Stochastic stability of a class of partial differential equations of thermoelasticity
Àn example of stability analysis of a class of partial differential equations (in terms of Lyapunov functional) is presented. Applying Kozin’s method to construction of Lyapunov functional the sufficient conditions of stochastic stability of the heat transfer in a strip-plate are established.
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Інститут прикладних проблем механіки і математики ім. Я.С. Підстригача НАН України
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irk-123456789-76902010-04-09T12:01:05Z Stochastic stability of a class of partial differential equations of thermoelasticity Krol, M. Àn example of stability analysis of a class of partial differential equations (in terms of Lyapunov functional) is presented. Applying Kozin’s method to construction of Lyapunov functional the sufficient conditions of stochastic stability of the heat transfer in a strip-plate are established. Виведено умови стохастичної стабільності для рівняння термопружності для тонкої пластинки. Для цього згідно з методом Козіна використано функціонал Ляпунова. Як приклад встановлено умови стохастичної стабільності рівняння термопружності для напівнескінченного стрижня. Выведены условия стохастической стабильности для уравнений термоупругости для тонкой пластинки. Для этого согласно методу Козина использован функционал Ляпунова. В качестве примера установлены условия стохастической стабильности уравнения термоупругости для полубесконечного стержня. 2008 Article Stochastic stability of a class of partial differential equations of thermoelasticity / M. Krol // Приклад. пробл. механіки і математики. — 2008. — Вип. 6. — С. 193-200. — Бібліогр.: 17 назв. — англ. 1810-3022 http://dspace.nbuv.gov.ua/handle/123456789/7690 539.3 en Інститут прикладних проблем механіки і математики ім. Я.С. Підстригача НАН України |
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Àn example of stability analysis of a class of partial differential equations (in terms of Lyapunov functional) is presented. Applying Kozin’s method to construction of Lyapunov functional the sufficient conditions of stochastic stability of the heat transfer in a strip-plate are established. |
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| author |
Krol, M. |
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Krol, M. Stochastic stability of a class of partial differential equations of thermoelasticity |
| author_facet |
Krol, M. |
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Krol, M. |
| title |
Stochastic stability of a class of partial differential equations of thermoelasticity |
| title_short |
Stochastic stability of a class of partial differential equations of thermoelasticity |
| title_full |
Stochastic stability of a class of partial differential equations of thermoelasticity |
| title_fullStr |
Stochastic stability of a class of partial differential equations of thermoelasticity |
| title_full_unstemmed |
Stochastic stability of a class of partial differential equations of thermoelasticity |
| title_sort |
stochastic stability of a class of partial differential equations of thermoelasticity |
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Інститут прикладних проблем механіки і математики ім. Я.С. Підстригача НАН України |
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2008 |
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http://dspace.nbuv.gov.ua/handle/123456789/7690 |
| citation_txt |
Stochastic stability of a class of partial differential equations of thermoelasticity / M. Krol // Приклад. пробл. механіки і математики. — 2008. — Вип. 6. — С. 193-200. — Бібліогр.: 17 назв. — англ. |
| work_keys_str_mv |
AT krolm stochasticstabilityofaclassofpartialdifferentialequationsofthermoelasticity |
| first_indexed |
2025-07-02T10:28:33Z |
| last_indexed |
2025-07-02T10:28:33Z |
| _version_ |
1836530651894906880 |
| fulltext |
ISSN 1810-3022. Ïðèêë. ïðîáëåìè ìåõ. ³ ìàò. – 2008. – Âèï. 6. – Ñ. 193–200.
UDK 539.3
M. Król
STOCHASTIC STABILITY OF A CLASS OF PARTIAL DIFFERENTIAL
EQUATIONS OF THERMOELASTICITY
Àn example of stability analysis of a class of partial differential equations (in
terms of Lyapunov functional) is presented. Applying Kozin’s method to construc-
tion of Lyapunov functional the sufficient conditions of stochastic stability of the
heat transfer in a strip-plate are established.
1. Introduction. Considerable progress has been made over the last four
decades, in the study of the problem of the stochastic stability of partial diffe-
rential equations. It has been initiated by the J. C. Samuels and Yu. M. Erin-
ger [12] and studies by mathematicians and engineers in the vibration analysis
of beams, plates, shells and heat transfer problems. Two basic models with
stochastic parametric excitations have been developed in the literature, name-
ly with stationary ergodic processes and white noise processes. For the models
with stationary ergodic processes the stability analysis has been initiated by T.
