CC BY
6
MSC 34K20, 34G10
DOI: 10.14529/ mmp230305
STABILITY OF A STATIONARY SOLUTION TO ONE CLASS OF NON-AUTONOMOUS SOBOLEV TYPE EQUATIONS
A.V. Buevich1, M.A. Sagadeeva1, S.A. Zagrebina1
1 South Ural State University, Chelyabinsk, Russian Federation E-mail: [email protected], [email protected]
The article is devoted to the study of the stability of a stationary solution to the Cauchy problem for a non-autonomous linear Sobolev type equation in a relatively bounded case. Namely, we consider the case when the relative spectrum of the equation operator can intersect with the imaginary axis. In this case, there exist no exponential dichotomies and the second Lyapunov method is used to study stability. The stability of stationary solutions makes it possible to evaluate the qualitative behavior of systems described using such equations. In addition to introduction, conclusion and list of references, the article contains two sections. Section 1 describes the construction of solutions to non-autonomous equations of the class under consideration, and Section 2 examines the stability of a stationary solution to such equations.
Keywords: relatively bounded operator; Lyapunov's second method; local stream of operators; asymptotic stability.
Introduction
The study of the stability of stationary solutions to abstract operator-differential equations allows us to obtain results on the qualitative behavior of a system described using equations of this type [1,2]. Using such results for specific models, it is possible to obtain values of characteristics that guarantee the stability of the simulated system.
In this article we investigate the properties of stationary solutions for a system of non-autonomous linear equations of the Sobolev type
where L and M are linear bounded operators acting from the space U to the space F, a vector-function f : R ^ F characterizes the external impact on the system, and a scalar function a : [0,T] ^ R+ characterizes the change in time of the parameters of this system. Here and below U and F are some Banach spaces. The properties of solutions to non-autonomous equations resolved with respect to the time derivative were studied, for example, in [3]. Sobolev type equations are understood as equations unresolved with respect to the highest derivative [1,2,4,5]. A characteristic feature of such equations is the fundamental unsolvability of the Cauchy problem
with an arbitrary initial data u0, which can be even from a dense set in U [1,6]. For the existence of solutions to the Sobolev type equation, it is necessary that the initial data
Li(t) = a(t)Mu(t) + f (t), ker L = {0},
(1)
u(0) = u0
belong to some set of permissible initial values, which is understood as the phase space of these equations [7].
The solvability of non-autonomous Sobolev type equations of the form (1) was first considered in [8] and the proposed methods were applied to study various problems. To construct a solution to non-autonomous equation (1), we use the technique proposed in [9]. The stability of solutions to Sobolev type equations with constant coefficients was studied by many authors, more details on this can be found in [8] and [10]. In this paper, the stability of a stationary solution to equation (1) is considered under general assumptions about the position of its relative spectrum. Thus, exponential dichotomies of solutions [8] do not necessarily exist for it, and therefore we use the second Lyapunov method [10,11] to study the stability.
1. Solvability of Non-Autonomous Sobolev Type Equation
Let U and F be Banach spaces, the operators L e L(U; F) (linear and continuous) and M e Cl(U; F) (linear, closed and densely defined).
Consider an L-resolvent set pL(M) = e C : (^L — M)-1 e L(F; U)} and an L-spectrum aL(M) = C \ pL(M) of the operator M ( [5], par. 2.1). The set pL(M) is always open, so the L-spectrum aL(M) is always closed. Also, the operator (^L — M)-1 is a holomorphic function of the variable ^ on the set pL(M).
Definition 1. [7] The operator M is called spectral bounded relatively to the operator L (or simlpy (L, a)-bonded), if 3a > 0 V^ e C (|^| > a) ^ (^ e pL(M)).
Let the operator M be (L, a)-bounded then we choose the loop y=(^eC: |^|=r>a} and construct the operators P = - / R^(M)dfi and Q = - / L^(M)dfi,
2ni Jy 2ni JY
where the operator RL(M) = (^L — M)-1 L is a right L-resolvent of the operator M, and the operator LL(M) = L(^L — M)-1 is a left L-resolvent of the operator M. Here the integrals are understood as Riemann integrals. So the operators P e L(U) and Q e L(F).
Lemma 1. Let the operator M be (L, a)-bounded then the operators P e L(U) and Q e L(F) are projectors.
Denote U0 = kerP, F0 = kerQ, U1 = imP, F1 = imQ. Denote the restriction of the operator L to the set Uk by Lk, and the restriction of the operator M to the set domMfiUk, k = 0,1, by Mk. Due to the properties of operators, linear sets domMk = domM if Uk are dense in Uk, k = 0,1.
