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Stability for determining the principal coefficient of parabolic equation
Applied Mathematics and Computation 219 (2012) 3826–3830
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Stability for determining the principal coefficient of parabolic equation Ali Demir ⇑, Arzu Erdem Department of Mathematics, Kocaeli University, Umuttepe Kampusu, 41380 Izmit-Kocaeli, Turkey
a r t i c l e
i n f o
a b s t r a c t This work investigates stability of an inverse problem for a backward linear parabolic heat problem by using an initial temperature measurement. This kinds of problem has been widely used in a various field of pure and applied science. The necessary condition of the minimizer for the cost functional are constructed by using the optimal control theory. Stability of the minimizer for the cost functional is proved based on the necessary condition. Ó 2012 Elsevier Inc. All rights reserved.
Keywords: Inverse parabolic problem Optimal control Source term Uniqueness
1. Introduction In this paper we deal with the stability for the inverse problem of determining the principal coefficient cðtÞ satisfying the linear parabolic heat equation
ut ¼ uxx cðtÞu þ f ðx; tÞ;
x 2 ð0; lÞ;
t 2 ð0; T
ð1Þ
along with the terminal time condition
uðx; TÞ ¼ uðxÞ x 2 ½0; l
ð2Þ
the boundary conditions
ux ð0; tÞ ¼ uðl; tÞ ¼ 0 t 2 ½0; T
ð3Þ
and the additional condition
uðx; 0Þ ¼ gðxÞ x 2 ½0; l;
ð4Þ
where f ðx; tÞ; uðxÞ; gðxÞ are all continuously differentiable. Problems of this type are some active area of research, such as heat conduction, optical medical imaging, geophysics of exploration. When the function cðtÞ is given the problem (1)–(3) has been analyzed as a direct problem in [4]. However, the identification of time dependent coefficient cðtÞ in (1)–(4) which is referred as the inverse problem lead to ill-posed or improperly posed problem in the sense of Hadamard [1,6,7,11]. Inverse coefficient problem has already been studied by several authors. Choulli and Yamamoto [2] proved, under some conditions, the inverse problem of recovering the coefficient qðxÞ , appearing in an initial-boundary value problem for the equation ut ¼ Du þ qðxÞu from overdetermined final data is locally wellposed in L2 around 0 when q is assumed to be a priori supported in some suitable subset. Shamsi and Dehghan [10] proposed a Legendre pseudospectral method for solving approximately an inverse problem of determining an unknown control parameter pðtÞ in the semilinear time-dependent three-dimensional diffusion equation ut ¼ Du þ pðtÞu þ kðx; tÞ subject to initial condition, Dirichlet boundary conditions and the integral overspecification over the spacial domain. Trucu et al. ⇑ Corresponding author. E-mail addresses: [email protected] (A. Demir), [email protected] (A. Erdem). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.10.011
A. Demir, A. Erdem / Applied Mathematics and Computation 219 (2012) 3826–3830
3827
