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UDC 517.929.4 Вестник СПбГУ. Прикладная математика. Информатика... 2019. Т. 15. Вып. 4 MSC 74J55
Stability of weak solutions of parabolic systems with distributed parameters on the graph
A. P. Zhabko1, A. I. Shindyapin2, V. V. Provotorov3
1 St. Petersburg State University, 7—9, Universitetskaya nab., St. Petersburg, 199034, Russian Federation
2 Eduardo Mondlane University, 1, Julius Nyerere av., 3453, Maputo, Mozambique
3 Voronezh State University, 1, Universitetskaya pl., Voronezh, 394006, Russian Federation
For citation: Zhabko A. P., Shindyapin A. I., Provotorov V. V. Stability of weak solutions of parabolic systems with distributed parameters on the graph. Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 2019, vol. 15, iss. 4, pp. 457-471. https://doi.org/10.21638/11702/spbu10.2019.404
The analysis of the behaviour of the evolutionary equation solution with unlimited time variable has been a subject of discussion in scientific circles for a long time. There are many practical reasons for this when the initial conditions of the equation are specified with a certain error: how the small changes in the initial conditions affect the behaviour of the solution for large values of the time. The paper uses the classical understanding of the stability of the solution of a differential equation or a system of equations that goes back to the works of A. M. Lyapunov: a solution is stable if it little changes under the small perturbations of the initial condition. In the work specified the stability conditions for the solution of an evolutionary parabolic system with distributed parameters on a graph describing the process of transfer of a continuous mass in a spatial network are indicated. The parabolic system is considered in the weak formulation: a weak solution of the system is a summable function that satisfies the integral form identity, which determines the variational formulation for the initial-boundary value problem. By going beyond the classical (smooth) solutions and addressing weak solutions of the problem the authors aim not only to describe more precisely the physical nature of the transfer processes (this takes on particular importance when studying the dynamics of multiphase media) but also to the path analysis processes in multidimensional network-like domains. The used approach is based on a priori estimates of the weak solution and the construction (the Fayedo—Galerkin method with a special basis — the system of eigenfunctions of the elliptic operator of a parabolic equation) of a weakly compact family of approximate solutions in the selected state space. The obtained results underlie the analysis of optimal control problems for differential systems with distributed parameters on a graph, which have interesting analogies with multiphase problems of multidimensional hydrodynamics.
Keywords: evolutionary system of parabolic type, distributed parameters on the graph, a weak solution, stability of a weak solutions.
1. Introduction. Today have a lot of the results on the mathematical theory of stability, however, as we know they are all overwhelmingly oriented on the ordinary differential equations and systems. In many applications because of the complexity of the mathematical models have to abandon the use of ordinary differential equations in behalf of considering the evolutionary equations with partial derivatives. Precisely this case is the object of the study in this work, in which represent the analysis of the stability of the
© Санкт-Петербургский государственный университет, 2019
weak solutions of the evolutionary systems with distributed parameters on the graph with unlimited growing of the temporary variable. By studying the relevant initial-boundary value problem, we to by beyond the scope a classical solutions and reduce to the weak solutions of the problem, reflecting more accurately the physical essence of appearance and processes (i. e. consider the initial-boundary value problem in weak formulation). This solutions reflects more exactly the physical essence of phenomena and processes. It is well known the choice of the class of weak solutions to be determined one way or the other functional space that is available to the researcher. This choice was made, first of all, the requirement of conservation of the existence theorem and the uniqueness theorem, if it is corresponds to the spirit of the studied phenomenon or process.
We are sufficiently detailed form here is the proof of the existence of weak solutions of initial-boundary value problem in distinct spaces (analogously reasoning shows in the works [1-4]). It is necessary to demonstrate the equivalent concept of Lyapunov stability of the undisturbed state differential system for differential equations with partial derivatives and show the ability to use known results [5, 6] in the studied case.
2. Notation, concepts and basic statement. We introduce the following concepts and symbols adopted in the works [3, 4]: r is the limited oriented geometric graph with edges y, parameterized the segment [0,1]; dr and J(r) are the many of boundary Z and internally £ nodes of graph respectively; r0 is the join of all the edges of the graph r, does not contain the endpoints; r = To x (0,t) Y = Yo x (0,t)), dr = dr x (0,t) (t e (0,T], T < x>).
In all the course of the work make use of the Lebesgue integral along r or rt:
ff (x)dx = f (x)Ydx or f f (x,t)dxdt = f (x,t)Ydxdt,
r Y rt Yt
f (-)Y is narrowing function f (■) on the edge y.
