Journal of Mathematical Analysis and Applications 258, Ž doi: jmaa , available online at http:

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1 Journal of Mathematical Analysis and Applications 58, 35 Ž. doi:.6jmaa..7358, available online at on On the Regularity of an Obstacle Control Problem ongwei Lou Institute of Mathematics, Fudan Uniersity, Shanghai 433, People s Republic of China hwlou@etang.com Submitted by L. D. Beroitz Received November 9, 999 An optimal control problem of the obstacle for an elliptic variational inequality is considered, in which the obstacle is regarded as the control. To get the regularity of the optimal pair, a new related control problem is introduced. By proving the existence of an optimal pair to such a new control problem, the regularity of the optimal pair to the original problem is obtained. It turns out that the regularity obtained is sharp in general. Some other interesting properties of the optimal pair are also established. Acemic Press Key Words: regularity; obstacle; control.. INTRODUCTION In this paper, we consider the following optimal control problem: Problem C. Find a, such that where J inf J Ž., Ž.. 4 J Ž. T Ž. z dx, Ž.. n, is a bounded domain with C boundary, z L is a given target profile, and y TŽ. is the solution of the following varia- This wor is supported in part by the Science Foundation of Education Ministry of China. -47X $35. Copyright by Acemic Press All rights of reproduction in any form reserved. 3

2 REGULARITY OF OBSTACLE CONTROL 33 tional inequality: y Ž., a.e. 4, Ž.3. y Ž y. dx, Ž.. It is well nown that y satisfies Ž.3. if and only if y is the minimizer of the functional dx over the set Ž.. Moreover, if y satisfies Ž.3., then y dx, Ž.,, a.e. 4 Žcf. 5, for example.. In, Adams et al. introduced the above Problem Ž C.. It was shown that there exists a unique optimal pair Ž y,. to Problem Ž C. and y must be Ž equal to. Moreover, if z L or z C for some., then, Ž, y is of class C C for any, or C Ž... The main purpose of this paper is to obtain further regularity and characterization of the optimal pair. More precisely, we show that W, p - regularity of the optimal pair to Problem Ž C. in the case that the target profile zž. is only an L p Ž p. function. When z L or,, z C for some,, the C or C boundary regularity of the optimal state is also obtained. With the aid of the results on regularity and characterization of the optimal pair, some examples are constructed to reveal that such regularity is best possible in general. We also obtain some other interesting properties about the optimal pair and give some examples to show how to calculate the optimal pair by using the results in this paper. Our main idea of getting the regularity of the optimal pair is to establish the existence theorem for a new related optimal control problem. We now explain this in detail. Since the optimal pair y, satisfies y, and for any, we have T Žcf.., thus finding the optimal pair Ž y,. to Problem Ž C. is equivalent to finding the minimizer y such that with J Ž y. inf J Ž y., y J y y z y dx. 4 Now we introduce the following optimal control problem: 4 Problem C. Find a u L L, a.e. such that ul Ž. J u inf J u,

3 34 ONGWEI LOU where 4 J Ž u. y z uy dx, Ž.4. Ž. Ž Ž.. and y y ; u is the solution to the equation y u, ½ y. in, Ž.5. The following result reveals the relation between Problem Ž C. and Problem ŽC.. PROPOSITION.. If y, u is an optimal pair of Problem C, then Ž y, y. is the optimal pair of Problem Ž C.Conersely,. if Ž y, y. is an optimal pair of Problem C and y Ž., then y, and by setting u y L Ž., Ž y, u. is an optimal pair of Problem Ž C.. Proof. For any u L Ž., let y yž; u. be the corresponding solution of.5. We have y and 4 4 J Ž u. y z uy dx y z y dx JŽ y.. On the other hand, for any y Ž., we have y. Thus by defining u y, we see that yž. is the unique solution of Ž.5. corresponding to už.. Then 4 4 J y y z y dx y z uy dx J u. Now, suppose that y, u is an optimal pair of Problem C. By the definition of Ž y, u., y is the minimizer of J Ž y. over Ž.. Since is dense in, y must be the minimizer of J over. ence Ž y, y. is an optimal pair of Problem Ž C.. Similarly, suppose that Ž y, y. is an optimal pair of Problem Ž C. and y. Then we have y Ž.. Therefore, y is the minimizer of J y over. Let u y. Then u L Ž..SoŽ y, u. is an optimal pair of Problem Ž C.. Since the optimal pair of Problem Ž C. uniquely exists, Proposition. tells us that the -regularity of an optimal pair to Problem Ž C. is equivalent to the existence of an optimal pair to Problem ŽC.. In Problem Ž C., the relation between state and control is nonlinear. But in Problem ŽC., the relation between state and control is linear, which maes it much easier to deal with.

