CC BY
22
Серия «Математика» 2020. Т. 31. С. 96-110
Онлайн-доступ к журналу: http: / / mathizv.isu.ru
ИЗВЕСТИЯ
Иркутского государственного ■университета
YAK 517.929 MSG 34K20
DOI https://doi.org/10.26516/1997-7670.2020.31.96
On the Stability of Tubes of Discontinuous Solutions of Bilinear Systems with Delay
A. N. Sesekin1'2, N. I. Zhelonkina1
1 Ural Federal University, Ekaterinburg, Russian Federation,
2N. N. Krasovskii Institute of Mathematics and Mechanics UB RAS, Ekaterinburg, Russian Federation
Abstract. The paper considers the stability property of tubes of discontinuous solutions of a bilinear system with a generalized action on the right-hand side and delay. A feature of the system under consideration is that a generalized (impulsive) effect is possible non-unique reaction of the system. As a result, the unique generalized action gives rise to a certain set of discontinuous solutions, which in the work will be called the tube of discontinuous solutions.The concept of stability of discontinuous solutions tubes is formalized. Two versions of sufficient conditions for asymptotic stability are obtained. In the first case, the stability of the system is ensured by the stability property of a homogeneous system without delay; in the second case, the stability property is ensured by the stability property of a homogeneous system with delay. These results generalized the similar results for systems without delay.
Keywords: differential equations with delay, impulsive disturbance, stability.
1. Problem Statement
Consider the bilinear system of the differential equations with delay
Dj(t)vj(t) x(t) +g(t)vm(t) + AT(t)x(t - r)+
о
+ G(t,s)x(t + s)ds + f(t), t^t0 (1.1)
x(t) = A(t) +
m— ]
E
Here A(t), AT(t), G(t,s), Dj(t), j € l,m are continuous of the bounded matrix functions of dimension n x n, Vj(t) (j € 1, m) are components bounded variation vector functions v(t) = (vi(t),v2(t),... ,vm(t))T, r > 0 is constant delay, <p(t) — is initial function, which is n-dimensional vector function of bounded variation, defined on [to — r, to], f(t) is n-dimensional vector function with the integrated elements, g{t) — is continuous n -dimensional vector function.
Characteristic of the system (1.1) is that its right part contains an incorrect multiplication operation discontinuous function to generalized one. This is due to the next fact. If the function v(t) is discontinuous at some moment time, then the system is subjected to impulse action at this moment. Therefore, the function x(t) appears breaking at the same
m
moment, and in the term Dj(t)vi(t) x(t) is an incorrect operation of
j=i
multiplying a generalized function by a discontinuous functions, which leads to the problem of formalizing the concept of decisions.
The formalization of the concept of a solution to the system (1.1) has been considered by various authors. The monograph [10] provides a fairly complete overview of possible approaches. Note that various formalizations of the concept of a solution lead to different trajectories. In this work we will take an approach, which is based on the approximation of generalized actions by smooth approximations and the determination of a solution based on the closure of the set of continuous trajectories resulting from the approximation of generalized actions by summable functions. For more information, see [2;4;10]. N.N. Krasovskii in the monograph [3] noted, that such definition is natural from the point of view of control theory. In the case where the matrices Dj are mutually commutative, for any admissible t > to, any sequence Vk(t), which converges pointwise to v(t), generates a sequence of solutions of the equation (1.1), which will have a single limit and will not depend on the method of approximating the function of the function of bounded variation v(t). This case for systems without delay was considered in [2; 4; 10], and for systems with delay was considered in
[7].
In the case when the sequence of smooth solutions is not convergent, in [6; 10] it is proposed to take all partial pointwise limits of such sequence. As in [6; 10] we will say that the sequence Vk(t) V - converges to v(t), if Vk(t) converges pointwise to v(t) and var Vk(-) converges pointwise to
[io, t]
V(t) € BV[to, 1?]. For this convergence we will use the symbol Vk(t) v(t).