We use classic space of functions: C(r) is a space continuous functions on r; Lp(r) (p = 1,2) is a Banach space of measurable functions on r0, integrable with degree of order p (similarly defined the space Lp(rT)); L2,1(rT) is the space of functions from L1(YT)
T
with norm defined by ratio ||w||i2 l(rT) = /(/u2dx)zdt.
o r
We also use analogues of Sobolev's spaces [7, 8]: W 1(r) is the space of functions from L2(r), with generalized derivative of order 1 also from L2(r); W 1,0(rT) is the space of functions from L2(rT) with generalized derivative of order 1 for x belonging to space L2(rT) (similarly defined the space WrT)).
Relabel V2(rT) the set of all functions u(x,t) e W21,0(rT) with finite norm
IM|2,rT = ma^ ||u(-,t)yi2(p) + \\ux\\L2{Tt) • (1)
In addition these functions are continuous on t in norm space L2(r), that is under At ^ 0 ||u(t + At) - u(t)|L2(r) ^ 0 uniformly on [0,T].
We will introduce the state space of parabolic system and the auxiliary space. To do this, consider the bilinear form
v) = f + b(x)ij>(x)is(x)j dx
with a fixed measurable and limited on r0 functions a(x), b(x) square integrable:
0 <a* < a(x) < a*, |b(x)| < /3, x e r. (2)
Following approval is valid [9, p. 92].
Lemma 1. Let the function u(x ) e W2(r) such that
i(u, v) — J f (x)v(x)dx = 0 r
for any v(x) e W2(r) (f (x) e L2(r) is the fixed function).
Then for any edge 7 C T the narrowing of the function a(x)7 ducontinuously in the endpoints of the edge 7.
Relabel Oa(r) the many of these functions u(x) satisfying the ratios
E a(l)7^= E <o)7d-A
YeR(Z) Yer(t)
in all nodes £ e J(r) (in here R(£) and r(£) as the sets of the edges 7 accordingly oriented "to node £" and "from node £") and u(x)\ar = 0. The closing of the set Oa(r) in norm W1 (r) relabel W¡(a, r).
Let the next Qa(rT) is the set of functions u(x,t) e V2(rT), whose traces are defined in sections of the domain rT the plane t = t0 (t0 e [0, T]) as a function of class W0(a, r) and satisfy a ratios
E E (3)
for all nodes £ e J(r). The closing of the set Qa(rT) in norm (1) relabel V 1,0(a, rT); it is clear that V0(a, rT) C W¡,0(rT).
Another the subspace of the space W2'°(rT) is W10(a, rT); W10(a, rT) is the closing in norm W 1,0(rT) the set of differentiable on r0 functions u(x,t) satisfy a ratios (3) for all nodes £ e J(r) and boundary condition u(x,t)\dr = 0 for any t e [0, T] (the derivatives in the nodes are defined as the one-sided derivative). The space W2'0(rT) is defined similarly. Remark 1. The space
V ¡'0( a, rT) describes the states set of parabolic system, W1 (a, rT) is the auxiliary space. The distinction of elements space V ¡'0( a, rT) from elements space W¡(a, rT) is the absence of elements of W¡(a, rT) continuity for the temporary variable t.
In spaces W¡(a, rT) and V 1,0(a, rT) consider the parabolic equation
(«W^1) + Kx)y(x,t) = f(x,t), (4)
represents a system of differential equations with distributed parameters on each edge 7 of the graph T; f(x,t) G ^iITt)- The state y(x,t) (x,t G Ft) of the system (4) in the domain is determined by a weak solution y(x,t) of the equation (4), satisfying the initial and boundary conditions
y \t=0 = ^(x), x e r, y |KedrT =0, (5)
where x) e L2(r). The suppositions about functions a(x) and b(x) to make mention above in (2). From y(x,t) e V1,0 (a, rT) should be noted, that the map y : [0,T] ^ W J (a, r) C L2(r) is a continuous function, so that the first equality (5) makes sense and is be understood almost everywhere.
Remark 2. In the work considers the initial-boundary value problem (4), (5) with Dirichlet boundary conditions in the ratios (5), for other types of boundary conditions is given any comment.
We present the basic statements and the basic fragments of proofs of their first in auxiliary space W 1,0(a, Гт), then in the space V 1,0(a, Гт), full proofs given in the work [4].