4 REGULARITY OF OBSTACLE CONTROL 35 The rest of the paper is organized as follows. In Section, we will prove the existence of optimal pairs to Problem ŽC. and give a characterization of the optimal control. In Section 3, we will use the result obtained in Section to explore some further interesting properties of optimal pairs. Some nontrivial examples will also be presented. For further information on optimal control problems for variational inequalities see,, 57, 93, 68,,, for examples.. EXISTENCE AND CARACTERIZATION OF OPTIMAL PAIR We first introduce some preliminary lemmas. m, p LEMMA.. Let C be a constant. If W Ž., p, m, then 4 Ž x., a.e. x x Ž x. C, m, n where,..., is an n-tuple of nonnegatie integers, Ý. n i i i Proof. If m, then the result is standard Žcf. 9.. m, If m, then W p Ž.,. Thus we can easily get the result by induction. The following lemma is a special case of the so-called strong maximum principle. LEMMA.. Suppose L I,, y satisfies Ly in. Ž. i Suppose for some ball B, we hae sup y sup y. B Then y must be a constant in. ii Suppose y C Ž., y on, and y. Let x be such that satisfies an interior sphere condition at x. Then y Ž x., where is the outward normal on. The proof of Lemma. can be found in 4. Let us now present the following lemma which is related to Problem ŽC..

5 36 ONGWEI LOU LEMMA.3. Suppose z L Ž.. Denote u :, b u is measurable 4, if b, U, b ½ L Ž., if b. Ž. i Suppose b, then there exists a unique u U, b such that J Ž u. inf J Ž u.. uu, b Let y be the solution of Eq. Ž.5. with u replaced by u. Then there exists a such that and z y u, in, Ž.., if Ž x., už x. zž x. yž x., if Ž x., a.e.. Ž.. b, if Ž x., ii Suppose b. If there exists a u U, b such that uu, b J u inf J u, then such a u is unique. Let y be the state corresponding to it. Then there exists a such that Ž.. holds and, a.e., Ž.3., if Ž x., už x. ½ a.e.. Ž.4. zž x. yž x., if Ž x., Proof. Ž. i Suppose b. Let u 4 be a minimizing sequence of J Ž. over L Ž.. We have u b C,, L where is the Lebesgue measure of. Therefore y C,, for some constant C, where y u. So we can suppose that is the state corresponding to the control y y, wealy in, strongly in, u u, wealy in L.

6 REGULARITY OF OBSTACLE CONTROL 37 Thus ½ 5 4 y z uy dx lim y z u y dx inf J Ž u.. uu, b Since U, b is convex, u U, b. On the other hand, it is easy to see that y must be the state corresponding to u. ence, J Ž u. inf J Ž u. u U, b. Suppose that u U, b satisfies J Ž u. J Ž u.. Let y be the state u u corresponding to u. Then U, b, and ž / ½ 5 u u y y y y J dx u u Ž J Ž u. J Ž u.. J Ž u. J. ž / Therefore y y, leing to u u, and we have the uniqueness. Now, let L such that for any Ž,., u u U, b. We have where Ž. ½ ž / ž / 5 J u J u Y y Y z y Y dx, Y, ½ Y. in, Ž.5. Letting, we have Set 4 YŽ y z. y dx. z y u, ½. in, Ž.6.