Definition 1. Any partial pointwise sequence limit Xk(t), k = 1,2,..., generated by arbitrary V-convergent sequence of absolutely continuous functions Vk(t), k = 1,..., we will call V - solution of the system (1.1), which satisfies the initial condition x{t) = <p(t), t € [to — r,to}-
Let z(0) = x(t), 0) = v(t) are the initial conditions of the system
m— 1
m = E пгСшыо + f(t)Vm(o m = v(o- a-2)
i= 1
According to [10] all V— solutions of the equation (1.1) will satisfy the following integral inclusion:
t t to
+ j №% + j Cte)x(Z-T)dt + j j G(C,s)x(C + s)dCds+
to to to —t
+ E 8(и,х(и-0),Аь(и-0),У(и-0),АУ(и-0))+
ti<t, ti€fl-
+ E Б(и,х(и),Аь(и + 0),У(и),АУ(и + 0)) (1.3)
ti<t, ti€fl +
where vc(t) - is the continuous component of the vector of the function of bounded variation v(t).
In (1.3) set S(t, x(t), Av(t), V(t), AV(t)) (where t = U - 0 U € and t = ti if ti € f2+) defined as a sectional shift (¡j,(AV(t)) = v(ti) if ti € and /j,(AVt) = v(ti + 0), if U € П+) are system reachability sets (1.2) at a size —x(t) at the moment £ = AV(t), where the control r?(£) satisfies the constraint MOW < 1, MOW = TZi 1^(61-
Thus, to each discontinuity point (left or right) of the function v(t) and every possible jump in the trajectory of the system (1.1) at moment t the function defined on the segment [0, AV(t)\, which, by solving the
system of equations (1.2) will determine the jump value of Ax(t) of the trajectory at the time t.
Definition 2. Continuing solutions of integral inclusion (1.3) for [to, сю) will be called the solution of the equation (1.1) on the interval [¿o,oo).
We denote the tube section of the solutions of the integral inclusion (1.3) by X(t,(po( ),v(-), V(-)) which is generated by the initial condition ^o(') and a pair of functions v(-), V(-)
Definition 3. We say that the solution tube for the integral inclusion X(t,ipo(-),v(-), V(-)) is stable ¿/Ve > 0 35(e) > 0, what if
sup 1Ы0-¥>i(OI < <*>
?€[i0-T,i0]
that
p(X(t,M-),v(-), V(-)),X(t,Vl(0,v(-), V(-)))<£,
for any t > to, where p(A,B) is the Hausdorff distance between the sets A and B.
Definition 4. We say that the solution tube for the integral inclusion X(t,Lpo(-),v(-), is asymptotically stable, if it is stable, and also equal-
ity is validity
lim p(X(t,x0,v(-), V(-)),X(t,x,v(-), V(-)))=0.
t—>-oo
2. Stability of discontinuous solutions tubes
The results obtained below are a generalization of [9] for systems with delay.
Theorem 1. Let the fundamental matrix Y(t, s) of the system x = A(t)x satisfy the estimate
\\Y(t,s)\\ < ce-(a(i"s)), (2.1)
where a and c are some constants such that a > 0, c > 1, t > s > to- In addition, suppose that the estimates
\\Dj(t)\\<K, ||G(i,s)|| <K Vt € [i0,oo),s € [t0,t],j € l,m-l (2.2)
Here K - is a positive constant. Then ifx(t) andx(t) are integral inclusion solutions generated by the initial conditions <po(t) and <fi(t), as well as the same system of functions ?y®(£) which generates jumps of the trajectory s x{t) andx{t), then the following estimate holds:
\\x(t) - x(t)\\ < sup \\^(0-MO\\><e~{a{t~to)~cK<t~to+vm (2.3)
çe[i0-r,i0]
Proof According to [6], x{t) and x{t) will satisfy the integral equation
J. m-1 * *
x(t) = <p(to) + J A(0x(0d(+J2 J Dj(0x(0dvcj(0 + J g(Odvcm(0+
to j=1 to to
t t
+j MtMÇ-T)dÇ+j /(£)d£+ £ S(U,x(U-o), F(i,-0),
to to ti<t,ti&n
AV(U-0))+ Y, S(tt,x(tt),r,^(-),V(U),AV(tt)), (2.4)
ti<t, ti&Qj
where <p(to) will be equal to ^o(io) and <fi(to) for x(t) and x(t) respectively,
S(tMV,V®(-),V(t),AV(T)) = z(AV(t))-x(t), where z(£) is a solution to the equation
m— 1
¿(0 = E D^(0rif\0+gM\0, z(0) = x(t). (2.5)
3=1
Using the Cauchy formula [10] we get that the solutions x(t) and x(t) will satisfy the integral equation
t t
TO—1
it)=Y(t,toMtQ)+J2 J Y(t,0Dj(0x(0dvcj(0+jY(t,0g(0dvcm(0 +
TO— 1
j=1to to
t 0
+ j Y(t,OMtiMZ-T)dt+J I Y(t,OG&s)x(Z-s)dZds+
to to —t
+ E y(t,U)S(U,x(U-0),r]^-°\-),V(U-0),AV(U-0))+
ti<t, ti&Q-
+ E Y{t,U)S{tl,x{ti),r}^\-),V{ti),AV{ti)) + jY(t,£)№(%.