Theorem 1. [7] (Sviridyuk's splitting theorem)
Let the operator M be (L, a)-bounded then
(i) the operators Lo e L(U0; F0) and Lx e ; F1);
(ii) the operators M0 e Cl(U0;F0) and M1 e ^(U1;F1);
(iii) there exist the operator L-1 e ^(F1;U1) and M—1 e ^(F0;U0).
Denote H = M0-1 L0 e L(U0), S = L-1MX e ^(U1).
Definition 2. For the L-resolvent (^L — M)-1 of the operator M, the infinity point is called
(i) a disposable singular point, if H = O;
(ii) a polar of the order p G N, if Hp = O and Hp+1 = O;
(iii) an essentially singular point, if Hp = O for all p G N.
Remark 1. For further discussion, it is more convenient to refer to the disposable singular point as a "pole of the zero order". Then the operator M is called (L,p)-bounded, p G {0} U N = No, if M is (L, a)-bounded, and the point œ is a pole of the order p G No of its L-resolvents.
A vector-function u G Cœ (R; U) is called a solution u = u(t) to the equation
if u satisfies this equation. The solution u(t) to equation (2) is called the solution to the Cauchy problem for equation (2) if u additionally satisfies the Cauchy condition
for some vector u0 G U.
Definition 3. A set P is called a phase space of equation (2), if
(i) any solution u = u(t) of (2) belongs to P as a trajectory (i.e. u(t) G P Vt G R);
(ii) for any u0 G P there exists a unique solution to problem (2), (3).
Theorem 2. [7] Let the operator M be (L/p)-bounded (p G N0) then the phase space of (2) is the subspace U1 .
Definition 4. [5] We refer to the operator-function U• G C^(R; U) as a group of resolving operators (or, in short, as a group) of equation (2) if
(i) UsU* = Us+* for all s, t G R ;
(ii) for any u0 G U the vector-function u(t) = U*u0 is a solution to (2).
Let us identify the group and its graph {U* : t G R}. The group {U* : t G R} is called holomorphic if {U* : t G R} is analytically continuous into the entire complex plane C under Properties (i), (ii); and is degenerate if its unit U0 is a projector. For a holomorphic degenerate group, the concepts of kernel and image are correct, and ker U• = ker U0 = ker U* for any t G R, and im U• = im U0 = im U* for any t G R. A holomorphic degenerate group {U* : t G R} is called the resolving group of equation (2) if its image im U• coincides with the phase space of equation (2).
Theorem 3. [7] Let the operator M be (L,pp)-bounded (p G N0) then there exists a unique resolving group of equation (2), which has the form,
Lu = Mu,
(2)
u(0) = «о
(3)
(4)
where 7 = {^ G C : = r > a} is the closed loop. Remark 2. It is clear that the identity of group (4) is U0 = P. On the interval J C R, consider the Cauchy problem (t0 G J)
u(t0) = u0,
(5)
for the homogeneous non-autonomous equation
Lu(t) = a(t)Mu(t).
(6)
Definition 5. [9] The vector-function u e C 1(J; U) is called a solution to equation (6) if u satisfies this equation on J. A solution to (6) is called a solution to the Cauchy problem, (5), (6), if it additionally satisfies condition (5).
Theorem 4. [9] Let the operator M be (L/p)-bounded (p e N0) and the function a e C(R, R+) then a phase space of equation (6) is the subspace U1.
Definition 6. [9] The two-parameter family U(•, •) : R x R ^ L(U) is called a family of resolving operators, if the following conditions hold :
(i) U(t, t) = P for all t e R;
(ii) U(t, s)U(s, t) = U(t, t) for all t,T,s e R.
A family of resolving operators is called analytic if its operators admit an analytic continuation into the entire complex plane C under Properties (i) and (ii) from Definition 6. The two-parameter family of operators U(•, •) : R x R ^ L(U) is called a family of resolving operators of equation (6) if for any u0 e U the vector function u(t) = U(t, t0)u0 is a solution to equation (6) (in the sense of Definition 5).
Let the operator M be (L,p)-bounded (p e N0) and the function a e C(R; R). By analogy with group (4), consider the operators
with s,t e R and the closed loop y = (^ e C : = r > a}
Theorem 5. [9] Let the operator M be (L,p)-bounded (p e N0) and the function a e C(R; R), then the family of operators (U(t,s) e L(U) : t,s e R}, given by formula (7), is an analytic degenerate family of resolving operators.