7 38 ONGWEI LOU We have, and Thus we have Y Ž u. y 4 dx Y Ž y. y 4 dx Y Ž y. y 4 dx dx. Ž.7. 4, a.e. x už x.,, a.e. x už x. b 4, Ž.8., a.e. x už x. b 4. Consequently, and u, a.e. x Ž x. 4, Ž.9. u b, a.e. x Ž x. 4. Ž.. 4 To get the value of u on the set x x, noting that Ž., and by Lemma., we have, a.e. x Ž x. 4. Ž.. Thus z y u, a.e. x Ž x. 4. Ž.. Combining. with.9. we get.. ii Suppose b and there exists a u U, b such that uu, b J u inf J u. Ž. Lie in i, we can get the uniqueness of u and find a which satisfies Ž.. and Ž.7.. Since u U, b, Ž,., L L, a.e. 4, we have Ž.3. by Ž.7.. Similarly, we have Ž.9., Ž.. Ž.., and Ž.4. follows.

8 REGULARITY OF OBSTACLE CONTROL 39 We now would lie to show that if lacs W, p -regularity, then it is quite difficult to determine u on the set E x Ž x. 4. Certainly, if the Lebesgue measure of E were zero, then u would be a bang-bang control Žsee Ž.... In, it was proved that E really has zero measure in some contexts. But for Problem Ž C., as we will see below, the set E usually has positive measure. Therefore, to determine u on E is very important. By introducing an approximation problem, i.e., by restricting the missible control to a good space, we are able to get the, p W -regularity of which will le to a description of u on E by Lemma.. Because we have explicit expressions of optimal controls in approximation problems, the uniform W, p -boundedness of the optimal states can be obtained, which enable us to go further. Let us mae this more precise. LEMMA.4. If z L Ž., then Problem ŽC. mits a unique optimal pair Ž y, u. and y C z L, u z L L. Proof. Let z L. By Lemma.3, denote u to be the minimizer of J Ž. over U,, and y to be the corresponding state. Then we have such that and z y u, in,, if Ž x., u Ž x. zž x. y Ž x., if, a.e.., if Ž x., Since u, so y by the maximum principle for Ž.5.. Thus we have 4 z y z L, a.e. x x., p On the other hand, since u L, we have y W for any p. Thus y L. Therefore z y u L Ž., which, implies C for all Ž,.. Thus x Ž x. 4 is an open subset of. If the Lebesgue measure of is positive, then and by the wea maximum principle Žcf. 4., we have Ž x., a.e.,. This contricts the definition of. Therefore, we have, i.e.,, a.e.. Consequently, u Ž x. Ž z y., a.e.. 4

9 4 ONGWEI LOU L j L This yields u x z, a.e.. Similarly, we have u z, j. Therefore u also minimizes J Ž. over U, j. Then by the uniqueness, we must have u u, j a.e. Ž. This means that u u minimizes J over U, j, j z L. Ž. Consequently, u must minimize J over L Ž.. Thus Lemma.4 follows from Lemma.3. Ž. Now we establish the existence theorem of Problem C, which also gives a characterization of the optimal pair. TEOREM.5. Suppose z L Ž.. Then Problem ŽC. Žand so does. Problem C mits a unique optimal state y Ž., and there exists a such that Ž x., a.e., Ž.3. y Ž z y. 4, Ž.4. Ž z y. 4. Ž.5. Ž. Ž. Moreoer, the pair y, satisfying Ž.3. Ž.5. is unique. Proof. Let z Ž x., if z Ž x., zž x. ½, if zž x., and y, u be an optimal pair to Problem C with z replaced by z.by Lemma.4 we have y C z C z L L, u z z L L L. By choosing subsequences, if necessary, we have Then y y, wealy in, strongly in, u u, wealy in L. y u, in.

10 REGULARITY OF OBSTACLE CONTROL 4 Since z z, we have L Then it is easy to see J Ž u. J Ž u., u L Ž.. J Ž u. J Ž u., u L Ž.. So we have the existence, and Ž.3. Ž.5. follows from Lemma.3. Ž Suppose y, u Ž.. Ž Ž. Ž.. satisfies Ž.3. Ž.5.. We have ½ 5 y y y y dx Ž y y. y y y y dx y y dx Ž y y.ž. dx y y dx. ence, y y. Consequently,, proving the uniqueness. 3. FURTER PROPERTIES OF TE OPTIMAL STATE In this section, we let y be an optimal state of Problem C corre- sponding to z L, and be a function satisfying Ž.3. Ž.5. as in Theorem.5. We see that y is the optimal state to Problem C. For any measurable function f in, f and f will be denoted the positive part and the negative part of f, respectively, i.e., f max f, and f maxž f,.., It is proved in that if z L, then y C C for any,. Moreover, if z C for some Ž,., then y,, W C Ž.. We strengthen these results in the following theorem. loc TEOREM 3.. Suppose z L. Ž. p., p i If z L for some p,, then y W and, p y W C z L p.