[12] discussed the retrieval of the time-dependent coefficient along with initial boundary value problem from various types of measured noisy and exact data. Pourgholi et al. [8] have suggested the numerical method combining the use of the finite difference method with the solution of ordinary differential equation for the determination of unknown coefficient in an inverse heat conduction problem. Fernandez and Pola [5] have studied the uniqueness of the inverse coeffcient problem under appropriate assumptions. Egger et al. [3] derived global uniqueness of a solution ðu; qÞ the inverse problem of determining the function qðÞ for the equation ut ðx; tÞ þ uxx þ qðuÞ ¼ f ðx; tÞ with the initial condition and Dirichlet boundary conditions nd Hölder stability of the functions u and q with respect to errors in the measurements of the Neumann boundary data the initial condition and the a priori knowledge of the function q. However, this method requires, assumptions on the functions q, boundary conditions and solution u. In this paper, we use optimization technique to deal with backward-inverse heat conduction problem with the initial time observation. These method works by minimizing an objective function such that its minimum corresponds to the ideal design configuration. The optimization methods replace the ill-conditioned problem with a well-posed problem that must be solved repetitively through a systematic approach to an optimum solution. Yang et al. [13] investigated an inverse problem of identifying the spacewise dependent coefficient of parabolic equation when the final observation is given by using the optimal control framework. Nevertheless, publications have not been found so far to use this method for solving the inverse problem of determining the principal coefficient cðtÞ with the final time conditon, Neumann–Dirichlet boundary data and initial time observation. In this work, first by using a transformation the backward problem (1)–(4) is reduced into the identification of source function in backward problem. Based on optimal control framework we form the necessary condition for the minimizer of the cost functional. Finally stability of the minimizer is established under this condition. The outline of the manuscript is as follows: In Section 2, the inverse problem (1)–(4) has been replaced by the identification of source function in backward problem. An optimal control problem is obtained from the backward inverse problem. The necessary condition of the minimizer is deduced in Section 3. In the final section, the global uniqueness and stability of the minimizer are presented. 2. Mathematical formulation The method begins with using the following transformation:
v ðx; tÞ ¼ uðx; tÞrðtÞ;
ð5Þ
where
rðtÞ ¼ exp
Z
t
cðgÞdg :
ð6Þ
0
Transformations (5), (6) changes the problem (1)–(4) into the identification of source function in backward problem, namely
v t ¼ v xx þ rðtÞf ðx; tÞ; v ðx; TÞ ¼ uðxÞrðTÞ; v x ð0; tÞ ¼ v ðl; tÞ ¼ 0; v ðx; 0Þ ¼ gðxÞ
x 2 ð0; lÞ; t 2 ð0; T;
ð7Þ
x 2 ½0; l;
ð8Þ
t 2 ½0; T;
ð9Þ
x 2 ½0; l:
ð10Þ kþa
The proof of the solvability of the inverse problem (7)–(10) in spaces C , with a fixed in ð0; 1Þ and k 2 N, is a continuous functions with Hölder continuous derivatives, has been constructed in [9], as follows. Theorem 1. If gðxÞ; uðxÞ2 C 2þa ½0; 1; gðxÞ P 0, and the compatibility conditions up to first order are satisfied, then there exists a unique solution v 2 C 2þa;1þa=2 ð½0; 1 ½0; TÞ; c 2 C a=2 ½0; T of the inverse problem (7)–(10) which is continuously dependent upon data. We now introduce the following optimal control problem. The optimization-based formulation of this inverse problem consists of the minimization of the objective functional
Jðer Þ ¼ min JðrÞ;
ð11Þ
r2R0
where
JðrÞ ¼
1 2
Z 0
l
½v ðx; 0; rÞ gðxÞ2 dx þ
a 2
Z
T
r 2 ðtÞdt; 0