Definition 1. A weak solution of the initial-boundary value problem (4), (5) class W 1>0(ГТ) is the function y(x,t) G W 1,0(a, Гт), that satisfies an integral identity
— f y(x, t) dr,tQf^ dxdt + tт(у, rj) = / f(x)r](x,0)dx + J f(x,t)r](x,t)dxdt ^
Г^ Г r^
for any function n(x,t) G W 1(a, Гт) equal to zero under t = T; It(y,n) is bilinear form defined by the ratio of
et(y,n) = J (aW^^+W,«)^«)) dxdt, t e (О,Т].
rt y 7
Definition 2. A weak solution of the initial-boundary value problem (4), (5) class W2'°(ГТ) is the function y(x,t) G V 1,0(a, Гт), that satisfies an integral identity
f y(x,t)rj(x,t)dx - f y(x, t)ar,<ft'^ dxdt + £t(y,r]) = Г Г
= J ^(x)n(x, 0)dx + J f (x,t)n(x,t)dxdt г rt
under any t G [0, T] and for any function n(x, t) G W 1(a, Гт).
In proving solvability the problem (4), (5) in the space W 1,0(a, Гт) (and V 1,0(a, Гт)) is a special basis of space W¿(a, Г) — system of generalized eigenfunctions of boundary value problem on the eigenvalues (the spectral problem)
(а(х)^г) + Kx)u(x) = Ам(ж), и(х)\дг = 0, (7)
in class W 1(a, Г) [10, 11]. This problem consists in finding many such numbers (eigenvalues boundary value problem (7)), each of which corresponds to at least one nontrivial generalized solution u(x) G W0(a, Г) (the generalized eigenfunction), satisfies the integral identity
£(u,n)= A(u,n) (8)
for any function n(x) G W0(a, Г) (here and everywhere below through (■, ■) designated scalar product in L2(Г) or L2(ГТ)). Install the necessary further properties of eigenvalues and generalized eigenfunctions of spectral problem (7). To do this, we will introduce in the space W^(a, Г) of new scalar product
[u,v] = f (a(x)^^-^j^- + (A0 + b(x))u(x)v(x)j dx = £(u,v) + X0(u,v), where A0 defined by inequalities A0 > в. Then given (2), correctly ratio
MI > ii2 + (Ao - mm2 > «(lit ii2 + \\вп
a > 0 is fixed constant. The latter means equivalence of norms |H|[V] = л/[и, и] induced by the scalar product [■, ■], and norm of space W^(a, Г). The generalized eigenfunctions of spectral problem (7) satisfy the integral identity
[u,n] = (A0 + A)(u, n) (9)
for any function n(x) G W0(a, Г).
In space W¡(a, r) we define the operator B by using identities
[Bu, n] = (u, n) Vri(x) e W¡(a, r),
then identity (9) is equivalent to
[Bu, r] = X[u, r] Vr(x) e W0(a, r),
where A = 1/(A0 + A). It is easy to verify that the operator B is completely, self-adjoint and positive. Whence it follows that the eigenvalues {Aj}, i > 1, of its valid, positive and they can be considered numbered in descending order, taking into account their multiplicity, where Ai ^ 0 under i ^ <x>. Note that the point A = 0 not its eigenvalue, because under A = 0 from the definition of the operator B derives u = 0. The generalized eigenfunctions {ui(x)}i>1, that correspond to their eigenvalues {Ai}i>1, are real and mutually orthogonal: [ui, uj ] = 0 under i = j .In view of the foregoing, the system of generalized eigenfunctions {ui(x)}i>1 form a basis in space W0(a, r), as Wq(a, r) densely in L2(r), then {ui(x)}i>1 is basic in L2(r) (you can take that the basis {ui(x)}i>1 is orthonormalized in L2(r)). Taking into account the connection Ai = —A0 + 1/Ai of the eigenvalues Ai of spectral problem (7) and the eigenvalues Ai of the operator B, as well as the coincidence by him corresponding generalized eigenfunctions, come to the following statement.
Lemma 2. Let met supposition (2). Then the spectral problem (7) has the denumerable set of real eigenvalues {Ai}i>1 (indexed in ascending order, taking into account their multiplicity) with limit point at infinity (the eigenvalues Ai are positive, with the except ion of maybe finite number first). The system of generalized eigenfunctions {ui(x)}i>1 form a basis in W0(a, r) and L2(r), orthonormalized in L2(r) and orthogonal in meaning of the scalar product [•, •].
Remark 3. If 0 < b(x) < /, as is usually the case in the applications, then all eigenvalues of the spectral problem (7) is positive. Indeed, this follows from the integral identities (8), when u = r = ui(x), A = Ai and a chain of equalities
l(ui,ui) = Ai(ui,ui) = Ai||ui||2 = Ai
for i = 1,2,... (see also [9, p. 98]). The positiveness of the eigenvalues is the determining factor for establishing the stability condition and the possibility of the stability of evolutionary systems of parabolic equations with distributed parameters on the graph.
Theorem 1. If any f (x) e L2,Q(rT), ^(x) e L2(r) and for any 0 < T < <x> the initial-boundary value problem (4), (5) is weak solvability in space W q'0( a, Tt).
