. UrO RAN, 2013. Vol. 19, no. 1. P. 284{299.3. Sesekin A.N., Nepp A.N. Impulse position control algorithms for nonlinear systems // AIP Confer-ence Proceeding, 2015. Vol. 1690, 040002. P. 1{6. http://dx.doi.org/10.1063/1.49367094. Andreeva I.Yu., Sesekin A.N. Degenerate linear-quadratic optimization with time delay // Autom.Remote Control, 1997. Vol. 58, no. 7. Part 1. P. 1101{1109.5. Fetisova Yu.V., Sesekin A.N. Discontinuous solutions of dierential equations with time delay //WSEAS transactions on systems, 2005. Vol. 4, no. 5. P. 487{492.6. Zavalishchin S.T, Sesekin A.N. Dynamic impulse systems: theory and applications. Kluwer Aca-demic Publishers. Dolrbrecht, 1997.7. Sesekin A.N. Dynamic systems with nonlinear impulse structure // Proceedings of the Steklov Instituteof Mathematics (Supplementary issues), 2000. Suppl. 2, P. S158{S172.8. Cartan H. Formes dierentielles. Calcul dierentiel. Hermann Paris, 1997.9. Lukojanov N.Yu. Functional equations of Hamilton-Jacobi and control tasks with hereditary infor-mation. Ural Federal University. Ekaterinburg, 2011.10. Bellman R., Cooke K. Dierential-dierence equations. Academik Press. New Yourk, London, 1963.11. Filippov A.F. The existence of solutions of generalized dierential equations // Mathematical Notes.Vol. 10, no. 3. P. 307{313.

IMPULSE–SLIDING REGIMES IN SYSTEMS WITH DELAY

Institutional repository of Ural Federal University named after the first President of Russia B.N.Yeltsin

URAL MATHEMATICAL JOURNAL, Vol. 2, No. 2, 2016IMPULSE–SLIDING REGIMES IN SYSTEMS WITH DELAY1Alexander N. SesekinKrasovskii Institute of Mathematics and Mechanics,Ural Branch of Russian Academy of Sciences, Ekaterinburg, Russia,sesekin@imm.uran.ruNatalya I. ZhelonkinaKrasovskii Institute of Mathematics and Mechanics,Ural Branch of Russian Academy of Sciences, Ekaterinburg, Russia,312115@mail.ruAbstract: The paper is devoted to the formalization of a concept of impulse-sliding regimes generated bypositional impulse controls for systems with delay. We define the notion of impulse-sliding trajectory as a limitof a sequence of Euler polygonal lines generated by a discrete approximation of the impulse position control.The equations describing the trajectory of impulse-sliding regime are received.Key words: Impulse position control, Systems with delay, Impulse-sliding regime, Euler polygonal lines.IntroductionUsually, the positional control algorithms are introduced by substitution in the program controlmodels the initial time and the initial model position to an arbitrary time moment and to anarbitrary state. Such replacement may result in that we will need to realize an impulse controlaction at each time moment. This fact leads to the appearance of a moving or a so called slidingimpulse. Such phenomenon from the point of view of the theory of differential equations requiresan appropriate formalization. In addition, this motion type in the space of positions creates themotion sliding on some functional manifold. Impulse-sliding regimes in systems without delay wereconsidered in [1–3]. Impulse-sliding regimes for linear systems with delay were studied in [4]. Thereaction of nonlinear systems with delay to impulse actions is understood here as