11 4 ONGWEI LOU p, n p Consequently, if z L for some p n,, then y C Ž.., In particular, if z L, then y C for any Ž,.. In the, one-dimensional case, if z L, then y C Ž.., ii If z C, and is of class C for some Ž,., then,, y W C Ž.. Proof. Ž. i This follows from Theorem.5 immediately. ii For L I,, we denote L z to be the solution of the equation L z, ½. in,, Let z y, z. We have y by Theorem.5. Since z C Ž.,so y C by Ž i.. We have z, z y, C. Thus,, C Ž.. By Theorem.5, we now that, a.e., and y z, a.e.. So y. Then it is easy to see that y is the solution of the following bilateral-obstacle problem: Ž. 4 y,, a.e., y Ž y. dx, Ž,..,, By the result of 8, we have y W C Ž.., In some sense, the C -regularity of the optimal state y is the best possible result which we can obtain. Even if z is analytic on, y may have no C -regularity. To illustrate this, we present an example. EXAMPLE. Consider,, z x x. Let y, u be the optimal pair of the corresponding Problem Ž C., and be the function as in, Theorem.5. Then y C, by Theorem 3.. Consequently,, C,. Noting that z is even, y,, u must also be even by the uniqueness. Therefore y Ž.. Since y, we can see that y is nondecreasing on, and nonincreasing on,. By Theorem 3. Ž ii. which will be established below, we have y. So yž. sup yž x. x,. Thus there exists a unique a Ž,. such that and z y, in Ž, a. Ž a,., z y, in Ž a, a.. Ž. 4 By Theorem 3. i, we have a, a x u x.

12 REGULARITY OF OBSTACLE CONTROL 43 Moreover, we have Ž a. Ž a.. Otherwise, suppose Ž a., Ž a.. Since and C,, we have Ž a. Ž a.. On the other hand, z y in the set Ž a, a.. By Lemma., we have ona, a. This contricts z y inž a, a.. Similarly, we have Ž x., x a, a. Thus, there exists a b Ža, such that and Therefore Ž b. Ž b., Ž x., x Ž b, b.. y Ž x., a.e. Ž b, b.. Since y, b a,. By.5, we have zž x. yž x. zž b. yž b., a.e. Ž b, Noting that b, we have in b,. Thus by.3, So, in Ž b,.. y Ž x. zž x. yž x. zž b. yž b., x Ž b,.. Since y Ž x. for x Ž, b., we find that y is not continuous at b. In fact, y b does not exist. So y is not a C function. Now, we will go further to get the precise expression of the optimal state y. We have seen that Ž x. when x b, and Ž x. when b x. So we have ½ Ž y y.. b x 4 y x y, a.e.,,, Since y C,, we can solve the equation and get Ce x Ce x x, x b, y C3 b x b, Ce x C e x x, b x,

13 44 ONGWEI LOU where 3e b be b C, b 3 e be b 3e C, b e e be b 6e b be C3 b. b e e The only thing we need to do now is to determine b. We have So x C, b x b. 3, x b, ½ C x x 4 C C x, x b., Noting that C,, we have b b. Therefore So C Cb 3 3b C5Cb 3 3b C5. b. ence b b e 3e b b. Ž b. b 3 3 Then we get y. Numerical calculations give the following results: b , C , C , C Next, we will give some further interesting results about y below. Then we will construct another example to show that z C is not sufficient, to obtain the C -regularity of y when the space dimension n. TEOREM 3.. Suppose z L. Ž. 4 i There exists a measurable set E x z x y x such that y E Ž z y. E Ž z y.,