P r o o f. In proving theorems construct the Faedo—Galerkin approximation on the basis {un(x)}n^1: the approximate solutions (N is fixed natural number) (4), (5) have the
N
form yN(x,t) = E °N(t)ui(x), where cN(t) is absolutely continuous on [0,T] functions
i=i
(ci(t) e L2(0,T)), defined from the system
+ J ('a(x)dy du£x) + b(x)yN(x,t)Ui(x)j dx = (f,Ui),
cf(0) = (cp,Ui), i = uv.
Further reasoning based on a priori estimates of norm of weak solutions (4), (5) and construction the subsequence {yNk} k>1 of sequence {yN}N >q, weakly converge to solution
y(x,t) G W 1,0(a,rT) in a norm of W 1,0(rT) (weak compactness {yNk}k>1). Namely [4, 7], for approximate solutions yN(x,t) the inequality by
\\yN II 2,rt < C (t) (\\yN (x, 0)||i2(r) + 2\\f yLa>l(rt)) ,
is valid for any t G [0, T]; here the function C(t) is limited to t G [0, T] (C(t) < C*,C* > 0), not depend on N, is determined by the value T and permanent a*, /3. From this inequality, taking into account the ratio cf^(O) = (y, «¿) (i = 1, N) and by virtue of inequalities
N / N
\\yN (x, 0)||l2(f) = \\ E &,щ)щ(х)\\ < WE < МЫГ)
i=i V i=i
is Euclidean norm: ||w|| = ^Jj2iLi ^ > it should be
\\yN\\2,rt < c(t) (Ml2(D +2\\f \\L2Art)), (11)
it means independent of N estimate
\\yN\\2,rt < c (C> 0).
The latter means: from the sequence {yN}N>1 with limited totality elements yN can be distinguish the subsequence {yNk}k>1, that converge weakly to certain element y G
W 1'0( a, r^) at a norm T/F1'°(Ft) ({yNk }fc>1 converge weakly to y together with at a norm L2(rT)). As a result of the consequent reasoning become clear that the all sequence {yN}N>1 is weakly converges to an element y G W 1,0(a, rT) (so as \\ • \\Wi,o(rT) ^ \\ • \\2,rT). Element y(x,t) is a weak solution (4), (5). This is established by multiplying the first relation (10) to the absolutely continuous on [0, T] function di(t) (di(T) = 0) and next by summing along i = 1, N and integrating results along t away from 0 up to T. After integrating the first term by parts to t obtain identity
- / yN(x,t)^^dxdt + £T(yN,Y) = JyN(x,0)Y(x,0)dx +
r
N (12)
+ f f (x,t)Y (x, t)dxdt (Y (x,t) = di(t)ui(x)).
rT i=1
The set E (E is the set of all functions Y(x,t) with arbitrary coefficients di(t) possessing the above property, and arbitrary natural N) densely into the subspace of functions belonging to W 1(a, rT) and equal to zero at t = T. This follows from the density of the set {un(x)}n>1 in W^(a, r) and properties Y(x,t) (continuity Y(x,t) G E along t G [0,T], Y(x,t) G W 1(a, r) for each fixed t G [0,T] and Y(x,T) = 0) as follows. Fix in (12) function Y(x,t) = Y*(x,t) G E (Y*(x,t) = d*(t)ui(x)^ and on selected above subsequence {Nk}k>1, corresponding subsequence {yNk}k>1, we pass to the limit, starting with numbers Nk > N*. As a result we obtain be a integral ratio (6) for the limit function y(x,t) when n(x,t) = Y*(x,t), and that means, by virtue of the density of many E in the subspace of functions belonging to W 1(a, rT) and equal to zero when t = T, y(x,t) is a weak solution of initial-boundary value problem (4), (5) of W 1,0(a, rT). Note that linearity of problem (4), (5) and estimate (11) guarantee the uniqueness of weak solutions y(x,t). The statement of the theorem is proved.
Turn to the study of the resulting in proving theorems 1 weak solutions y(x,t) G W 1,0(a, rT): show that y(x,t) belongs to space V 1,0(a, rT).