Finally, we describe the solution for the inhomogeneous equation
where a : [0,T] ^ R+ is a scalar function that characterizes the change in time of the parameters of the mutual influence of the states of the system under study, the vector function g : [0,T] ^ F characterizes the external impact. Denote (% — Q)g(t) = g0(t).
Theorem 6. [9] Let [0,T] e J, the operator M be (L,p)-bounded (p e N0) and the function a e Cp+1([0,T]; R+). Then for an arbitrary vector-function g : [0,T] ^ F such that Qg e C 1([0,T]; F1), g0 e Cp+1([0,T]; F0) and under the condition of approval
s <t
(7)
Lu(t) = a(t)Mu(t) + g (t),
(8)
and for any initial data u0 G U there exists a unique solution u G C 1([0,T]; U) to Cauchy problem (5), (8), which has the form,
u(t) = U(t,0)Puo + J U{t,8)LT1Qg{8)d8-^HkMô1 (9)
2. Second Lyapunov Method in Normed Spaces
Let V be a normed space.
Definition 7. A local two-parameter stream on V (in shortly, stream) is a map S such that for all u G V and some t = t(u) G R+, the following conditions hold:
(i) S = SS u G V for all t, s G (—t ; t ) ; S°u = u;
(ii) SS = SZSzsu for all t, s, z G (-t, t). A point u G V such as
(iii) SS u = u for all t, s G (—t ; t ) ,
is called a stationary point of the stream S.
Definition 8. A stationary point u G V of the stream S is called
(i) stable (according to Lyapunov), if for any neighborhood Ou of the point u G V there exists a neighborhood O'u (may be, another) of this point such that SSv G O'u for all v G Ou and t, s G R+;
(ii) asymptotically stable (according to Lyapunov), if it is stable and for any point v from some neighborhood Ou of the point u the following is true: SS v ^ u with t ^ œ.
Definition 9. A functional V G C(V; R) is called Lyapunov functional of the stream S,
(v (S0 «) - V («))
if for all u G 2J it has the form V(u) = lim- < 0.
Theorem 7. Let u G V be the stationary point of the stream S on V. This point u G V is stable if for the stream S there exists a Lyapunov functional, which satisfies the following two conditions:
(i) V(u) = 0;
(ii) V(v) > p(||v — u||) with some strictly increasing continuous function p such that p(0) = 0 and p(r) > 0 for r G R+.
Proof. We follow [10,11], where a similar theorem is proved in the case of a stationary Sobolev type equation.
So, for every r G R+ we put Or = {v G V : V(v) < r}. Each of the sets Or is a neighborhood of the point u, and V G Or ^ V(S0v) < V(v) < r for all t G R+.
If V(v) > p(||v — u||) then for any e G R+ there exists r = p(e) > 0 such that V(v) < r ^ ||v — u|| < e. Due to the continuity of V, there exists 8 G R+ such that with ||v — u|| <8 we have v G Or, and we get S0v G Or such that ||S0v|| < e for all t G R+.
□
Theorem 8. Let the conditions of Theorem 7 be fulfilled, and suppose that there exists a strictly increasing continuous function ^ such that -0(0) = 0 and ^(r) > 0 for r G R+, and U(v) < ——(||v — u||), then the point u G V is asymptotically stable.
Proof. Let the point be v e Ou, then by virtue of Theorem 7 V(S0v) is a non-increasing
non-negative function with t e R+.
Let l = lim V(S0v) and suppose that l > 0, then inf ||S0v|| > 0. And we can t—teR+
conclude that sup F(S0v) < —m for some m e R+, which contradicts the non-negativity teR+
of V(S0v). So V(S0v) and ||S0v|| tend to zero at t ^
□
Theorem 9. Let the operator M be (L,p)-bounded (p e N0) and the function a e C(R; R), then a family of operators (Siu e L(U) : t, s e R} is a local stream of operators. The zero point is the stationary point of this stream. And the operators of this stream have the form,
^ = ^d / exp a(0d(^j dfi, s,teR, s = 0 < t, (10)
where the closed contour y bounds the area containing the L-spectrum aL(M) of the operator M.
Proof. It is obvious that the zero point is a stationary point of this stream due to the linearity of its operators.
Let us show that Situ is a local stream of operators. Statement (i) of Definition 7 follows from the method of specifying the operator Sig using (7), Property (i) of Definition 6 and Remark 2.