in the paper [5].The definition of a solution of nonlinear systems with delay given in [5] is a generalization for thenotion of solution for systems without delay in [6, 7].1. Formalization of impulse-sliding regimeConsider a dynamic system with impulse controlẋ(t) = f(t, x(t), x(t− τ))+B(t, x(t))u, t ∈ [t0, ϑ], (1.1)with the initial conditionx(t) = φ(t), t ∈ [t0 − τ, t0], (1.2)where f(·, ·, ·) is a function with value in Rn, B(·, ·) is a m × n-matrix function. Elements off and B are continuous functions and satisfy the conditions, which guarantee the existence and1The research was supported by Russian Science Foundation (RSF) (project No.16–11–10146).142 Alexander N. Sesekin and Natalya I. Zhelonkinauniqueness of a solution for any summable function u(t). Let xt(·) be a function-prehistory xt(·) ={x(t+s); −τ ≤ s < 0}. The function φ(t) here is a function of bounded variation for t ∈ [t0−τ, t0].We will assume that the function B(t, x) satisfies the well-known Frobenius condition [8],n∑ν=1∂bij(t, x)∂xνbνl(t, x) =n∑ν=1∂bil(t, x)∂xνbνj(t, x). (1.3)According to [5, 6] this condition ensures the uniqueness of the system response to the impactof a generalized control u(t) (the generalized derivative of a bounded variation function). We notethat there are various ways of defining a solution for the equation (1.1) which lead generally tovarious implementations of the trajectories [6]. We will use the definition that is based on theclosure of the set of smooth trajectories in the space of functions of bounded variation [6]. Thisdefinition is the most natural from the point of view of control theory. This is due to the fact thatimpulse controls are often some control idealizations operating in short time intervals and withgreat intensity.By an impulse positional control we will mean an operator t, xt(·) −→ U(t, x(t)) mapping thespace of extended states {t, x(t)} into the space of m-vector-valued distributionsU(t, x(t)) = r(t, x(t)) δt. (1.4)In this paper we assume that a delay is only in f(t, x(t), x(t− τ)) and a control function doesnot contain a delay.Here r(t, x(t)) is m-dimensional vector function, δt is the Dirac impulse function concentrated att. The system reaction to the impulse position control U(t, x(t)) (which we call an impulse-slidingregime) is defined as the set of Euler polygonal functions xh(·), h = max(tk+1 − tk) correspondingto all decompositions t0 < t1 < ... < tp = ϑ of the interval [t0, ϑ]. The Euler polygonal function(Euler line) xh(·) is constructed as a left continuous function of bounded variation such that theequation holdsẋh(t) = f(t, xh(t), xh(t− τ))+p∑i=1B(t, xh(t))r(ti, x(ti))δti (1.5)with the initial condition x(t) = φ(t), t ∈ [t0 − τ, t0].The Euler line satisfies the equationxh(t) = φ(t0) +∫ tt0f(ξ, xh(ξ), xh(ξ − τ))dξ +∑ti<tS(ti, xh(ti), r(ti, x(ti)))(1.6)and the jump functions are defined by the equationsS(ti, xh(ti), r(ti, xh(ti)))= z(1)− z(0), (1.7)ż(ξ) = B(t, z(ξ)) r(ti, xh(ti)), z(0) = xh(ti). (1.8)Here the jump function S(t, x, µ) is the solution of the equation∂y∂µ= B(t, y). (1.9)We will assume that the equalityr(t, x(t) + S(t, x(t), r(t, x(t))))= 0. (1.10)is true.This equality means that after the action of impulse at the system