Optimal Boundary Control of a Nonlinear Di usion Equation y

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1 AppliedMathematics E-Notes, (00), c Availablefreeatmirrorsites ofhttp://math.math.nthu.edu.tw/»amen/ Optimal Boundary Control of a Nonlinear Di usion Equation y Jing-xueYin z,wen-meihuang x Received6 April 00 Abstract We discuss the optimal boundary control governed bya nonlinear di usion equation, andestablishthe existence andstabilityof the optimal control. In this paper, we are concerned with the optimal boundary control governed by the following nonlinear heat conduction equation subject to the initial value condition and the boundary + div ~ J + u = 0; (x; t) = (0; ); () u(x; 0) = u 0 (x); x ; () ~J ~n = h(u ); (x; (0; ); (3) where J ~ = jruj p ru is the heat ux, p >, ½ R N is a bounded domain with smooth boundary, ~n denotes the outward normal to the is a positive constant, u 0 (x) is a nonnegative bounded function and h is the heat transfer coe± cient which we take as our control. he cost functional is chosen as ( J(h) = ) (u d ) dxdt + h dsdt ; h U M ; (0;) where U M is the admissible set, namely, U M = fhj 0 h M; h L (@ (0; )); h 0 n g : Here is a partial boundary of with mes > 0, d is the desired temperature distribution, the coe± cient and are per unit costs associated with failing to achieve the MathematicsSubjectClassi cations: 35K99 y SupportedbytheMinistryofScienceandechnologyofChina z DepartmentofMathematics,JilinUniversity,Changchu,Jilin300,P.R.China x DepartmentofAppliedMathematics,BeijingInstituteofechnology,Beijing0008,P.R.China 97

2 98 Nonlinear Di usion Equation desired temperaturedistribution and with imposinga heat transfer coe± cient di erent from zero. According to di erent requests, and can take di erent values. hen the optimal control problem of the temperature system is o nd a h U M ; s:t: J(h ) = inf hu M J(h): (5) hus, the state equation () with the initial and boundary value condition (), (3), together with the cost functional (4), and question (5) compose a mathematical model of the optimal boundary control of the heat transfer system. If u >, the system releases heat, while if u <, the system absorbs heat. It was Lenhart and Wilson [] who rst studied the optimal control for such kind of system with p =, established the existence, uniqueness and stability of the optimal control, and proved that the optimal control can be formulated by h = qu, where q is a nonnegative function independent of u. Later on, similar results were obtained by several authors, see for example [5], [6] and [7]. It is well known that for the classical heat conduction equation, i.e., the case where p =, the speed of propagation is in nite. However, for the case where p >, the state equation becomes the p- Laplace equation, whose solutions possess the property of nite speed of propagation of disturbances. Hence, it is more natural to consider the heat transfer system governed by the p-laplace equation. Due to the degeneracy of our equations, we are only interested in weak solutions to our problem (){(3) in the following sense: A nonnegative function u C(0; ; L ( )) L p (0; ;W ;p ( )) is said to be a weak solution of the problem (){(3) if the following integral equality holds u(x; )'(x; )dx + jruj p ru r'dxdt + h(s; t)(u(s; t) )'(s; t)dsdt + u(x; t)'(x; (0;) u 0 (x)'(x; 0)dx u(x; t)' t (x; t)dxdt = 0; (6) where ' is an arbitrary test function in C ( )), (0; ) and = (0; ). he main results of this paper are as follows. HEOREM. Assume that d L ( ), u 0 C ( ) and satis es the following compatibility condition jru 0 j p ru 0 ~n = hu 0 ; : hen there exists an optimal control h U M which minimizes the cost functional J(h) de ned by (4). As for the stability of the optimal control h, we have HEOREM. Suppose u = u(h) and u " = u(h + "l) are solutions of problem (){(3), corresponding to h U M, h + "l U M respectively. hen ku " uk L ( ) = O("); "! 0:

3 J. X. Yin and W. M. Huang 99 We need several lemmas which will be used in the proof of our main results. LEMMA. For any u, v L p (0; ; W ;p ( )), the following inequality holds: (jruj p ru jrvj p rv) r(u v)dxdt 0: Indeed, the above inequality follows easily from the convexity of (X) = jxj p. LEMMA (Feng [3]). Let B 0, B and B be re exive Banach spaces which satisfy C B 0,! B,! B, where,! denotes imbedding and,! C denotes compact imbedding. hen we have L r0 (0; ; B 0 ) \ fájá t L r (0; ; B )g C,! L r0 (0; ; B); L (0; ; B 0 ) \ fájá t L r (0; ; B )g C,! C(0; ; B); L r0 (0; ; B) \ fájá t L r0 (0; ; B)g C,! C(0; ; B): Here r 0 ; r, and < r. LEMMA 3. Under the assumption in heorem, there exists a unique solution u h of the problem (){(3) for any h U M. PROOF. For the existence of solution u h, we refer to []. he uniqueness can also be proved in a rather standard way as follows. Let u, u be solutions of the problem (){(3). From the de nition of a solution of ()-(3), we have (jru j p ru jru j p ru )r'dxdt (u u )' t dxdt + h(u u )'dsdt + (u u (0;) = (u (x; ) u (x; ))'(x; )dx; for all (0; ): Choosing ' = u u, we obtain = for all (0; ): Noticing that (jru j p ru jru j p ru ) r(u u )dxdt + h(u u ) dsdt + (u u ) (0;) (u u )(u u ) t dxdt (u (x; ) u (x; )) dx; (jru j p ru jru j p ru ) r(u u ) 0;