R0 :¼ frðtÞ 2 C½0; T : rðTÞ ¼ c0 ; constantg: Here v ðx; 0; rÞ denotes the initial solution of the problem (7)–(9) for given rðtÞ.
ð12Þ
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A. Demir, A. Erdem / Applied Mathematics and Computation 219 (2012) 3826–3830
3. Necessary condition of optimal control problem For the solution of the optimization (11), (12) we derive the necessary condition. Theorem 2. Let rðtÞ be the solution of the optimal control problem (11), (12) and v ðx; tÞ be the solution of (7)–(9) corresponding to this optimal coefficient. Then for any hðtÞ 2 R0 the following integral inequality holds:
Z
l
½v ðx; 0; rÞ gðxÞnðx; 0Þdx þ a
0
Z
T
rðtÞ½hðtÞ rðtÞdt P 0:
ð13Þ
0
Here, nðx; tÞ is the solution of the problem given by
8 > < nt ¼ nxx þ ðhðtÞ rðtÞÞf ðx; tÞ; x 2 ð0; lÞ; t 2 ð0; T nðx; TÞ ¼ 0; x 2 ½0; l > : nx ð0; tÞ ¼ nðl; tÞ ¼ 0; t 2 ½0; T: Proof. For any hðtÞ 2 R0 and d 2 ½0; 1 let us set
rd :¼ ð1 dÞr þ dh: Then we have
J d :¼ Jðrd Þ ¼
1 2
Z
l
½v ðx; 0; r d Þ gðxÞ2 dx þ
0
Z
a 2
0
T
r 2d ðtÞdt:
Differentiation of J d with respect to d at d ¼ 0 gives the following inequality since rðtÞ is the solution of the optimal control problem (11), (12):
Z
l
½v ðx; 0; rÞ gðxÞ
0
dv ðx; 0; rd Þ dx þ a dd
Z
T
rðtÞðhðtÞ rðtÞÞdt P 0:
ð14Þ
0
vd e d ¼ @@d e d is the solution of the Let us denote by v d ¼ v d ðx; t; r d Þ the corresponding solution of problem (7)–(10). Taking v then v following problem:
8 e d;xx þ ðhðtÞ rðtÞÞf ðx; tÞ; x 2 ð0; lÞ; t 2 ð0; T e d;t ¼ v > :e v d;x ð0; tÞ ¼ ve ðl; tÞ ¼ 0; t 2 ½0; T:
ð15Þ
e d jd¼0 then n satisfies the following problem: Let us set n ¼ v
8 > < nt ¼ nxx þ ðhðtÞ rðtÞÞf ðx; tÞ; x 2 ð0; lÞ; t 2 ð0; T nðx; TÞ ¼ 0; x 2 ½0; l > : nx ð0; tÞ ¼ nðl; tÞ ¼ 0; t 2 ½0; T:
ð16Þ
Taking into account (16) in (14) we obtain
Z
l
½v ðx; 0; rÞ gðxÞnðx; 0Þdx þ a
0
Z
T
rðtÞ½hðtÞ rðtÞdt P 0:
0
Lemma 3. Let rðtÞ be the solution of the optimal control problem (11), (12). Then for any hðtÞ 2 R0 the following integral inequality holds:
Z
T 0
Z
l
½hðtÞ rðtÞf ðx; tÞuðx; tÞdxdt þ a 0
Z
T
rðtÞ½hðtÞ rðtÞdt P 0;
ð17Þ
0
where uðx; tÞ is the solution of the adjoint problem given by
8 > < ut uxx ¼ 0; x 2 ð0; lÞ; t 2 ð0; T uðx; 0Þ ¼ v ðx; 0Þ gðxÞ; x 2 ½0; l > : ux ð0; tÞ ¼ uðl; tÞ ¼ 0; t 2 ½0; T: Proof. The unique solution uðx; tÞ of (18) belongs to C 2þa;1þa=2 ð½0; 1 ½0; TÞ. Multiplying each side of equation (18) by nðx; tÞ satisfies (16), integrating it by parts, we obtain
ð18Þ
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A. Demir, A. Erdem / Applied Mathematics and Computation 219 (2012) 3826–3830
0¼
Z
T
Z
0
0
l
nðut uxx Þdxdt ¼
Z
T
Z
0
l
nt udxdt 0
Z
T
Z
0
l
nxx udxdt
Z
0
l
0
nujt¼T t¼0 dx
Z
T
0
nux jx¼l x¼0 dt þ
Z 0
T
nx ujx¼l x¼0 dt
and using the conditions of (16), (18), the above integral identity implies:
Z
Z
T
0
l
½hðtÞ rðtÞf ðx; tÞuðx; tÞdxdt þ
0
Z
l
½v ðx; 0; rÞ gðxÞnðx; 0Þdx ¼ 0
ð19Þ
0
Combining (19) and (13),we obtain the integral inequality (17). h
4. Uniqueness
Theorem 4. Suppose that g 1 ðxÞ and g 2 ðxÞ are two functions which satisfy (7)–(10). Let r 1 ðtÞ and r 2 ðtÞ be the solutions of the optimal control problem (11), (12) then we have the following estimate:
1 kr 1 r 2 kL2 ð0;TÞ 6 pffiffiffiffiffiffi kg 1 g 2 kL2 ð0;lÞ : 2a Proof. Denote by v i ¼ v i ðx; t; ri Þ; i ¼ 1; 2 the solutions of the problem (7)–(10) corresponding to ri ðtÞ. Let n1 ¼ nðx; t; r2 r1 Þ and n2 ¼ nðx; t; r 1 r2 Þ be two solutions of (16). Defining by w ¼ v 1 v 2 and p ¼ n1 þ n2 ; wðx; tÞ and pðx; tÞare the solutions of the following problems, respectively:
8 < wt ¼ wxx þ ðr 1 ðtÞ r 2 ðtÞÞf ðx; tÞ; x 2 ð0; lÞ; t 2 ð0; T wðx; TÞ ¼ 0; x 2 ½0; l : wx ð0; tÞ ¼ wðl; tÞ ¼ 0; t 2 ½0; T;
ð20Þ
8 < pt ¼ pxx ; x 2 ð0; lÞ; t 2 ð0; T pðx; TÞ ¼ 0; x 2 ½0; l : px ð0; tÞ ¼ pðl; tÞ ¼ 0; t 2 ½0; T:
ð21Þ
Note that we have w ¼ n1 and w ¼ n2 so we conclude pðx; tÞ ¼ 0. Now, taking h ¼ r 3i and r ¼ ri ; i ¼ 1; 2 into (13), respectively, there holds
Z
l
½v 1 ðx; 0Þ g 1 ðxÞn1 ðx; 0Þdx þ a
0
Z
l
½v 2 ðx; 0Þ g 2 ðxÞn2 ðx; 0Þdx þ a
0
Z
T
r 1 ðtÞ½r 2 ðtÞ r 1 ðtÞdt P 0;
ð22Þ
r 2 ðtÞ½r 1 ðtÞ r 2 ðtÞdt P 0:
ð23Þ
0
Z
T
0
Combining (22) and (23), we obtain
a
Z
T
½r 1 ðtÞ r 2 ðtÞ2 dt 6
0
¼
Z Z
½v 1 ðx; 0Þ g 1 ðxÞn1 ðx; 0Þdx þ
0
0
þ ¼
l
Z
l
½v 1 ðx; 0Þ g 1 ðxÞn1 ðx; 0Þdx Z
Z Z
l
½v 2 ðx; 0Þ g 2 ðxÞn2 ðx; 0Þdx
0
0
l
½v 2 ðx; 0Þ g 2 ðxÞn1 ðx; 0Þdx þ
l
½v 2 ðx; 0Þ g 2 ðxÞn2 ðx; 0Þdx
0 l
½wðx; 0Þ ðg 1 ðxÞ g 2 ðxÞÞn1 ðx; 0Þdx
0
then let us use wðx; tÞ ¼ n1 ðx; tÞ and Young’s inequality in the above inequality:
a
Z
T 0
Z
l
n21 ðx; 0Þdx þ
1 2
Z
l
1 ðg 1 ðxÞ g 2 ðxÞÞ2 dx þ 2 0 0 Z Z l 1 l 1 ðg ðxÞ g 2 ðxÞÞ2 dx n2 ðx; 0Þdx ¼ 2 0 1 2 0 1 Z 1 l ðg ðxÞ g 2 ðxÞÞ2 dx: 6 2 0 1
½r 1 ðtÞ r 2 ðtÞ2 dt 6
Finally, we obtain
1 kr 1 r 2 kL2 ð0;TÞ 6 pffiffiffiffiffiffi kg 1 g 2 kL2 ð0;lÞ : 2a
Z 0
l
n21 ðx; 0Þdx
Z 0
l
½v 2 ðx; 0Þ g 2 ðxÞn1 ðx; 0Þdx
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A. Demir, A. Erdem / Applied Mathematics and Computation 219 (2012) 3826–3830
References [1] S. Avdonin, T.I. Seidman, Identification of qðxÞ in ut ¼ Du qu from boundary observations, SIAM J. Control Optim. 33 (1995) 1247–1255. [2] M. Choulli, M. Yamamoto, An inverse parabolic problem with non-zero initial condition, Inverse Problems 13 (1997) 19–27. [3] H. Egger, H. W Engl, M.V. Klibanov, Global uniqueness and Holder stability for recovering a nonlinear source term in a parabolic equation, Inverse Problems 21 (2005) 271–290. [4] L.C. Evans, Partial differential equations, American Mathematical Society Providence, 1998. [5] L.A. Fernandez, C. Pola, Identification of a quasilinear parabolic equation from final data, Int. J. Appl. Math. Comput. Sci. 11 (4) (2001) 859–879. [6] V. Isakov, Inverse Problems for Partial Differential Equations, Springer, New York, 1998. [7] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problem, Springer, New York, 1999. [8] R. Pourgholi, A. Tahmasbi, A.H. Borzabadi, S.A. Ketabi, A numerical approach to solving an inverse parabolic problem using finite differential method, Journal of Information and Computing Science 3 (3) (2008) 233–240. [9] A.I. Prilepko, V.V. Sololev, Solvability of the inverse boundary-value problem of finding a coefficient of a lower-order derivative in a parabolic equation, Differential Equations 23 (1987) 101–107. [10] M. Shamsi, Mehdi Dehghan, Determination of a control function in three-dimensional parabolic equations by Legendre pseudospectral method, Numer. Methods Partial Differ. Equat. 28 (1) (2012) 74–93. [11] A. Tikhonov, V. Arsenin, Solutions of Ill-posed Problems, Geology Press, Beijing, 1979. [12] D. Trucu, D.B. Ingham, D. Lesnic, Inverse time-dependent perfusion coefficient identification, J. Phys. Conf. Ser. 124 (1) (2008). [13] L. Yang, J.N. Yu, Z.C. Deng, An inverse problem of identifying the coefficient of parabolic equation, Appl. Math. Modell. 32 (2008) 1984–1995.