K. Caughey and A. H. Gray (jr) [6] and developed by F. Kozin [8, 9], F. Kozin
and C. M. Wu [10]. A. Tylikowski has obtained several interesting results [16, 17].
The first work for the models with white noise processes has been pub-
lished by U. G. Haussmann [7] who applied the obtained results to the study
of a heat-transfer problem. T. Caraballo, K. Liu, and X. Mao [5, 11] genera-
lized these results for instance. The new criteria of stability for string, stick
and plate models with parametric white noise excitations have been obtain by
A. Tylikowski [14, 15].
In this paper we will deal with problem of stability of the partial diffe-
rential equation, which describe the heat-transfer in strip-plate. We use the
equation of motion derived by Yu. M. Kolyano, Ya. S. Podstryhacz [4] and we
assume that the parametric excitations are in the form of stationary ergodic
process. We apply the Kozin’s method to determine sufficient conditions of
almost sure asymptotic stability.
2. Mathematical preliminaries. The common stability properties of sto-
chastic systems that have been studied in the literature have generally been
related to Lyapunov stability [13]. Recognizing that stability in the Lyapunov
sense is merely a uniform convergence, which respect to the initial conditions,
various concepts of stability for stochastic systems can be immediately defined
by invoking one of the usual modes of probability theory. That is, for instan-
ce, convergence in probability and almost sure convergence.
In that follows, 0 0u t x t( ; , ) will denote the n -dimensional vector solution
at time t , with initial state 0x at time 0t , u will denote a suitable norm,
such as an absolute value or Euclides norm, and we shell be testing the stabi-
lity of the equilibrium solution 0u ≡ of the partial differential equations
i
i
u
F u x t
t
∂
=
∂
( , , ) , 11i M= , , , (1)
0kA u x t =( , , ) , 0t t ≥:{ } , 21k M= , , . (2)
That system has the following distinctive marks [14]:
1) occupies some region (coherent, open set) NΩ ⊂ is N -dimensional
Euclidean space with a boundary 1NB − = ∂Ω , [ ]1
N
Nx x x= ∈, , ;
2) describes the set M of the function [ ]1 Mu x t u x t u x t= ( , ) ( , ), , ( , ) , which
belongs to a function space of X XΩ ≡( ) , which is called phase space and
194 M. Król
satisfies the system of partial differential equations (1), (2) in region
1 1N NC + += Ω × ∆ ⊂ , where iF and kA are the differential operators,
which are described by continuous, differentiable functions and related to
spatial values. The space X of the points may be the set of variable
parameters, which characterize the condition of the system;
3) the function u x t( , ) which characterize the system are assumed to take
the values
0 0 0 0u u x t X X= ∈ Ω ⊃ Ω( , ) ( ) ( ) (3)
in the plane 10 Nt +Ω × = ⊂ { } (initial conditions) and the boundary
conditions
1 1 1u u x t X X= ∈ Ω ⊃ Ω( , ) ( ) ( ) . (4)
In stochastic case instead of equation (1) we consider the following sto-
chastic differential equation
i
i
u
F u x t
t
∂
= ω
∂
( , , , ) , 1i n= , , , (5)
where ω is an element of probability space B PΓ( , , ) . Similarly, the initial and
boundary conditions are determined [14].
In this paper we will use the following definitions of stochastic stability
[1, 2, 17].
Definition 1. (Almost Sure Lyapunov Stability). The equilibrium solution
of system (5) is said to be almost surely stable if
{ }
0 0
0 0
0
0 1
x t t
P u t x t
→ ≥
= =lim sup ( ; , ) .
Definition 2. (Almost Sure Asymptotic Stability). The equilibrium solution
of system (5) is said to be almost surely asymptotically stable if definition 1
holds and
{ }0 0 0 1
T t T
P u t x t
→∞ ≥
= =lim sup ( ; , ) .