ti<t,ti£ Q^ £
(2.6)
According to (2.6), x(t) — x(t) will satisfy the integral equation x(t) — x(t) =
m-1 *
= Y(t,to)(Mto) - yi(io)) + E / Y^Om)(x(0 - x(0) dvt(0+
ti
+ J Y(t,OMO№-T)-x(t-T))d{ + J Y(t,0 J G(t,s)№-s)-
to to —t
-x{£-s)) ds E Y(t, ti) {S(ti,x(U-o),rfu-°X-), V(ti-o), AV(U-o))-
ti<t, ti€fl-
- S(u, x{u - 0),rfu-°\-), V{u - 0), AV(ti - 0))) +
+ E Y(t,U)(S(U,x(U),r]^\-),V(U),AV(U- 0))-ti<t, ti€fl+
- S(u, x(ti), r?(ii)(-), V(ti),AV(ti + 0))). (2.7)
As shown in [9] (the equation describing the jump of the trajectory does not depend on the delay) fair inequality
a V(t)
\\z(AV(t)-x(t)-(z(AV(t))-x(t)M <K\\x(t)-x(t)\\- j
0
a V(t)
+K J \\z(0-x(t)-(z(0-xm\\\v{t)(ad0-0
According to the Gronwall-Bellman lemma [1] rom the last inequality we
get
a V(t)
\\z(AV(t))-x(t)-(z(AV(t))-x(t))\\<K J HiOmm-xiMx
0
A V(t)
k j ihwiolld? x(e o ). (2.8)
Using the obvious estimate aea < e/3<l — 1 for all a > 0 and (3 > e the inequality (2.8) with < 1 leads to the inequality
\\z(AV(t))-x(AV(t))-(z(l)-x(t))\\ < \\x(t)-x(t)\\(eeKAV^-l). (2.9)
We introduce the notation
y{t) =x{t)-x{t). (2.10)
We calculate the norms of the left and right sides in (2.7) taking into account (2.1), (2.2), (2.8), (2.9) and obvious inequality (c > 1) c(ea - 1) < eca - 1, from (2.8) we get
t
< c[e-a^\\y(t0)\\ +K [ e-^-^MOWd var vc{-)+
J [Í0,?]
to
t to
+K j e-a^\\y((-T)\\d( + K j e-^-Ü j ||y(£ - s)|| ds} +
to to —t
+ Y, -l)MU-0)\\ +
ti<t, í¿en_
+ Y e-^-V (ecKeWAVMW -l)||í/(í¿)||.
í¿<í,í¿€ H+
We multiply the last inequality by ea(í-í°) and introduce the notation
q(t) = ea^\\y(t)\\, (2.11)
we get:
t
q(t)<c\\y(to)\\+cK Í q(Od vaT Vc(-)+ J 1*0,51
to
t
+cK f eaTq(( - t) di + E (ecKe||Ay(íi"0)l1 - 1 )q{U - 0) + t0 ti<t,ti&n_
ti<t, íiSHa
We introduce another notation h(t) = sup[t_r t] q(-). Then the last inequality can be rewritten as
t
h(t) < ch(to) + cK [ h(0 d((eaT + r)£ + var vc(-)) +
J [io,?]