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Stability for determining the principal coefficient of parabolic equation Ali Demir ⇑, Arzu Erdem Department of Mathematics, Kocaeli University, Umuttepe Kampusu, 41380 Izmit-Kocaeli, Turkey
a r t i c l e
i n f o
a b s t r a c t This work investigates stability of an inverse problem for a backward linear parabolic heat problem by using an initial temperature measurement. This kinds of problem has been widely used in a various field of pure and applied science. The necessary condition of the minimizer for the cost functional are constructed by using the optimal control theory. Stability of the minimizer for the cost functional is proved based on the necessary condition. Ó 2012 Elsevier Inc. All rights reserved.
Keywords: Inverse parabolic problem Optimal control Source term Uniqueness
1. Introduction In this paper we deal with the stability for the inverse problem of determining the principal coefficient cðtÞ satisfying the linear parabolic heat equation
ut ¼ uxx cðtÞu þ f ðx; tÞ;
x 2 ð0; lÞ;
t 2 ð0; T
ð1Þ
along with the terminal time condition
uðx; TÞ ¼ uðxÞ x 2 ½0; l
ð2Þ
the boundary conditions
ux ð0; tÞ ¼ uðl; tÞ ¼ 0 t 2 ½0; T
ð3Þ
and the additional condition
uðx; 0Þ ¼ gðxÞ x 2 ½0; l;
ð4Þ
where f ðx; tÞ; uðxÞ; gðxÞ are all continuously differentiable. Problems of this type are some active area of research, such as heat conduction, optical medical imaging, geophysics of exploration. When the function cðtÞ is given the problem (1)–(3) has been analyzed as a direct problem in [4]. However, the identification of time dependent coefficient cðtÞ in (1)–(4) which is referred as the inverse problem lead to ill-posed or improperly posed problem in the sense of Hadamard [1,6,7,11]. Inverse coefficient problem has already been studied by several authors. Choulli and Yamamoto [2] proved, under some conditions, the inverse problem of recovering the coefficient qðxÞ , appearing in an initial-boundary value problem for the equation ut ¼ Du þ qðxÞu from overdetermined final data is locally wellposed in L2 around 0 when q is assumed to be a priori supported in some suitable subset. Shamsi and Dehghan [10] proposed a Legendre pseudospectral method for solving approximately an inverse problem of determining an unknown control parameter pðtÞ in the semilinear time-dependent three-dimensional diffusion equation ut ¼ Du þ pðtÞu þ kðx; tÞ subject to initial condition, Dirichlet boundary conditions and the integral overspecification over the spacial domain. Trucu et al. ⇑ Corresponding author. E-mail addresses: [email protected] (A. Demir), [email protected] (A. Erdem). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.10.011
A. Demir, A. Erdem / Applied Mathematics and Computation 219 (2012) 3826–3830
3827
[12] discussed the retrieval of the time-dependent coefficient along with initial boundary value problem from various types of measured noisy and exact data. Pourgholi et al. [8] have suggested the numerical method combining the use of the finite difference method with the solution of ordinary differential equation for the determination of unknown coefficient in an inverse heat conduction problem. Fernandez and Pola [5] have studied the uniqueness of the inverse coeffcient problem under appropriate assumptions. Egger et al. [3] derived global uniqueness of a solution ðu; qÞ the