2. The strip-plate equation of heat transfer. Let Θ be a bounded do-
main in d , where 3d ≤ , with 2C boundary. We will study the following
heat transfer equation of a thickness ( )2Z = δ = const isotropic homogenous
infinite strip-plate, which is averages by a integrals characteristic of tempera-
ture [4]
2 2 2
2
2 2 2 2
1 1
q
T T T TT
aX Y c
∗ ∗ ∗
∂ ∂ ∂ ∂+ − = +
∂ ∂ ∂τ ∂τ
æ , (6)
where T T X Y ∗= τ( , , ) , ∗ +τ ∈ (where + denotes the interval [ )0 ∞, ),
X Y ∈ Θ, , [ ]10X d∈ , , [ ]20Y d∈ , .
In this equation we use the following notation: T is the temperature,
X Y Z, , are the space coordinates, ∗τ is the time, qc is the propagation of
speed of the heat, 2 z
t
∗
α
=
λ δ
æ , t
v
a
c
λ
= is the coefficient of thermal conductivity,
vc is the coefficient of volumetric heat contents, zα is the coefficient of the
given up the heat by flank, tλ is the coefficient of heat conduction.
Stochastic stability of a class of partial differential equations of thermoelasticity 195
Now we assume the following initial conditions
0T = , 0T
∗
∂ =
∂τ
for 0∗τ = , (7)
0T = , 0T
Y
∂ =
∂
for 0Y = and 2Y d= . (8)
We introduce in this equation the following notations
x
Xx
k
= ,
y
Yy
k
= ,
z
Zz
k
= ,
k
∗
∗
τ
ττ = (9)
are dimensionless coordinates and time, where x y zk k k, , are scale coefficients,
and assume that
2
2
qc
a
β = , (10)
2
0∗ = + τ( )æ æ æ , (11)
where τ( )æ is a stationary ergodic stochastic process, whose samples are con-
tinuous functions with the probability one, 0æ is a constant.
If we assume, that 1xk d= , 2yk d= in coordinates (9) and using notation
(10) and (11) in equation (6) we obtain
2 2 2
02 0q xx q yy qT T c T c T c k Tττ τ+ β − − + + τ =( )( )æ , (12)
where
2
2
TTττ
∂=
∂τ
, TTτ
∂=
∂τ
,
2
2xx
TT
x
∂=
∂
,
2
2yy
TT
y
∂=
∂
,
1 20 0
x y
d d
x y
k k
∈ × = Ω
( , ) , , ,
z z
z
k k
δ δ ∈ −
, , 0τ ∈ +∞, )[ .
3. Stochastic stability analysis. In this section we use the Kozin’s method
to the equation (6). We investigate the stability of the trivial solutions of this
equation.
We set the initial conditions
0T = , 0xx yyT T= = for x y ∈ ∂Ω( , ) . (13)
We assumed that τ( )æ is the stationary ergodic process with the diffe-
rentiable realizations with probability 1. We shell study asymptotic stability of
the trivial solution 0T Tτ= = of equation (12) via a Lyapunov functional ap-
proach, using Kozin’s method [8].
We define the Lyapunov functional
2 2 2
2 2 2
T T T T T TV Q T d
x y x yΩ
∂ ∂ ∂ ∂ ∂ ∂ = Ω ∂ ∂ ∂τ ∂ ∂ ∂τ∫ , , , , , , ,
where Q is a quadratic form of its variables.
We shell use the following approach. Upon expanding T x y τ( , , ) into its
modes, we have
1 1 1 1
nm nm
n m n m
T x y T x y W nx my
∞ ∞ ∞ ∞
= = = =
τ = τ = τ π π∑ ∑ ∑ ∑( , , ) ( , , ) ( ) sin ( ) sin ( ) . (14)
Substituting (14) into (12) yields the model equations
2 2 2 2 2 2
02 0nm nm nm q nmW W W c n m W+ β + π + π + τ + = ( ) ( ) ( )( )æ æ ,
1 2 3n m = , , , , .
196 M. Król
Using the methodology proposed by F. Kozin and C. M. Wu [10] we will
deal with quadratic Lyapunov function for studying the almost sure stability
properties of nmW τ( ) in the form
nm nm nmV W Wτ = ×( )
2 2 2 2 2
02
1
nmq
nm
Wc n m
W
β + π + π + τ + β× ⋅
β
( ) ( ) ( )( )æ æ
.