at time t, the state {t, x(t)}will belong to the manifold r(t, x(t)) = 0.Impulse-sliding regimes in systems with delay 1432. Properties of the impulse-sliding regimeLemma 1. Assume that for all admissible values t1, t2, x1, x2, y1 and y2 the following inequal-ities are true∥f(t, x, y)∥ ≤ C(1 + sup[t0−τ ]∥x(·)∥), (2.1)∥S(t1, x1, r(t1, x1))− S(t2, x2, r(t2, x2))∥ ≤ L(|t1 − t2|+ ∥x1 − x2∥). (2.2)Then for all decompositions h and all t ∈ [t0, ϑ] the set of Euler polygonal functions xh(·) is bounded,what means that there exists a constant M such that∥xh(t)∥ ≤ M. (2.3)P r o o f. From (1.6) and (2.1) the following inequality follows∥xh(t)∥ ≤ ∥φ(t0)∥+ C∫ tt0(1 + sup[t0−τ,ξ]∥xh(·)∥)dξ +∑ti<t∥∥S(ti, xh(ti), r(ti, xh(ti))∥∥. (2.4)Due to the fact thatS(ti−1, xh(ti−1 + 0), r(ti−1, xh(ti−1 + 0))= 0,in view of (2.2), we have the inequalities∥S(ti, xh(ti), r(ti, xh(ti))∥ = ∥S(ti, xh(ti), r(ti, xh(ti))∥−−S(ti−1, xh(ti−1 + 0), r(ti−1, xh(ti−1 + 0))≤ L(ti − ti−1 + ∥xh(ti)− xh(ti−1 + 0)∥).(2.5)At the same time, in view of (2.1),∥xh(ti)− xh(ti−1 + 0)∥ ≤∫ titi−1∥f(ξ, xh(ξ), xh(ξ − τ))∥dξ≤ C(ti − ti−1 + L∫ titi−1(1 + sup[t0−τ,ξ]∥xh(·)∥)dξ).(2.6)In consequence, from (2.4) in view of (2.5) and (2.6) we get the following inequality∥xh(t)∥ ≤ ∥φ(t0)∥+ (L+ C)(t− t0) + L(1 + C)∫ tt0sup[t0−τ,ξ]∥xh(·)∥dξ. (2.7)As in [9], from the last inequality we getsup[t0−τ,t]∥xh(·)∥ ≤ R+ (L+ C)(t− t0) + L(1 + C)∫ tt0sup[t0−τ,ξ]∥xh(·)∥dξ, (2.8)whereR = sup[t0−τ,t0]∥φ(·)∥.Applying the result of [10] we get from (2.8) the estimatesup ∥xh(·)∥ ≤(R+ (L+ C)(ϑ− t0))eL(1+C)(ϑ−t0),which completes the proof of the lemma. 144 Alexander N. Sesekin and Natalya I. ZhelonkinaNote that as a constant M we can take the following numberM =(R+ (L+ C)(ϑ− t0))eL(1+C)(ϑ−t0).Let D be a bounded closed set which contains all xh(·). By continuity we may assume that allfunctions f(t, x, y), B(t, x) and r(t, x) are bounded.DenoteM1 = max[t0,ϑ]×D×D∥f(t, x, y)∥, M2 = max[t0,ϑ]×D∥B(t, x)∥, M3 = max[t0,ϑ]×D∥r(t, x)∥. (2.9)Lemma 2. Under the above assumptions from each confinal sequence of Euler functions {xh(·)}we can select a subsequence {xhp(·)} uniformly at (t0, ϑ] converging to absolutely continuous func-tion x(·). Moreover for all t ∈ (t0, ϑ] we have r(t, x(t)) = 0(x(t) = φ(t) for t ∈ [t0−τ, t0]), in otherwords the limit element of the impulse-sliding regime moves over the manifold which is describedby the equation r(t, x(t)) = 0.P r o o f. The proof of convergence of xh(·) uses the generalization of Arzela’s lemma from [11].Let xhi(·) be a confinal sequence. Then according to (1.6) we have∥xhi(t′′)− xhi(t′)∥ ≤∫ t′′t′∥f(t, xh(t), xh(t− τ)∥ds+m(t′′)∑k=m(t′)+1∥S(tk, xhi(tk), r(tk, xhi(tk)))∥, (2.10)where m(t) is the nearest point on the left in the decomposition which generates the polygonal linexhi(·). In accordance with (1.6) we have∥S(tk, xhi(tk), r(tk, xh(tk)))∥ = ∥S(tk, xh(tk), r(tk, xh(tk)))∥−−∥∥S(tk−1, xhi(tk−1 + 0), r(tk−1, xhi(tk−1 + 0)))∥∥.Considering (2.2) we get∥S(tk, xhi(tk), r(tk, xh(tk)))∥ ≤ L(tk − tk−1 + ∥xhi(tk)− xhi(tk−1 + 0)∥).At the same timexhi(tk)− xhi(tk−1 + 0) =∫ tktk−1f(ξ, xh(ξ))dξ.By taking into account (2.8), we obtain∥∥S(tk, xhi(tk), r(tk, xh(tk)))∥∥ ≤ L(tk − tk−1 +M1(tk − tk−1)) = L(1 +M1)(tk − tk−1). (2.11)From (2.10) and (2.11) it follows that∥xhi(t′′)− xhi(t′)∥ ≤(M1 + L(1 +M1))(t′′ − t′) + L(2 +M)(t′ − ttihi), (2.12)where ttihi is the nearest point at the left in partition hi to the point t′. The last inequality allowsto apply the generalization of Arzela’s lemma from [11] and ensures the existence of a subsequencexhi(·) which uniformly converges to the function x(·).Now we pass to the limit in the inequality (2.12) as i → ∞. As a result we have∥x(t′′)− x(t′)∥ ≤ (M1 + L(1 +M1))(t′′ − t′).This means that x(t) is an absolutely continuous function at (t0, ϑ].Impulse-sliding regimes in systems with delay 145Now let us show that the limit element xh(·) belongs to the manifold r(t, x) = 0. Let tmthi bethe nearest point from the left in partition hi by the time t. The following inequality holds∥r(t, x(t)∥ ≤ ∥r(t, x(t))− r(t, xhi(t)) + r(t, hi(t))∥≤ ∥r(t, x(t))− r(t, xhi(t))∥+ ∥r(tmthixhi(tmthi + 0))− r(t, xhi(t))∥≤ L[∥x(t)− xhi(t)∥+ (t− tmthi) + ∥xhi(tmthi + 0)− xhi(t)∥]≤ L[∥x(t)− xhi(t)∥+ (L+M)(t− tmihi)].By the uniform convergence of the sequence xhi(·) the first term at the right hand part at thelast inequality tends to zero. The second one tends to zero because hi → 0 when i → ∞. Thereforer(t, x(t)) ≡ 0 when t ∈ (t0, ϑ], this completes the proof of lemma. Lemma 3. Let r(t, x) be a vector function continuously differentiable in all variables. Thenthe following equality holdsS(tk, xh(tk), r(tk, xh(tk)))− S(tk−1, xh(tk−1 + 0), r(tk−1, xh(tk−1 + 0)))=∫ tktk−1[∂S(ξ, xh(ξ), r(t, xh(ξ)))∂ξ+∂S(ξ, xh(ξ), r(ξ, xh(ξ)))∂xf(ξ, xh(ξ), xh(ξ − τ))+ (2.13)+∂S(ξ, xh(ξ), r(ξ, xh(ξ)))∂r(∂r(ξ, xh(ξ))∂ξ+∂r(ξ, xh(ξ))∂xḟ(ξ, xh(ξ), xh(ξ − τ)))]dξ.The lemma follows from the the formula for differentiating a composite function.Theorem 1. Let all assumptions given above hold. Then an impulse-sliding regime on (t0, ϑ]is described by the equationẋ(t) =∂S(t, x(t), r(t, x(t))))∂t+∂S(t, x(t), r(t, x(t)))∂r∂r(t, x(t))∂t++[E +∂S(t, x(t), r(t, x(t)))∂x+∂S(t, x(t), r(t, x(t)))∂r× ∂r(t, x(t)))∂x]f(t, x(t), x(t− τ)),(2.14)x(t0 + 0) = x(t0) + S(t0, x(t0), r(t0, x(t0))).P r o o f. According to (1.6) and Lemma 3 xh(t) satisfies the equationxhi(t) = φ(t0) +∫ tt0f(ξ, xhi(ξ), xhi(ξ − τ))dξ +∫ tmhit0[∂S(ξ, xhi(ξ), r(t, xhi(ξ)))∂ξ++(∂S(ξ, xhi(ξ), r(ξ, xhi(ξ)))∂x+∂S(ξ, xhi(ξ), r(ξ, xhi(ξ)))∂r× ∂r(ξ, xhi(ξ))∂x)f(ξ, xhi(ξ), xhi(ξ − τ))++∂S(ξ, xhi(ξ), r(ξ, xhi(ξ)))∂r· ∂r(ξ, xhi(ξ))∂ξ]dξ.Passing to the limit at the last equation and bearing in mind that x(t) is an absolutely continuousfunction, we can see that the theorem is true. 146 Alexander N. Sesekin and Natalya I. Zhelonkina3. ConclusionThe formalization of the impulse-sliding regime for a nonlinear system with time delay is made.The equation to describe the limiting element of impulse-sliding regime is obtained.REFERENCES1. Zavalishchin S.T., Sesekin A.N. Impulse-sliding regimes of nonlinear dynamic systems // Differ.Equations, 1983. Vol. 19, no. 5. P. 562–571.2. Finogenko I.A., Ponomarev D.V. On differential inclusions with positional discontinuous and pulsecontrols // Trudy Inst. Mat. i Mekh. UrO RAN, 2013. 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Impulse–Sliding Regimes in Systems with Delay

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