inverse problem of determining the function qðÞ for the equation ut ðx; tÞ þ uxx þ qðuÞ ¼ f ðx; tÞ with the initial condition and Dirichlet boundary conditions nd Hölder stability of the functions u and q with respect to errors in the measurements of the Neumann boundary data the initial condition and the a priori knowledge of the function q. However, this method requires, assumptions on the functions q, boundary conditions and solution u. In this paper, we use optimization technique to deal with backward-inverse heat conduction problem with the initial time observation. These method works by minimizing an objective function such that its minimum corresponds to the ideal design configuration. The optimization methods replace the ill-conditioned problem with a well-posed problem that must be solved repetitively through a systematic approach to an optimum solution. Yang et al. [13] investigated an inverse problem of identifying the spacewise dependent coefficient of parabolic equation when the final observation is given by using the optimal control framework. Nevertheless, publications have not been found so far to use this method for solving the inverse problem of determining the principal coefficient cðtÞ with the final time conditon, Neumann–Dirichlet boundary data and initial time observation. In this work, first by using a transformation the backward problem (1)–(4) is reduced into the identification of source function in backward problem. Based on optimal control framework we form the necessary condition for the minimizer of the cost functional. Finally stability of the minimizer is established under this condition. The outline of the manuscript is as follows: In Section 2, the inverse problem (1)–(4) has been replaced by the identification of source function in backward problem. An optimal control problem is obtained from the backward inverse problem. The necessary condition of the minimizer is deduced in Section 3. In the final section, the global uniqueness and stability of the minimizer are presented. 2. Mathematical formulation The method begins with using the following transformation:
v ðx; tÞ ¼ uðx; tÞrðtÞ;
ð5Þ
where
rðtÞ ¼ exp
Z
t
cðgÞdg :
ð6Þ
0
Transformations (5), (6) changes the problem (1)–(4) into the identification of source function in backward problem, namely
v t ¼ v xx þ rðtÞf ðx; tÞ; v ðx; TÞ ¼ uðxÞrðTÞ; v x ð0; tÞ ¼ v ðl; tÞ ¼ 0; v ðx; 0Þ ¼ gðxÞ
x 2 ð0; lÞ; t 2 ð0; T;
ð7Þ
x 2 ½0; l;
ð8Þ
t 2 ½0; T;
ð9Þ
x 2 ½0; l:
ð10Þ kþa
The proof of the solvability of the inverse problem (7)–(10) in spaces C , with a fixed in ð0; 1Þ and k 2 N, is a continuous functions with Hölder continuous derivatives, has been constructed in [9], as follows. Theorem 1. If gðxÞ; uðxÞ2 C 2þa ½0; 1; gðxÞ P 0, and the compatibility conditions up to first order are satisfied, then there exists a unique solution v 2 C 2þa;1þa=2 ð½0; 1 ½0; TÞ; c 2 C a=2 ½0; T of the inverse problem (7)–(10) which is continuously dependent upon data. We now introduce the following optimal control problem. The optimization-based formulation of this inverse problem consists of the minimization of the objective functional
Jðer Þ ¼ min JðrÞ;
ð11Þ
r2R0
where
JðrÞ ¼
1 2
Z 0
l
½v ðx; 0; rÞ gðxÞ2 dx þ
a 2
Z
T
r 2 ðtÞdt; 0
R0 :¼ frðtÞ 2 C½0; T : rðTÞ ¼ c0 ; constantg: Here v ðx; 0; rÞ denotes the initial solution of the problem (7)–(9) for given rðtÞ.