Taking into account the properties
2
2
2
mn
nm
T
n T
x
∂
π =
∂
( ) ,
2
2
2
mn
nm
T
m T
y
∂
π =
∂
( ) ,
we can apply conditions of the ortogonality of the functions nxπsin ( ) and
mxπsin ( ) on 0 1,[ ] for 1 2 3n m = , , , , in order to obtain the desired Lyapu-
nov functional
2 2 2 2 2 2 2 2 2 2
02 2 q x q y qV T TT T c T c T c T dτ τ
Ω
τ = + β + β + + + Ω∫( ) ( )æ . (15)
The time-derivative of the functional (15) along the solution of the
equation (12) is given by:
2 2 2 2
02 2 q
dV
T T T T TT c
d ττ τ τ τ
Ω
τ = + + β + β + +τ ∫( ) ( ) ( )æ
2 2
q x x q y yc T T c T T dτ τ + + Ω , (16)
where
2
x
TT
xτ
∂=
∂ ∂τ
, x
TT
x
∂=
∂
,
2
y
TT
yτ
∂=
∂ ∂τ
, y
TT
y
∂=
∂
.
Substitution Tττ , determined by equation (12) into (16), yields the time-
derivative of the functional (15) in the form
2 2
dV
V U
d
τ = − β τ + τ
τ
( ) ( ) ( ) , (17)
where new functional U τ( )has the form
2 2 3 2 2 2 2 2 2 2 2
02 2 q x q y qU T TT T c T c T c Tτ τ
Ω
τ = β + β + β + β + β + β −∫( ) æ
2 2 2 2 2 2
q q x q yc TT T c T c Tτ τ− τ − β − β − β −( )æ
2 2 2 2 2 2
0q qc T c T d− β τ − β Ω( )æ æ . (18)
Using the dependencies of integrating by parts as for every pair of ele-
ments T and Tτ :
x x xxT T d T T dτ τ
Ω Ω
Ω = − Ω∫ ∫ ,
2
x xxT d TT d
Ω Ω
Ω = − Ω∫ ∫ ,
y x xxT T d T T dτ τ
Ω Ω
Ω = − Ω∫ ∫ ,
2
y yyT d TT d
Ω Ω
Ω = − Ω∫ ∫ ,
Stochastic stability of a class of partial differential equations of thermoelasticity 197
and taking into account the fact, that the V τ( ) satisfies the initial conditions
(13), we can introduce the functional U τ( ) in the form
2 2 2 2 2 2 22 2q qU c TT c T dτ
Ω
τ = β − τ + β β − τ Ω ∫( ) ( ) ( )( ) ( )æ æ . (19)
We look for a function λ , which satisfies the following inequality
U Vτ ≤ λ τ( ) ( ) . (20)
The function λ is defined as a minimum of the ratio /U V with respect
to permissible functions T and Tτ , which satisfy the initial conditions (13).
Since the minimum is the particular case of the stationary point, we can ap-
ply the calculus of variations and we consider the variation problem
0U Vδ − λ =( ) . After solving the equation of the variations we obtain
2 2 22 2 2qc T T T Tτ τ
Ω
β − τ − λ + β δ + ∫ ( ) ( )( )æ
2 2 2 2 2 22 2 2q qc T c Tτ+ β − τ + β β − τ − ( ) ( )( ) ( )æ æ
2 2 2 2 2
02 4 2 2 2 0q xx q yy qT T c T c T c T v dτ − λ β + β − − + δ Ω =( )æ . (21)
Taking into account the independence of variations appearing in integrals
(18) we find that the square brackets in (21) are equal zero
2 2 22 2 2 0qc T T Tτβ − τ − λ + β =( ) ( )( )æ , (22)
2 2 22 2qc T Tτβ − τ + β −( ) ( )( )æ
2 2 2 2 2
02 4 2 2 2 0q xx q yy qT T c T c T c Tτ− λ β + β − − + =( )æ . (23)
We calculate Tτ from equation (22) and substitute it into equation (23).