to
+ ^ (eeke||ay(tl-0)|| _ _ q) + £ (ecke|| av(tl)|| _
ti<t, ii€n_ ti<t, ti€fl+
Multiply the integral on the right-hand side of the last inequality by e and replace var vc(-) to Vc(t). As a result, we obtain the inequality
t
h(t) < ch(to) + cKe J h(0 d((ear + r)£ + Fc(£)) +
to
+ £ (eeke||ay(tl-0)|| _ ^^ _ q) + ^ {ecK e\\AV (U)\\ _ ti<t,ti€Cl- ti<t,ti&Q+
(2.12)
According to Lemma 5.4.3 from [10] every solution to the inequality (2.12) will satisfy the estimate
h(t) < ecKe(t-t0+V(t))c gup _
?e[i0-r,i0]
Multiplying this inequality by and taking into account the designa-
tion (2.11) we obtain the estimate (2.3). □
ON THE STABILITY OF TUBES OF DISCONTINUOUS SOLUTIONS 103 Theorem 2. Under the assumptions of the theorem 1 inequality is fair p(X(t, p(-), *(•), <), V(■)), X(t, p(.)X(0, <), V(■))) < < c sup ) - (£)|| x e-(a(t-t0)-cK<t-to+vm. (2.13)
?e[io-T,to ]
Proof. Between the sets of V — solutions X(t, p(-),x(-), v(-), V(■)) and X(t, p(-),X(-), v(-), V(■)) one-to-one correspondence is established: every trajectory from X(t, p(-), x(-), v(-), V(■)) is associated with a trajectory from X(t, </?(•), X(-), v(-), V(■)) by the rule - the initial conditions are different (p(-) and p(-)), and the system of functions , that defines jumps is the same.
First, note that according to [5] the sets X(t, p(-), x(-), v(-), V(■)) and X(t, (p(-)x(-), v(-), V(■)) are closed. Their boundedness follows from the previous theorem. Then
p(x (t, p(-), «(•), <), V (■)), X (t, ¥>(•)*(•), v(-), V (■))) =
= max{ max min ||x — y||;
xex(t,v(-),x(-),v(-),v(■)) yex(t,<p(-)x(-),v(-),v(■))
max min I|x — y||j. (2.14)
y€X(t,<p()x(),v(),V(■)) x&X(t,<p()x()v()V(■))"
Let the extremum in (2.14) achieved when X € X(t, p(-),x(-), v(-), V(■)) and y € X(t, ■ )X( ■ ), v(■ ), V(■)) i.e.
P(X(t, ■ ), x( ■ ), v(■ ), V( ■ )), X(t, ■ )X(■ ), v(■ ), V( ■ ))) =
= max min x y =
xex(t,<p(),x(),v(),v(■)) yex(t,<p№),v()v(■))
= min ||X — y|| = ||X — y||.
yex (t,<p(W)v()v (■))
The element X can be matched X € X(t, (p(-)X(-), v(-), V(■)). It's obvious that
min ||X — y|| = ||X — y|| < ||X — X||.
yex(t,<p№),v()v (■))
Then from (1.2), (1.3), (2.1) and the theorem 1 implies the validity of the theorem 2.
□
Corollary 1. Let the assumptions of theorem 1 holds. Then if
(a — cKe)(t — to + V(t)) > q,
for all t € [t0, to), where q — some constant, then the tube of solutions of the unperturbed motion X(t,p0(-),v(-), V(■)) will be stable, and if
lim (a — cKe)(t — t0 + V(t)) = to, t—^^
then the tube of solutions of the unperturbed motion X(t,(po(),v(-), V(-)) will be asymptotically stable.
Next, we consider another variant of the sufficient stability conditions for the solution tubes. First we give the Cauchy formula for a linear system with delay [1]
o
x(t) = A(t)x(t) + AT(t)x(t - t) + J G(t, s)x(t + s)ds + p(t), (2.15)
—r
with the initial condition <p(t), defined on the interval [to — r, to], p(t) — is an integrable function. The matrix functions A(t), AT(t), G(t,s) satisfy the same conditions as in the equation (1.1)
According to [1] in the case when p(t) — is an integrable function, , the solution of the equation (2.15) with the initial condition tp(t) = x(t), to — t < t < to exists and is unique.
Denote Q as the square in the plane s and t, where to<s<'dto<t<'d. According to [1] under the assumption that p(t) — is an integrable function the solution of the equation (2.15) can be represented as
to t
x(t) = F(t, toMto) + J F(t, s + t)At(s + t)v(s) ds + j F(t, s)p(s) ds.