We show the fulfillment of Statement (ii) from Definition 7. To do this, consider SlSzsu = JR^(M) exp (vfa(0cK) dfi JRLx(M)uexp (af a(()d(^) d\ =
Y Y'
1 RL{M)R^{M)uexp ( Af a(()d( ) dX I exp ( fij a(()d( ) dfi
(2ni)2
Y \Y'
1
(2ni)2
^ exP fA/ a(()dA d\
J J - \_fi -i?Î(M)wexp ( Mfa(()d( ) d/x+
Y Y'
t x exp ( a(()d( J d^ ^
+ j R^(M)uexp (^Xf a(()d(^j J-—^--dX
/
where the point ^ G y lies inside the area bounded by the contour yand the point A G y' is located outside the area bounded by the contour y• Then by Deduction Theorem
exP ( A/ a(( )dU dA , t . exP ( M.f a(( )d()
J X-, =exp(ia(0d() ■ j .-a ^«■
Y' Y
Y
Y
and we have the fulfillment of Statement (ii) of Definition 7
SlSzsu = J exp R%(M)uexp (^ij a(()d(^j dfi =
y
= ¿7 RL,{M)ueW(^i (f a(()d( + f a(()d(^ dfi = Slu.
y
□
Thus, using a family of resolving operators, it is always possible to construct a local stream of operators in the sense of Definition 7. For a specific type of operators, using Theorems 7 and 8, based on information about the points of the relative spectrum aL(M), it is possible to investigate stationary zero solutions for Lyapunov stability.
Conclusion
The results obtained in this paper are planned to be used to study the stability of the null solution in non-autonomous Hoff models on geometric graphs. These models describe structures made of I-beams. In such models, at high temperatures, the parameter on the right side of the equation ceases to be stationary, which explains the appearance of the time function in the equation. The stability of the solution of such models makes it possible to more accurately determine the time of maintaining the stability of the structure, which is an urgent problem when carrying out work to eliminate fires.
References
1. Sviridyuk G.A., Manakova N.A. Nonclassical Mathematical Physics Models. Phase Space of Semilinear Sobolev Type Equations. Bulletin of the South Ural State University. Series: Mathematics. Mechanics. Physics, 2016, vol. 8, no. 3, pp. 31-51. DOI: 10.14529/mmph160304 (in Russian)
2. Al'shin A.B., Korpusov M.O., Sveshnikov A.G. Blow-up in Nonlinear Sobolev Type Equations. Berlin, de Gruyter, 2011.
3. Sell G.R. Topological Dynamics and Ordinary Differential Equations. London, Van Nostrand Reinhold, 1971.
4. Demidenko G.V., Uspenskii S.V. Partial Differential Equations and Systems Not Solvable with Respect to the Highest-Order Derivative. New York, Basel, Hong Kong, Marcel Dekker Inc., 2003.
5. Sviridyuk G.A., Fedorov V.E. Linear Sobolev Type Equations and Degenerate Semigroups of Operators. Utrecht, Boston, VSP, 2003.
6. Keller A.V. The Leontief's Type Systems: Classes of Problems with the Showalter - Sidorov Intial Condition and Numerical Solving. The Bulletin of Irkutsk State University. Series Mathematics, 2010, vol. 3, no. 2, pp. 30-43. (in Russian)
7. Sviridyuk G.A. Quasistationary Trajectories of Semilinear Dynamical Equations of Sobolev Type. Russian Academy of Sciences. Izvestiya Mathematics, 1994, vol. 42, no. 3, pp. 601-614. DOI: 10.1070/IM1994v042n03ABEH001547
8. Sagadeeva M.A. Investigation of Solutions Stability for Linear Sobolev Type Equations. PhD (Math) Thesis. Chelyabinsk, 2006. (in Russian)
9. Sagadeeva M.A. Degenerate Flows of Solving Operators for Nonstationary Sobolev Type Equations. Bulletin of the South Ural State University. Series: Mathematics. Mechanics. Physics, 2017, vol. 9, no. 1, pp. 22-30. DOI: 10.14529/mmph170103 (in Russian)
10. Zagrebina S.A., Moskvicheva P.O. Stability in Hoff Models. Saarbrucken, LAMBERT Academic Publishing, 2012. (in Russian)
11. Moskvicheva P.O. Stability of the Evolutionary Linear Sobolev Type Equation. Bulletin of the South Ural State University. Series: Mathematics. Mechanics. Physics, 2017, vol. 9, no. 3, pp. 13-17. (in Russian) DOI: 10.14529/mmph170302
Received May 25, 2023
УДК 517.9 DOI: 10.14529/mmp230305
УСТОЙЧИВОСТЬ СТАЦИОНАРНОГО РЕШЕНИЯ ОДНОГО КЛАССА НЕАВТОНОМНЫХ УРАВНЕНИЙ СОБОЛЕВСКОГО ТИПА
Л.В. Буевич1, M.A. Сагадеева1, С.А. Загребина1
1 Южно-Уральский государственный университет, г. Челябинск,
Российская Федерация
Статья посвящена исследованию устойчивости стационарного решения задачи Ко-ши для неавтономного линейного уравнения соболевского типа в относительно ограниченном случае. А именно рассматривается случай, когда относительный спектр оператора уравнения может пересекаться с мнимой осью. В этом случае не существуют экспоненциальные дихотомии и для исследования устойчивости применяется второй метод Ляпунова. Устойчивость стационарных решений позволяет оценить качественное поведение систем, описываемых с помощью таких уравнений. Статья кроме введения, заключения и списка литературы содержит две части. В первой из них описывается построение решений неавтономных уравнений рассматриваемого класса, а во второй исследуется устойчивость стационарного решения таких уравнений.