Then we obtain a partial differential equation for the function T x y τ( , , ) :
2 2 2
2 2 2
2 2
2 2
2
q
q
c T T
c T
β − τ − λβ
β − τ + β − λ
( )
( )
( )
( )
æ
æ
2 2 22 2
2
2
qc T Tβ − τ − λβ
− λ β + λ
( )( )æ
2 2 2 2 2
04 2 2 2 0q xx q yy qT c T c T c T
+ β − − − =
æ . (24)
It is easy to notice, that the n -order approximation of the solution of (24)
in the form
1 1
nm
n m
T T nx my
∞ ∞
= =
= τ π π∑ ∑ ( ) sin ( ) sin ( ) (25)
satisfies initial conditions (13). Substituting (25) into equation (24), we obtain
an algebraic equation with respect to the variable λ for every pair n m( , ) .
We denote these variables by nmλ . As a function λ we select the maximum
of variables nmλ , which satisfy inequality (20).
Finally the function λ has the form
2 2 2
1/22 2 2 2 21 2
0
2
2
q
n m
q
c
c n m=
β − τ λ =
β − π + π + , , ,
( )
max
( ) ( )( )
æ
æ
. (26)
198 M. Król
After substituting inequality (20) to equation (17), we obtain the differen-
tial inequality of functional V τ( ) . Solving this inequality, we obtain the upper
estimation of the functional V τ( ) (lemma 2.1 [3])
0
10 2V V s ds
τ τ ≤ − β − λ τ τ
∫( ) ( ) exp ( ) .
If τ( )æ is an ergodic and stationary process and we replace the mean va-
lue of the temperature by the corresponding mean value of the set of realiza-
tions, we obtain the condition of almost sure asymptotic stability with respect
to the following measure of temperature
T V= , (27)
where τ → ∞ in the form Eβ ≥ λ[ ] .
To derive the sufficient conditions of the almost sure instability we look
for a function η , which satisfies the inequality U Vτ ≥ η τ( ) ( ) . Watching the
derivation of the function can check it λ . The function η equals to the mini-
mum of nmλ given by (26) and suitable estimation of the functional V τ( ) has
the form
0
10 2V V s ds
τ τ ≥ η − β τ τ
∫( ) ( ) exp ( ) .
The condition of almost sure asymptotic instability for the stationary and
ergodic process has the form
Eβ ≤ η[ ] .
The obtained results can be summarized in the following criteria.
Criterion 1. The trivial solution of the partial differential equation (12) is
almost sure asymptotic stable with respect to the norm (27) if the following
conditions are satisfied
(i) process τ( )æ is stationary and ergodic;
(ii) Eβ ≥ λ[ ] ,
where
2 2 2
1/22 2 2 2 21 2
0
2
2
q
n m
q
c
c n m=
β − τ λ =
β − π + π + , , ,
( )
max
( ) ( )( )
æ
æ
.
Criterion 2. The solution of the partial differential equation (12) is almost
sure asymptotic unstable with respect to the norm (27) if the following condi-
tions are satisfied
i) process τ( )æ is stationary and ergodic;
(ii) Eβ ≤ η[ ] ,
where
2 2 2
1/22 2 2 2 21 2
0
2
2
q
n m
q
c
c n m=
β − τ η =
β − π + π +
, , ,
( )
min
( ) ( )( )
æ
æ
.
Example. We consider a particular case of system (6).
If we assumed that
2
2
T
X
∂
∂
is equal zero in equation (6), then this equation
we can rewrite in the form
Stochastic stability of a class of partial differential equations of thermoelasticity 199
2 2
2
2 2 2
1 1
q
T T TT
aY c
∗ ∗ ∗
∂ ∂ ∂− = +
∂ ∂τ ∂τ
æ , (28)
where T T Y ∗= τ( , ) , ∗ +τ ∈ (where + denotes the interval 0 ∞, )[ ), Y ∈ Θ .
Now we assume the following initial conditions
0T = , 0T
∗
∂ =
∂τ
for 0∗τ = ,
0T = , 0T
Y
∂ =
∂
for 20Y d∈ ( , ) .
Now we introduce the following notations
y
Yy
k
= ,
z
Zz
k
= ,
k
∗
τ
ττ = (29)
are dimensionless coordinates and time, where y zk k, are scale coefficients,
like in equation (6).
Using notation (10), (11) and (29) in equation (28) we obtain
2 2
02 0q yy qT T c T c k Tττ τ+ β − + + τ =( ( ))æ (30)
where
2
2
TTττ
∂=
∂τ
, TTτ
∂=
∂τ
,
2
2yy
TT
y
∂=
∂
,
20y d∈ = Ω( , ) ,
z z
z
k k
δ δ∈ −
, , 0τ ∈ +∞, )[ ,
with the simplified initial conditions (13).