ð12Þ
3828
A. Demir, A. Erdem / Applied Mathematics and Computation 219 (2012) 3826–3830
3. Necessary condition of optimal control problem For the solution of the optimization (11), (12) we derive the necessary condition. Theorem 2. Let rðtÞ be the solution of the optimal control problem (11), (12) and v ðx; tÞ be the solution of (7)–(9) corresponding to this optimal coefficient. Then for any hðtÞ 2 R0 the following integral inequality holds:
Z
l
½v ðx; 0; rÞ gðxÞnðx; 0Þdx þ a
0
Z
T
rðtÞ½hðtÞ rðtÞdt P 0:
ð13Þ
0
Here, nðx; tÞ is the solution of the problem given by
8 > < nt ¼ nxx þ ðhðtÞ rðtÞÞf ðx; tÞ; x 2 ð0; lÞ; t 2 ð0; T nðx; TÞ ¼ 0; x 2 ½0; l > : nx ð0; tÞ ¼ nðl; tÞ ¼ 0; t 2 ½0; T: Proof. For any hðtÞ 2 R0 and d 2 ½0; 1 let us set
rd :¼ ð1 dÞr þ dh: Then we have
J d :¼ Jðrd Þ ¼
1 2
Z
l
½v ðx; 0; r d Þ gðxÞ2 dx þ
0
Z
a 2
0
T
r 2d ðtÞdt:
Differentiation of J d with respect to d at d ¼ 0 gives the following inequality since rðtÞ is the solution of the optimal control problem (11), (12):
Z
l
½v ðx; 0; rÞ gðxÞ
0
dv ðx; 0; rd Þ dx þ a dd
Z
T
rðtÞðhðtÞ rðtÞÞdt P 0:
ð14Þ
0
vd e d ¼ @@d e d is the solution of the Let us denote by v d ¼ v d ðx; t; r d Þ the corresponding solution of problem (7)–(10). Taking v then v following problem:
8 e d;xx þ ðhðtÞ rðtÞÞf ðx; tÞ; x 2 ð0; lÞ; t 2 ð0; T e d;t ¼ v >
ð15Þ
e d jd¼0 then n satisfies the following problem: Let us set n ¼ v
8 > < nt ¼ nxx þ ðhðtÞ rðtÞÞf ðx; tÞ; x 2 ð0; lÞ; t 2 ð0; T nðx; TÞ ¼ 0; x 2 ½0; l > : nx ð0; tÞ ¼ nðl; tÞ ¼ 0; t 2 ½0; T:
ð16Þ
Taking into account (16) in (14) we obtain
Z
l
½v ðx; 0; rÞ gðxÞnðx; 0Þdx þ a
0
Z
T
rðtÞ½hðtÞ rðtÞdt P 0:
0
Lemma 3. Let rðtÞ be the solution of the optimal control problem (11), (12). Then for any hðtÞ 2 R0 the following integral inequality holds:
Z
T 0
Z
l
½hðtÞ rðtÞf ðx; tÞuðx; tÞdxdt þ a 0
Z
T
rðtÞ½hðtÞ rðtÞdt P 0;
ð17Þ
0
where uðx; tÞ is the solution of the adjoint problem given by
8 > < ut uxx ¼ 0; x 2 ð0; lÞ; t 2 ð0; T uðx; 0Þ ¼ v ðx; 0Þ gðxÞ; x 2 ½0; l > : ux ð0; tÞ ¼ uðl; tÞ ¼ 0; t 2 ½0; T: Proof. The unique solution uðx; tÞ of (18) belongs to C 2þa;1þa=2 ð½0; 1 ½0; TÞ. Multiplying each side of equation (18) by nðx; tÞ satisfies (16), integrating it by parts, we obtain
ð18Þ
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A. Demir, A. Erdem / Applied Mathematics and Computation 219 (2012) 3826–3830
0¼
Z
T
Z
0
0
l
nðut uxx Þdxdt ¼
Z
T