to—T to
(2.16)
The function F(t, s) is a solution to the equation
^^ = -F(t, s)A(s) - F(t, s + t)At(s + r) (2.17)
with initial condition
F(t, t — 0) = E] F(t, s) = 0 s > t. (2.18)
Now apply the formula (2.16) to the equation (1.1) under the assumption that v(t) is an absolutely continuous function. As a result, we get:
ft 0
x(t) = F(t,t0)¡p(t0) + / F(t,s + t)Ar(s + t)ip(s)ds+
Jto-T
rn—l * *
+ E J F(t,0D,(0x(0v,(0ds + J F(t,0g(0vm(0ds+
t
+ f F(t,0№<%. (2.19)
to ¿0
We substitute in (1.2) v(t) = v where v is a sequence of absolutely continuous functions that converge pointwise to the function of bounded variation v(t). By x
we denote the sequence of absolutely continuous solutions of the equation (1.1), generated by the sequence v^k\t). It is not difficult to show that the sequence x^it) is bounded and the sequence of variations of these functions will also be uniformly bounded. Then according to Helly's theorem, from this sequence we can distinguish a subsequence x^ (t) which converges pointwise to some function of bounded variation x(t). In (1.2) we pass to the limit for hi —> oo.
The main difficulty in performing this limit transition takes place in the t
expression f Dj(£)x(£)vj(€) ds. The passage to the limit in
to
this expression can be done by replacing the time £ = t + var v^ in the
[to,t]
same way as in [6; 7; 10]. As a result, we get that x(t) will satisfy the integral equation
x{t) =
m— 1 * *
= F(t,toMtQ) + Y J F(t,0D,(0x(0dvcj(0 + J F(t,0g(0dvcm(0+
j=1 to to
+ £ F(t,U)S(U,x(U-0),r]^-°\-),V(U-0),AV(U-0))+
ti<t, ti€Cl-
t
+ Y F{t,ti)S{tl,x{ti),r}^\-),V{ti),AV{ti)) + jF(t,0№d£.
ti<t,ti£ Q^ ^
(2.20)
Suppose that, as in the theorem 1 x(t) and x(t) are integral inclusion solutions generated by the initial conditions <po(t) and <pi(t) as well as the same system of functions r?x(£)which generates jumps in the trajectories x(t) and x(t). Then for the difference x(t) — x(t) the expression is true
m— 1 *
x(t)-x(t) = F(t,t0)&i(to)-Vi(to)) + Yl / F(t,0Dl(0(x(0~
to
ii<i,ii€n_
AV(U-0)) ~ S(U,x(U - 0V{U - 0), AV(ti - 0))) +
+ £ F(t,U)(s(U,x(U),r]^\-),V(U),AV(U-0))-ti<t,ti€fl +
-S(u, X{U), rfu\-), V(U),AV(U)j). (2.21)
Further we will assume that the Cauchy matrix F(t, s) satisfies the inequality
\\F(t,s)\\ <ce-a^-s\ (2.22)
and also evaluations are performed
H^iCOII < K,j e I,m,t e [t0,oo). (2.23)
As in the theorem 1, we calculate the norms of the left and right sides in (2.21), using the estimates (2.22), (2.23) and using notation (2.10). As a result, we obtain
< ce~a<yt~to^
i
+ cK [ e"a(i"?) llym lid var vc(-)+ J [io,?]
to
+ e-»{t-ti) (есКе\\^{и-0)\\ _ _ 0)|| +
ti<t,ti€fl-
+ ^ е~а(-г~и)(ecK^Av{ti+m - l)\\y(ti)\\ (2.24)
ti<t,ti€fl +
Multiply (2.24) by and introduce the notation
p{t)=ea(t-to)bm (2_25)
As a result, we obtain
t
p(t) < c\\y(t0)\\+cK f p(0d var vc{-)+ V (ecXe"A^-°)ll-l)p(U-0)+
1 [i0'?1 u<t,u^-
+ J- (eeKe||A,(ii+0)|| _ 1)р{и) ti<t, ti€fl+
Now, as in the theorem 1, we introduce the notation
h{t) = sup p{-). (2.26)
[t-T,t]
Given the notation (2.26) the last inequality can be written as
t
h(t) < ch(t0)+cK [ h(0d var vc{-)+ V (ecKe||Ai,(ii"0)l1-1)^-0)+ i [i0,?1 U<t,ue n_
ON THE STABILITY OF TUBES OF DISCONTINUOUS SOLUTIONS 107 + £ (eeKe||A„(ti+0)|| _ l)v{ti) (2_27)
ti<t, ti€fl+
Applying Lemma 5.4.3 from [10] to (2.27) we obtain the estimate
h(t) < ec
Multiply the last inequality by e_a(i_i°) and then take into account (2.25), (2.26) and (2.10). As a result we will receive
\\x(t) - x(t)\\ <c sup - MO\\e~{a{t~to)~cKent))■ (2.28)
?e[io-r,io]
As a result, the following theorem is proved.