Now we shell look for the Lyapunov functional using the method
describe above as defined in (15).
The functionals V τ( ) and U τ( ) for equation (30) have the form as follows
2 2 2 2 2 2 2 2
02 2 q y qV T TT T c T c T dτ τ
Ω
τ = + β + β + + Ω∫( ) ( )æ ,
2 2 2 2 2 2 22 2q qU c TT c T dτ
Ω
τ = β − τ + β β − τ Ω ∫( ) ( ) ( )( ) ( )æ æ .
Again we wish to determine the λ so that satisfies the inequality
U Vτ ≤ λ τ( ) ( )
and we apply the variational calculus to solve the problem
0U Vδ − λ =( ) . (31)
After applying the straightforward computations, we find the sequence
of mλ that satisfies (31) to be
2 2 2
1/22 2 2 21 2
0
2
2
q
n m
q
c
c m=
β − τ λ =
β − π + , , ,
( )
max
( )( )
æ
æ[ ]
.
4. Conclusions. The major conclusions are that the Lyapunov’s method is
an effective tool of solving the stability problem of strip-plate. The explicit
criteria developed in the paper define the stability region in terms of the
excitation process and physical characteristic of strip-plate. The analytical for-
mulas defining the stability regions are obtained using the calculus of varia-
tions.
200 M. Król
1. Ãèõìàí È. È., Äîðîãîâöåâ A. ß. Îá óñòîé÷èâîñòè ðåøåíèé ñòîõàñòè÷åñêèõ äèô-
ôåðåíöèàëüíûõ óðàâíåíèé // Óêð. ìàò. æóðí. – 1968. – 17, ¹ 6. – Ñ. 3–21.
2. Ãèõìàí È. È., Ñêîðîõîä A. Â. Ñòîõàñòè÷åñêèå äèôôåðåíöèàëüíûå óðàâíåíèÿ. –
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СТОХАСТИЧНА СТАБІЛЬНІСТЬ ДЕЯКОГО КЛАСУ ДИФЕРЕНЦІАЛЬНИХ
РІВНЯНЬ ТЕРМОПРУЖНОСТІ У ЧАСТИННИХ ПОХІДНИХ
Âèâåäåíî óìîâè ñòîõàñòè÷íî¿ ñòàá³ëüíîñò³ äëÿ ð³âíÿííÿ òåðìîïðóæíîñò³ äëÿ
òîíêî¿ ïëàñòèíêè. Äëÿ öüîãî çã³äíî ç ìåòîäîì Êîçiíà âèêîðèñòàíî ôóíêö³îíàë
Ëÿïóíîâà. ßê ïðèêëàä âñòàíîâëåíî óìîâè ñòîõàñòè÷íî¿ ñòàá³ëüíîñò³ ð³âíÿííÿ
òåðìîïðóæíîñò³ äëÿ íàï³âíåñê³í÷åííîãî ñòåðæíÿ.
СТОХАСТИЧЕСКАЯ СТАБИЛЬНОСТЬ НЕКОТОРОГО КЛАССА ДИФФЕРЕНЦИАЛЬНЫХ
УРАВНЕНИЙ ТЕРМОУПРУГОСТИ В ЧАСТНЫХ ПРОИЗВОДНЫХ
Âûâåäåíû óñëîâèÿ ñòîõàñòè÷åñêîé ñòàáèëüíîñòè äëÿ óðàâíåíèé òåðìîóïðóãîñòè
äëÿ òîíêîé ïëàñòèíêè. Äëÿ ýòîãî ñîãëàñíî ìåòîäó Êîçèíà èñïîëüçîâàí ôóíêöèî-
íàë Ëÿïóíîâà.  êà÷åñòâå ïðèìåðà óñòàíîâëåíû óñëîâèÿ ñòîõàñòè÷åñêîé ñòà-
áèëüíîñòè óðàâíåíèÿ òåðìîóïðóãîñòè äëÿ ïîëóáåñêîíå÷íîãî ñòåðæíÿ.
Politechnika of Rzeszów, Rzeszów, Poland Received
12.08.08
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