Theorem 2. If the conditions of the theorem 1, then initial-boundary value problem
(4), (5) weakly solvability in space V 1'0( a, rT) for any 0 <T <
Proof. For proof sufficiently to establish that every fixed t G [0, T] track of belonging to space W 1,0(a, rT) weak problem solving (4), (5) is an element W^(a, r) and continuously depends from t in norm W2(r) (that is in norm L2(r)). The tool of analysis uses the generalized Fourier method with base {ui(x)}i>1, ortonormalization in L2(r) and orthogonal in sense of the scalar product [•, •] (statement of the lemma 2). For simplicity of presentation, several narrow down space L2,1(rT), replacing it on CL2j1(rT) c L2,1(rT) (is the space of functions of L2j1(rT), continuous along t in norm of L2(r)), when this f (x,t) G CL2j1(rT) (the latter is easy condition in applications). In other words the functions f (x,t), summable on the variable t G (0,T) in norm of L2(r) are replaced functions f (x,t), that are continuous-variable t G [0,T] in norm of L2(r)). Consider a series
y(x, t) = E (fie-^ + } fi(r)e-Xi(t-T)dr) m(x) (13)
¿=1 V 0 /
and decompositions
f(x) = E fiUi(x), fi = / f(x)ui(x)dx, i=1 r
f (x,t) = E fi(t)ui(x), fi(t) = / f (x,t)ui(x)dx, t G [0,T],
i=1 r
It is easy to verify that any from a finite segments of series (13) is a weak solution of problem (4), (5) that satisfy (6), boundary and initial conditions (5) (formally, ignoring the convergence). Further study of the character of the convergence of series (13) based on the properties of many eigenvalues and systems of generalized eigenfunctions, submitted statements of lemma 2, as well as on the analysis of norms i(r), i(r),
t t
/||utm)||^i(r)dt, ||u(-,t)||M, / ||u(-,t)||2. .idt of series (13) [4] (see also [12, pp. 180,
0 2 0 ['1
185]). The latter represent a series of uniformly convergent about t G [0, T]. Where should that the sum y(x,t) of the series (13) belongs to V 1,0(a, rT) and is a weak solution (4),
(5) for any T. Checking integral identity of definition 2 is similar to how it was done in
N ( t \ proving the theorem 1: partially sum yN(x,t) = E ( fie-Xit + / fi(r)e-Xi(t-T)dr I ui(x)
i=1\ 0 J
of series (13) is substituted in the integral identities (10) and conversions are performed, resulting in (12). Following passage to the limit under N ^ ^ conclude the reasoning. This means that the entire sequence {yN}N>1 is weakly converges to an element (see proof of theorem 1).
Remark 4. When improvement the property of function f (x,t) along variable t (f (x,t) G CL2j1(rT)) raise the level of smoothness by y(x,t) along t > 0 (compare under t = 0 y(x, 0) G L2(r)). This stipulate to differentiability y(x,t) under t when t > 0, as is characteristically for equations of parabolic type [4] (see also [12, p. 184]).
Theorem 3. Initial-boundary value problem (4), (5) has only a weak solution in the space V 1,0(a, rT) for any 0 <T < <x>.
Proof. Proof of uniqueness by virtue of linearity problem (4), (5) is the standard way: assumes the existence of two different solutions y1(x,t), y2(x,t) of class V 1,0(a, rT) and shows (see (11) and the consequent reasoning) correctness of inequality ||y||2,rT ^ 0 (y(x,t) = y1(x,t) — y2(x,t)) for any T > 0, and that means, coincidence solutions y1(x,t), y2(x,t) in space V1,0(a, rT) (y1(x,t) similarly y2(x,t) almost everywhere with a fixed t e [0,T] for any T > 0).
Corollary 1. A weak solution of initial-boundary value problem (4), (5) continuously depends on the source data f (x,t) and x). Thus, shows the correctness of Hadamard initial-boundary value problem (4), (5) in the space V 1,0(a, rT) for any 0 <T < to.
Remark 5. Statements of theorems 1-3 are preserved under substitution [0, T] on [t0, T] (t0 > 0), the initial condition in the ratio (5) is replaced by y \t=t0 = p(x).
Remark 6. Boundary condition in (5) can be non-homogeneity:
y(x,t) \Kear= ^(x,t).
Proof of theorems in this case literally repeat the above reasonings. For this as a preliminary introduces a new unknown function y(x,t) = y(x,t) — Q(x,t) (here Q(x,t) is a arbitrary function of L2(Yt), having generalized derivative g L2(Yt) and satisfying (almost everywhere) non-homogeneity boundary condition (5)). The integral identities in definitions 1 and 2 be changed respectively.
In many applications where it is necessary to investigate the properties of solutions y(x,t) e V1,0(a, rT) of problem (4), (5) for an arbitrary finitely T, it is important to know the behavior y(x,t) under t ^ +to, i. e. when t e [0, to).
Let, as above f (x,t) e CL2,1(rT), and
t+1
I If(^)lll2rd < A (14)
for any t > 0 (A is fixed constant); the latter means that the function f (x,t) is defined in the domain r x [0, T) for any finite T.