Z
0
l
nt udxdt 0
Z
T
Z
0
l
nxx udxdt
Z
0
l
0
nujt¼T t¼0 dx
Z
T
0
nux jx¼l x¼0 dt þ
Z 0
T
nx ujx¼l x¼0 dt
and using the conditions of (16), (18), the above integral identity implies:
Z
Z
T
0
l
½hðtÞ rðtÞf ðx; tÞuðx; tÞdxdt þ
0
Z
l
½v ðx; 0; rÞ gðxÞnðx; 0Þdx ¼ 0
ð19Þ
0
Combining (19) and (13),we obtain the integral inequality (17). h
4. Uniqueness
Theorem 4. Suppose that g 1 ðxÞ and g 2 ðxÞ are two functions which satisfy (7)–(10). Let r 1 ðtÞ and r 2 ðtÞ be the solutions of the optimal control problem (11), (12) then we have the following estimate:
1 kr 1 r 2 kL2 ð0;TÞ 6 pffiffiffiffiffiffi kg 1 g 2 kL2 ð0;lÞ : 2a Proof. Denote by v i ¼ v i ðx; t; ri Þ; i ¼ 1; 2 the solutions of the problem (7)–(10) corresponding to ri ðtÞ. Let n1 ¼ nðx; t; r2 r1 Þ and n2 ¼ nðx; t; r 1 r2 Þ be two solutions of (16). Defining by w ¼ v 1 v 2 and p ¼ n1 þ n2 ; wðx; tÞ and pðx; tÞare the solutions of the following problems, respectively:
8 < wt ¼ wxx þ ðr 1 ðtÞ r 2 ðtÞÞf ðx; tÞ; x 2 ð0; lÞ; t 2 ð0; T wðx; TÞ ¼ 0; x 2 ½0; l : wx ð0; tÞ ¼ wðl; tÞ ¼ 0; t 2 ½0; T;
ð20Þ
8 < pt ¼ pxx ; x 2 ð0; lÞ; t 2 ð0; T pðx; TÞ ¼ 0; x 2 ½0; l : px ð0; tÞ ¼ pðl; tÞ ¼ 0; t 2 ½0; T:
ð21Þ
Note that we have w ¼ n1 and w ¼ n2 so we conclude pðx; tÞ ¼ 0. Now, taking h ¼ r 3i and r ¼ ri ; i ¼ 1; 2 into (13), respectively, there holds
Z
l
½v 1 ðx; 0Þ g 1 ðxÞn1 ðx; 0Þdx þ a
0
Z
l
½v 2 ðx; 0Þ g 2 ðxÞn2 ðx; 0Þdx þ a
0
Z
T
r 1 ðtÞ½r 2 ðtÞ r 1 ðtÞdt P 0;
ð22Þ
r 2 ðtÞ½r 1 ðtÞ r 2 ðtÞdt P 0:
ð23Þ
0
Z
T
0
Combining (22) and (23), we obtain
a
Z
T
½r 1 ðtÞ r 2 ðtÞ2 dt 6
0
¼
Z Z
½v 1 ðx; 0Þ g 1 ðxÞn1 ðx; 0Þdx þ
0
0
þ ¼
l
Z
l
½v 1 ðx; 0Þ g 1 ðxÞn1 ðx; 0Þdx Z
Z Z
l
½v 2 ðx; 0Þ g 2 ðxÞn2 ðx; 0Þdx
0
0
l
½v 2 ðx; 0Þ g 2 ðxÞn1 ðx; 0Þdx þ
l
½v 2 ðx; 0Þ g 2 ðxÞn2 ðx; 0Þdx
0 l
½wðx; 0Þ ðg 1 ðxÞ g 2 ðxÞÞn1 ðx; 0Þdx
0
then let us use wðx; tÞ ¼ n1 ðx; tÞ and Young’s inequality in the above inequality:
a
Z
T 0
Z
l
n21 ðx; 0Þdx þ
1 2
Z
l
1 ðg 1 ðxÞ g 2 ðxÞÞ2 dx þ 2 0 0 Z Z l 1 l 1 ðg ðxÞ g 2 ðxÞÞ2 dx n2 ðx; 0Þdx ¼ 2 0 1 2 0 1 Z 1 l ðg ðxÞ g 2 ðxÞÞ2 dx: 6 2 0 1
½r 1 ðtÞ r 2 ðtÞ2 dt 6
Finally, we obtain
1 kr 1 r 2 kL2 ð0;TÞ 6 pffiffiffiffiffiffi kg 1 g 2 kL2 ð0;lÞ : 2a
Z 0
l
n21 ðx; 0Þdx
Z 0
l
½v 2 ðx; 0Þ g 2 ðxÞn1 ðx; 0Þdx
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A. Demir, A. Erdem / Applied Mathematics and Computation 219 (2012) 3826–3830
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