4 00 Nonlinear Di usion Equation we (0;) h(u u ) dsdt + (u u ) dxdt (u u ) (x; )dx 0; which implies that u (x; t) = u (x; t), a.e. (x; t). We are now in a position to present and prove our main results. First consider heorem. Without loss of generality, we assume that = 0. Let fh n g be a sequence in U M for which lim J(h n) = inf J(h): n! hu M By Lemma 3, for each n, we can de ne u n = u(h n ) as the solution of the problem (){(3) with h = h n, namely, u n satis es the following integral equality u n (x; )'(x; )dx + jru n j p ru n r'dxdt + h n (s; t)u n (s; t)'(s; t)dsdt + u n (x; t)'(x; (0;) u 0 (x)'(x; 0)dx u n (x; t)' t (x; t)dxdt = 0 (7) Using the regularity results in [4] for the p-laplace equation, we see that u nt L ( ) and satis es the following estimate ju nt (x; t)j dxdt C; (0; ); (8) where C is a positive constant independent of n. By virtue of this and the de nition of weak solutions, after an approximation process, we may always choose u n, or à u n for some smooth function Ã, as a test function in (7). First, take u n as the test function in (7) and obtain fu n(x; ) u 0(x)gdx + jru n j p dxdt + h n u ndxdt + u ndxdt = 0: (9) Noticing that the last three terms in (8) are nonnegative, we have u n(x; )dx u 0(x)dx; (0; ); (0) jru n (x;t)j p dxdt u 0(x)dx; (0; ): ()

5 J. X. Yin and W. M. Huang 0 From Lemma, there exists a subsequence of fu n g, denoted also by fu n g, u C(0; ; L ( )) \ L p (0;;W ;p ( )) and w L p=(p ) ( ), which satisfy u n! u a.e. ; u nt * u t in L ( ); ru n * ru in L p ( ); and jru n j p ru n * w in L p=(p ) ( ): We claim that w = jru j p ru. Indeed, in view of u ' t dxdt w i ' xi dxdt u 'dxdt = 0; () for ' C0 ( ); we need only to show the following jru j p ru r'dxdt = w i ' xi dxdt; ' C0 ( ): (3) Actually, for any v L p (0; ; W ;p ( )) \ C(0; ; L ( )), à C 0 ( ), 0 Ã, suppã ½, we have à (jru n j p ru n jrvj p rv) r(u n v)dxdt 0: (4) Choosing ' = à u n in (7), we obtain à u ndxdt + à jru n j p dxdt = à t u n dxdt u n jru n j p ru n rã dxdt: It follows from (4) that à t u ndxdt u n jru n j p ru n rã dxdt à jru n j p ru n rvdxdt à u ndxdt à jrvj p rv r(u n v)dxdt 0 (5) Letting n! in (5), we get à t u dxdt u w i à xi dxdt à w i v xi dxdt à u dxdt à jrvj p rv r(u v)dxdt 0 (6) ake à u as a test function in () to obtain u à t dxdt w i à xi u dxdt w i à u x i dxdt u à dxdt = 0 (7)

6 0 Nonlinear Di usion Equation Using (6), we have à (w i jrvj p v xi )(u x i v xi )dxdt 0: (8) Choosing v = u µ' in (8), where µ 0, ' C 0 (), we get Letting µ! 0, we have à (w i jr(u µ')j p (u µ') xi )' xi dxdt 0: à (w i jru j p u x i )' xi dxdt 0; ' C 0 ( ): Obviously, if we let µ 0, we can get another inequality which has reverse direction. herefore, we can choose a function Ã, with supp' ½ suppã, and à = on supp', such that (3) is true, which implies w = jru j p ru. By h n * h in L ); and the continuity of the mapping from H ( ) to L (@ ), we have u n! u in L ((0; ); L (@ )); and let n! in (7), we see that u is a weak solution of the problem (){(3) with h as the heat transfer coe± cient. At last, by the lower semicontinuity of the cost functional and using the weak convergencies derived above, we see that h is an optimal control. he proof of heorem is complete. We now turn to the proof of heorem. From the de nition of a solution to our problem (){(3), we see that u " and u satisfy the integral equality (7). Choosing ' = u " u in (7), we have (u " (x; ) u(x; )) dx + jru " j p ru " r(u " u)dxdt jruj p ru r(u " u)dxdt + (u " u) dxdt + h(u " u) dsdt + "lu " (u " u)dsdt = 0: Using we get (jru " j p ru " jruj p ru) r(u " u)dxdt 0; (u " u) dxdt "l he proof is complete. u " (u u " )dsdt C":

7 J. X. Yin and W. M. Huang 03 References [] S. Lenhart and D. G. Wilson, Optimal control of a heat transfer problem with convective boundary condition, J. Optimization heory and Appl., 79(3)(993), 58{597. [] J. Filo and J. Kacur, Local existence of general nonlinear parabolic systems, Nonlinear Anal. MA, 4()(995), 597{68. [3] X. B. Feng, Strong solutions to a nonlinear parabolic systems modeling compressible miscible displacement in porous media, NonlinearAnal. MA, 3()(994), 55{ 53. [4] E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, Inc [5] N. Aradaand J. P. Raymond, Optimalityconditions for state-constrained Dirichlet boundary control problem, J. Optim. heo. and Appl., 0()(999), 5{68. [6] J. E. Rubio, he global control of nonlinear di usion equations, Siam J. Contr. and Optim., 33()(995), 308{33. [7] A. Friedman and B. Hu, Optimal control of chemical vapor deposition reactor, J. Optim. heo. and Appl., 97(3)(998), 63{644.