Theorem 4. Let y(x,t) e V1,0 (a, rT) is a weak solution of problem (4), (5) for an arbitrary T > 0 which f (x,t), satisfies the condition (14). Then there is such positive
constant C, that
t+1
a) / ht^Wwwd ^ C for anyt > °
b) IIy(-,t))IlL2(r) < C, when t ^ +to.
P r o o f. In the conditions of theorems 2 and 3 the mapping t ^ y(x,t) (t ^ 0) is continuously. Split half-line [0, to) into sections [j — 1,j], j = 1,2,..., and relabel tj a number belonging to [j — 1,j], for which
M-^j )Il2L2(rT) = t^-ffl Wy(-,t)l2L2(rT), j = 1, 2,~.. (15)
For arbitrary positive s and t (s < t) from the integral identities of the definition 1 of weak problem solving (4), (5) should the ratio
f(^,y(x,t))dt + ff + b(x)y(x,t)y(x,t)) dxdt =
s г v '
t
J(f (x,t),y(x,t))dt,
if you put n(x, t) = y(x, t) (ipso the remark 4 of theorem 3 the function y(x, t) differentiable along t, € L2{Tt))- Of this ratio is obtained the inequality
hhi-Ml^v) - hM; s)lll2(r) + « / M-^Wwkd^ <
ft X1/2 ft s X1/2 (16)
||f (•,t)||2(r>d^ (/||y(^)||Wi(r)*J ,
where the positive constant a depends only on fixed a* and / (see the condition (2)).
Further reasoning relies on the idea presented in the monograph by J.-L. Lions [13, p. 519]: use the inequality (16) for an arbitrary segment [tj,tj+2], j = 1,2,..., because when condition (15) makes it possible the equality tj = tj+1. We show that for any j = 1,2,...
||y(-,tj+2)||L2(r) < max{||yO,tj)||l2(r)M}, (17)
where M = (^¡rX2 + Tr)1^ X is the inclusion constant the space W\(Y) in L2(Y): PHi2(r) < HWWj(r) foraany v G W 1(r).
Let ||y(•, tj+2)|L2(r) < h(^,tj)||L2(r), then the proof is complete. Suppose that
||y(•,tj+2)||L2(r) > Uy(;tj)|L2(r). (18)
By virtue of (16) for s = tj, t = tj+2 true the inequality
1 3 + 2
(ti + 2 ^ \ 1/2 (ti + 2 \1/2 (19) <Ш-л-)11|2(Г)+( I WfMWUr)*) ( / >
whence, comparing (19) to (18) it should be
fti + 2 V/2 (¿3 + 2 \1/2
« / (Г)^ Я ('^W)^ ■ (20)
' i + 2
Taking into account tj+2 — tj < 3 and the ratio (14), we obtain f \\f (-,я)\\2Ь2(Г)< 3A and the evaluation (20) takes a resultant form
¿3 + 2
f U^)\\2wl{T]<k<^- (21)
¿j
¿3 + 2
Since tj+2 — tj > 1, then exist т e [tj,tj+2] such that / \\y(-,s)\\Wl^d? = (tj+2 — tj)\\y(',t)\\Wi(Г) > \\y(',T)\Wi(Г), it means, from the evaluation (21) it should be
М;т)\\Ыг)^Х0. (22)
Let right how s = t, t = tj+2 in the ratio (16), then using (21), come to the inequality
h\\y(;ti+2)\\Ur) < iU-^)\\i2{T] + m1'2 (W>
or, taking into account (22),
_ 2
m-^wi^ < h {xy/¥) + m1/2 (^)1/2 = ^x2+¥
or
From here and the assumption (18) it should be (17) and further to any t > 0
(j = 1, 2,...) the inequality \\y(-,t))\\L2<r) < max{ max \\y(-,t)\\L2(r),M} = 9 is correctly.
te[o,2]
Considering the inequality (16) in the interval [t,t + 1] and given the assessment to \\y(-t))\\L2(r), get
t+1 f t+1 \1/2 (here ■d = ±<92 +
Thus, the assertion of the theorem is proved (C = max(9,$/a)), the solution y(x,t) defined in domain r^Q = To x [0, to).
3. Stability of system (4). Suppose 0 ^ b(x) ^ ¡3 for x e r then the eigenvalues Aj, i > 1, are positive (remark 3). Consider the system (4) on the set r^Q = r0 x (0, to). Relabel rto,t = ro x (to,t), d rt0,t = dr x (to,t) (0 <to <t< to), r^ =ro x (to, to), drto,TO = dr x (to, to); it is clear that rto,t C rt. As above f (x,t) e CL2,1(rT), moreover t+1
I \\f (■^nhrd ^ A for any t > 0.