Theorem 3. Suppose that the fundamental matrix F(t,s) of the system (2.15) satisfies the estimate (2.22) and the matrices Di(t) satisfy the estimates (2.23). Then the solutions of the integral inclusion (1.3), generated by the initial conditions <po(t) and <pi(t), as well as the same system of functions i?fc(£), which determines the jumps of the solutions x{t) andx(t), satisfy the inequality (2.28).
Similar to theorem 2 and corollary 1 we can state the following theorem
Theorem 4. Under the conditions of theorem 3, the inequality is fair
p(X(t, <p(-),x(-),v(-), V(-)),X(t, <p(-)x(-)M-), V(-)))
<c sup - MO\\e~{a{t~to)~cKent))■ (2.29)
?€[i0-T,i0]
Therefore, if
a(t - t0) - cKeV(t)) > q
for all t € [¿o,oo), where q — is some constant, then the tube of solutions X(t,<p(-),x(-),v(-),V(-)) is stable, and if
lim (a(t — to) — cKeV(t)) = oo,
t—>-oo
then the solution tube X(t,<p(-),x(-),v(-),V(-)) is asymptotically stable.
3. Conclusion
We investigated the stability property of solutions of a bilinear system with a generalized actions and delay. A distinctive feature of the system under consideration is that a non-unique reaction is possible on the generalized actions. In this regard, the paper gives a formalization of
the concept of stability of discontinuous solution tubes and two sufficient conditions are obtained that ensure the stability of discontinuous solution tubes. The results of the paper generalize the corresponding theorems for systems without delay obtained in [9].
References
1. Bellman R. Stability Theory of Differential Equations. Dover Books on Mathematics, 2008.
2. Dykhta V.A., Samsonyuk O.N. Optimum impulse control with applications. Moscow, Fizmatlit Publ., 2000. (in Russian)
3. Krasovsky N.N. Theory of motion control. Linear system. Moscow, Nauka Publ.,1968. (in Russian)
4. Miller B.M., Rubinovich E.Ya. Discontinuous solutions in the optimal control problems and their representation by singular space-time transformations. Automation and Remote Control, 2013, vol. 74, pp.1969-2006. https:// doi.org/10.1134/S0005117913120047
5. Sesekin A.N. The properties of the attainability set of a dynamical system with impulse control. Automation and Remote Control, 1994, vol. 55, no. 2, pp. 190-195.
6. Sesekin A.N. On sets of discontinuous solutions of nonlinear differential equations. Russ. Math., 1994, vol.38, no. 6, pp.81-87.
7. Sesekin A.N., Fetisova Yu.V. Functional Differential Equations in the Space of Functions of Bounded Variation. Proceeding of the Steklov Institute of Ma,them,a,tics. 2010, vol. 269, suppl. 2. pp. 258-265. https://doi.org/10.1134/S00fel543810060210
8. Sesekin A.N., Zhelonkina N.I. On the stability of linear systems with generalized action and delay. IFAC-PapersOnLine, Proceedings of the 18th IFAC World Congress Milano, Italy, 2011, pp.13404-13407. https://doi.org/10.3182/20110828-6-IT-1002.02426
9. Sesekin A.N., Zhelonkina N.I. Tubes of Discontinuous Solutions of Dynamical Systems and Their Stability. AIP. Conference Proceeding, 2017, vol.1895, pp.050011 1-7. https://doi.Org/10.1063/l.5007383
10. Zavalishchin S.T., Sesekin A.N. Dynamic Impulse Systems: Theory and Applications. Dordrecht, Kluwer Academic Publishers, 1997.
Alexander Sesekin, Doctor of Sciences (Physics and Mathematics), Professor, Ural Federal University, 19, Mir st., Ekaterinburg, 620002, Russian Federation, tel.: (343)375-41-40, Leading Researcher, N. N. Krasovskii Institute of Mathematics and Mechanics, Ural Division of Russian Academy of Sciences, 16, S. Kovalevskay st., Ekaterinburg, 620990, Russian Federation, e-mail: sesekin01ist.ru, ORCID iD https://orcid.org/0000-0002-1339-9044.