Ключевые слова: относительно ограниченный оператор; второй метод Ляпунова; локальный поток операторов; асимптотическая устойчивость.
Литература
1. Свиридюк, Г.А. Неклассические модели математической физики. Фазовые пространства полулинейных уравнений соболевского типа / Г.А. Свиридюк, Н.А. Манакова // Вестник ЮУрГУ. Серия: Математика. Механика. Физика. - 2016. - Т. 8, № 3. - C. 31-51.
2. Al'shin, A.B. Blow-up in nonlinear Sobolev type equations / A.B. Al'shin, M.O. Korpusov, A.G. Sveshnikov. - Berlin: de Gruyter, 2011.
3. Sell, G.R. Topological Dynamics and Ordinary Differential Equations / G.R. Sell. - London: Van Nostrand Reinhold, 1971.
4. Demidenko, G.V. Partial differential equations and systems not solvable with respect to the highest-order derivative / G.V. Demidenko, S.V. Uspenskii. - New York; Basel; Hong Kong: Marcel Dekker Inc, 2003.
5. Sviridyuk, G.A. Linear Sobolev Type Equations and Degenerate Semigroups of Operators / G.A. Sviridyuk, V.E. Fedorov. - Utrecht; Boston: VSP, 2003.
6. Келлер, А.В. Системы леонтьевского типа: классы задач с начальным условием Шо-уолтера - Сидорова и численные решения / А.В. Келлер // Известия Иркутского государственного университета. Серия: Математика. - 2010. - Т. 3, № 2. - С. 30-43.
7. Свиридюк, Г.А. Квазистационарные траектории полулинейных динамических уравнений типа Соболева / Г.А. Свиридюк // Известия РАН. Серия математическая. - 1993. -Т. 57, № 3. - С. 192-207.
8. Сагадеева, М.А. Исследование устойчивости решений линейных уравнений соболевского типа: дисс. ... канд. физ.-мат. наук / М.А. Сагадеева. - Челябинск, 2006.
9. Сагадеева, М.А. Вырожденные потоки разрешающих операторов для нестационарных уравнений соболевского типа / М.А. Сагадеева // Вестник ЮУрГУ. Серия: Математика. Механика. Физика. - 2017. - T. 9, № 1. - С. 22-30.
10. Загребина, С.А. Устойчивость в моделях Хоффа / С.А. Загребина, П.О. Москвичева. -Saarbrucken: LAMBERT Academic Publishing, 2012.
11. Москвичева, П.О. Устойчивость эволюционного линейного уравнения соболевского типа / П.О. Москвичева // Вестник ЮУрГУ. Серия: Математика. Механика. Физика. -2017. - Т. 9, № 3. - С. 13-17.
Антон Викоторович Буевич, магистрант кафедры уравнений математической физики, Южно-Уральский государственный университет (г. Челябинск, Российская Федерация) .
Минзиля Алмасовна Сагадеева, кандидат физико-математических наук, доцент, доцент кафедры математического и компьютерного моделирования, ЮжноУральский государственный университет (г. Челябинск, Российская Федерация), [email protected].
Софья Александровна Загребина, доктор физико-математических наук, профессор, заведующий кафедрой математического и компьютерного моделирования, Южно-Уральский государственный университет (г. Челябинск, Российская Федерация), [email protected].
Поступила в редакцию 25 мая 2023 г.