Let the state of system (4) describes the function y(x,t) G T/1'°(a, ri0iOO), what is a weak solution of equation (4) in the domain rto with initial and boundary conditions
y \t=t0= xeT, y Ueart0,oo= 0, (23)
and function y(x,t) e V 1,o(
a, rt0,^) is a weak solution of equation (4) in the domain rt0,^ with initial and boundary conditions
y \t=to = ¥(x), x e y Ueart0,^, =0 (24)
(initial-boundary value problem (4), (23) differs from problem (4), (24) the fact that the function Tp(x) of first ratio (23) replaced by a different function <p(x)). The state y(x,t) of system (4) call perturbed, and y(x, t) is non-perturbed. From the proof of the theorem 3 (presentation (13) of weak problem solving (4), (5)) implies that states y(x,t), y(x,t) are defined in the domain rto,x, satisfy the corresponding initial and boundary conditions (23), (24) and belong to the space V 1,o(a, rt0,x) for f (x,t) e CL2A(r^).
Below we offer the notion of stability solution to initial-boundary value problem (4), (5) (stability of the undisturbed system (4)) what analog for the Lyapunov stability.
Definition 3. The non-perturbed system (4) is stability, if for any to > 0 and e > 0 executed such S(t0,e) >0 that under ||y-^||L2(r) < S(t0, e) is ||y(-,t) — y(•, i)|| vk1 (a,r) < e under t > to, where y(x,t) is the perturbed status of system (4).
Remark 7. Similarly, the definition of stability of the non-perturbed system (4) you can enter the definition of uniformly stability of the non-perturbed system (4) in the domain ri0jTO.
Remark 8. For the system (4) you can enter the definition of asymptotic and exponential stability of the non-perturbed system (4) in the domain rt0jTO.
Remark 9. By virtue of the linearity of the system (4) all definitions can be reformulated for zero solution (trivial solution) of system (4).
Theorem 5. Let 0 ^ b(x) ^ в, x e Г, then the non-perturbed system (4) in the domain Г^ is stability.
P г о о f. By virtue of the linearity of equation (4) function 0(x,t) = y(x,t) —y(x,t) is element of space V 1'0( a, rt0jTO) and a weak solution of initial-boundary value problem for homogeneous equation (4) (f = 0), satisfies the initial and boundary conditions
в |t=to = Ф(х), x e Г, в \хедГо^ =0, (25)
where ф(х) = <p(x) — y>(x). Following the reasoning of proof of theorems 1-3, the initial-boundary value problem (4), (25) is uniquely weakly solvable, its solution has form
0(x,t) = E Фге—Х!1гщ(х), Фг = (Ф,щ), and is limit weakly converging sequence {0N}N>1 i=1 ^ of approximations
N
0N (x,t)= £ Фгв-^Щ^), i=1
moreover
N
\\0N\\2,Г* < E Ф2е-2Х^, N =1, 2,... ■
i=1
Passage to the limit in the last inequality under N ^ж and bearing in mind the remark 3 we come to the estimate (e-2Xit < 1, i = 1,2, ■■■)
\\в\\2,Г* < C* (\Ф\Ь2(Г)) for any t e [t0, ж); C* is constant, not dependent on t. The latter means that (see (1))
\\0(;t)\\
W1 (а,Г) < с* (\\Ф\\Ь2(Г)) ■ (26)
Fix e > 0, take 6 = then from (26) and
\\ф\\ь2(г) = < <5
it should be inequality ||0(-,i)||iyi(a,r) = II< e for any t > t0. Proof of completed.
Example. Let Г is graph-star with ribs Yk, k = 1,2, 3, and internal node £ (to simplification the formulas assume that the ribs Yk, k = 1,2, are parameterized of segment [0,п/2], 7з is parameterized of segment [n/2,n]). In the domain Г x [0, ж) we consider the initial-boundary value problem (4), (5) under a(x) = 1, b(x) = 0 and f (x,t) = 0:
dy{x,t) = a y{x,t)^ y \t=0=(fi(x), xer, y\xearT=0.
A weak solution y(x,t) e V 1,0(1, Гто) of a problem is determined by the identity
J y(x,t)r](x,t)dx — J y(x, t) dvdxdt + / ^af^^af—dxdt = Г Г Г
= J y(x)n(x, 0)dx
г
for any t > 0 and any function n(x,t) e W 1(1, rT). Easily to show [10, 11], that the spectral problem (7) (a(x) = 1, b(x) = 0) in the weak formulation (8) defines a set of eigenvalues {An}n-^1 (An = n2) and system of generalized eigenfunctions {un}n-^1, where eigenvalues when n = 2k — 1 is prime numbers, when n = 2k have multiplicity 2, the corresponding generalized eigenfunctions are determined by the relations (k = 1,2,...)