Natalia Zhelonkina, Senior Lecturer, Ural Federal University, 19, Mir st., Ekaterinburg, 620002, Russian Federation, tel.: (343)375-41-40, e-mail: 3121150mail.ru.
Received 19.11.19
Об устойчивости трубок разрывных решений билинейных систем с запаздыванием
А. Н. Сесекин1'2, Н. И. Желонкина1
1 Уральский федеральный университет, Екатеринбург, Российская Федерация
2Институт математики и механики им. Н. Н. Красовского УрО РАН, Екатеринбург, Российская Федерация
Аннотация. Исследуется свойство устойчивости трубок разрывных решений билинейной системы с обобщенным воздействием в правой части и запаздыванием. Особенностью рассматриваемой системы является то, что на обобщенное (импульсное) воздействие возможна неединственная реакция системы. В результате единственное обобщенное воздействие в качестве реакции системы порождает некоторую совокупность разрывных решений, которую в работе будем называть трубкой разрывных решений. Формализовано понятие устойчивости трубок разрывных решений. Получены два варианта достаточных условий асимптотической устойчивости. В первом случае устойчивость системы обеспечивается свойством устойчивости однородной системы без запаздывания, во втором случае свойство устойчивости обеспечивается свойством устойчивости однородной системы с запаздыванием. Эти результаты обобщают аналогичные результаты для систем без запаздывания.
Ключевые слова: стабилизация, обратная связь, децентрализованное управление.
Список литературы
1. Bellman R. Stability Theory of Differential Equations. Dover Books on Mathematics, 2008.
2. Дыхта В. А., Самсонюк О. H. Оптимальное импульсное управление с приложениями. М. : Физматлит, 2000.
3. Красовский Н. Н. Теория управления движением. Линейные системы. М. : Наука, 1968.
4. Miller В. М., Rubinovich E.Y a. Discontinuous solutions in the optimal control problems and their representation by singular space-time transformations // Automation and Remote Control. 2013. Vol. 74. P. 1969-2006. https://doi.org/10.1134/S0005117913120047
5. Sesekin A. N. The properties of the attainability set of a dynamical system with impulse control // Automation and Remote Control. 1994. Vol. 55, N 2. P. 190-195.
6. Sesekin A. N. On sets of discontinuous solutions of nonlinear differential equations // Russ. Math. 1994. Vol. 38, N 6. P. 81—87.
7. Sesekin A. N., Fetisova Yu. V.. Functional Differential Equations in the Space of Functions of Bounded Variation // Proceeding of the Steklov Institute of Mathematics. 2010. Vol. 269, suppl. 2. P. 258-265. https://doi,10.1134/S00&1543810060210
8. Sesekin A. N., Zhelonkina N. I. On the stability of linear systems with generalized action and delay. // IFAC-PapersOnLine, Proceedings of the 18th IFAC World
Congress. Milano, Italy, 2011. P. 13404-13407. https://doi.org/10.3182/20110828-6-IT-1002.02426
9. Sesekin A. N., Zhelonkina N. I. Tubes of Discontinuous Solutions of Dynamical Systems and Their Stability // AIP. Conference Proceeding. 2017. Vol. 1895. P.050011 1-7. https://doi.Org/10.1063/l.5007383 10. Zavalishchin S. T, Sesekin A. N. Dynamic Impulse Systems: Theory and Applications. Kluwer Academic Publishers, Dordrecht, 1997.
Александр Николаевич Сесекин, доктор физико-математических наук, профессор, Институт естественных наук и математики, Уральский федеральный университет им. первого Президента России Б. Н. Ельцина, Российская Федерация, 620002, г. Екатеринбург, ул. Мира, 19; ведущий научный сотрудник, Институт математики и механики им. Н. Н. Красовского УрО РАН, Российская Федерация, 620219, г. Екатеринбург, ул. Софьи Ковалевской, 16. тел.: (343)375-41-40, e-mail: [email protected], ORCID ÍD https://orcid.org/0000-0002-1339-9044.
Наталья Игоревна Желонкина, старший преподаватель, Уральский Энергетический институт, Уральский федеральный университет им. первого Президента России Б. Н. Ельцина, Российская Федерация, 620002, г. Екатеринбург, ул. Мира, 19, тел.: (343)375-41-40, e-mail: [email protected]
Поступила в редакцию 19.11.2019