{cos(2к — l)(x — j), x G 7i, cos(2/c - l)(x - x G 72, cos(2к — l)(x — j), x G 7з,
sin2k(x — x G 71, { 0, x G 71,
U2k,i{x) = { 0, x G 72, u2k,2{x) = I sin2k(x — §), x G 72,
sin2k(x — x G 73, [ sin2k(x — x G 73.
As above in proving theorem 3 forming approximation of solution yN(x,t) =
N _ 2 _ 2
J2 n tuj(x) weakly converging under N ^to to element y(x,t) = J2 e n tun(x),
n=1 n=1
yn = (p, un). Stability of the weak solution is obvious.
The work examined the initial-boundary value problem with Dirichlet boundary conditions in the ratios of (5). All estimates, reasoning and statements of theorem 1-5 persist for boundary conditions
(a(x)§l +ay) UearT= 0,
where a is a positive constant.
4. Conclusion. The approach presented by the statements of theorem 3-5 may be used to obtain conditions of stability (asymptotic stability) of weak solution of initial-boundary value problem (4), (5). The same approach applies to both tasks, space variable x which has a dimension of large 1 (x e Rn, n ^ 2) and for many-dimensional functions that describe the state of researched system [14-16]. The results are fundamental in tasks of optimal control and stabilization of differential systems with time-delay [17-28].
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Received: August 15, 2019.
Accepted: November 07, 2019.
Author's information:
Aleksei P. Zhabko — Dr. Sci. in Physics and Mathematics, Professor; [email protected]
Andrey I. Shindyapin — Dr. Sci. in Physics and Mathematics, Professor; [email protected]
Vyacheslav V. Provotorov — Dr. Sci. in Physics and Mathematics, Professor; [email protected]
Устойчивость слабого решения параболической системы с распределенными параметрами на графе
А. П. Жабко1, A. И. Шиндяпин2, В. В. Провоторов3
1 Санкт-Петербургский государственный университет, Российская Федерация, 199034, Санкт-Петербург, Университетская наб., 7—9
2 Университет Эдуардо Мондлане, Мозамбик, 3453, Мапуту, бульвар Джулиуса Ньерере, 1
3 Воронежский государственный университет, Российская Федерация, 394006, Воронеж, Университетская пл., 1
Для цитирования: Zhabko A. P., Shindyapin A. I., Provotorov V. V. Stability of weak solutions of parabolic systems with distributed parameters on the graph // Вестник Санкт-Петербургского университета. Прикладная математика. Информатика. Процессы управления. 2019. Т. 15. Вып. 4. С. 457-471. https://doi.org/10.21638/11702/spbu10.2019.404
Поведение решения эволюционного уравнения при неограниченном увеличении значений временной переменной давно является объектом обсуждения в научных кругах. Для этого есть немало причин прикладного характера, когда начальные условия уравнения задаются с определенной погрешностью: как малые изменения начальных условий влияют на поведение решения при больших значениях времени. Используется классическое понимание устойчивости решения дифференциального уравнения или системы уравнений, предложенное в работах А. М. Ляпунова: решение устойчиво, если оно мало изменяется при малых возмущениях начального условия и для любого момента времени. Установлены условия устойчивости решения эволюционной параболической системы с распределенными параметрами на графе, описывающей процесс переноса сплошной среды в пространственной сети. Параболическая система рассматривается в слабой постановке: слабым решением системы является суммируемая функция, удовлетворяющая интегральному тождеству, которое определяет вариационную постановку для начально-краевой задачи. Выход за рамки классических (гладких) решений и обращение к слабым решениям задачи продиктованы желанием авторов не только описать более точно физическую сущность процессов переноса (это приобретает особенное значение при изучении динамики многофазовых сред (multiphase media)), но и указать
пути анализа процессов переноса в многомерных сетеподобных областях. Используемый подход основывается на априорных оценках слабого решения и построении (метод Фаедо—Галеркина со специальным базисом — системой собственных функций эллиптического оператора параболического уравнения) слабо компактного семейства приближенных решений в выбранном пространстве состояний. Полученные результаты лежат в основе анализа задач оптимального управления дифференциальными системами с распределенными параметрами на графе, при этом выявлены интересные аналогии с многофазовыми задачами многомерной гидродинамики.
Ключевые слова: эволюционная система параболического типа, распределенные параметры на графе, слабое решение, устойчивость слабого решения.
Контактная информация:
Жабко Алексей Петрович — д-р физ.-мат. наук, проф.; [email protected] Шиндяпин Андрей Игоревич — д-р физ.-мат. наук, проф.; [email protected] Провоторов Вячеслав Васильевич — д-р физ.-мат. наук, проф.; [email protected]