14 REGULARITY OF OBSTACLE CONTROL 45 where E is the characteristic function of E. Ž ii. y z. Ž iii. y z y Ž I. z z. Ž iv. If x zž x. 4 has a positie measure, then y z y y z y. Ž. Proof. i Noting that y and y, we get the result from Theorem.5. ii Let y. Then z y u z, and so z. Let z. We have 4 J y J y y z y dx 4 y yz z y y dx 4 y yz z y z y dx 4 yz z y dx 4 yz z y dx 4 yz z dx 4 y z z dx z dx. Ž. Ž. Since J z dx, we have J J y. ence y. Ž iii. Let y z y. Then y Ž I. z, and since y, we have Ž I. z y z y z. Let z Ž I. z. We denote Z Ž I. z. It is easy Ž. to see that Z minimizes J over Ž.. Since Z z Z, there- fore Z and Z minimizes J over. ence y Z, or equivalently, y z y.

15 46 ONGWEI LOU Ž iv. Let y Ž z y.. Since y z y, we have z y. We want to prove z y, a.e., or equivalently,, a.e.. Otherwise, by Lemma. Ž. i,we have, in. This yields that y. So y and z y y, contricting our assumption. Thus we must have, a.e., and so y z y. Let y z y. Since y, so z y and we have z y z y. Therefore y z y. It is proved in that if is a wea solution of the equation Ž z., in, ½, and such that z, a.e., then y. Since these conditions are equivalent to Ž I. z z, we see that this result is equivalent to the result Ž I. z z y z y. Ž. Ž By Theorem 3. i, y on supp z.. It seems that y depends only on z. In fact y is not determined by z. For example, let be the unit ball B Ž., zž x. x M. Then M 4 z x x x. 4Ž n. n Let M nž n.; we have z. Thus, corresponding to z, the optimal state yž ; z.. On the other hand, by Lemma.3, we can easily see that z Ž x. in. Consequently, by Theorem 3. Ž ii., Ž. the optimal state y ; z corresponding to z is not. As is shown in, in two simple cases, we can get y easily. Case. If z, then y. Case. If z and z, then we have y I z. To see this, compare with z z z, ½ z, Z Z z, ½ Z, in, in, and we have Ž I. z z by the wea maximum principle Žcf. 4.. Therefore by Theorem 3. Ž iii., we have y Ž I. z.

16 REGULARITY OF OBSTACLE CONTROL 47 Moreover, we have the following corollary. COROLLARY 3.3. There exists a Ž,. such that if inf zž x. sup zž x., x then y Ž I. z. In particular, when B r, the ball of rius r in n Ž r centered at the origin, we can choose n n e.. Proof. Let Z Ž I., i.e., Z and Z Z, in. We have Z C Ž.. By the wea maximum principle, Z, in. Denote sup Z. It is easy to see that. We want to prove. Let ZŽ x.. Then x, and we have ZŽ x.. So Thus x ZŽ x. ZŽ x.. ZŽ x. ZŽ x. ZŽ x., x. We claim that ZŽ x.. Otherwise, suppose that ZŽ x.. Then denote W Z; we have ½ W W, W, in, and sup W WŽ x.. So sup W. By Lemma. Ž i., we see that W must be constant in. This is a contriction. Consequently,. Now, suppose Ž,., inf zž x. sup zž x. x x. Then z, in, and we have ž / I z x sup z I x sup z inf z z x. So we have y I z by Theorem 3. iii. When B, we have ZŽ x. hžx., where h C, r and satisfies r n h Ž s. h Ž s. hž s., in Ž, r, s hž r., h Ž..

17 48 ONGWEI LOU Moreover, we have with ž M 4 hž s. s s M M n nž n. 4 / s 6 nž n.ž n M r r r n nž n. 4 nž n.ž n So ž / ž / ½ 5 3 r r r r e. n! 3! n M n max hž s. hž., M r s, r n e Ž r. and we can choose n n e. As an application, we give another example to calculate y. EXAMPLE. Let n, Ž,., z 4 x sinž x., where is a given integer. Then we see that ž / ž / inf zž x. 3 5 sup zž x e e x, xž,. Therefore ence, we get y y 4 x sinž x.. 4 yž x. cosž x. sinž x. 4 Ž 4. 4 ž / e 4 e x e x 4. Ž 4. Now let us give another example to show that in high dimension,, z C is not sufficient for the C -regularity of y.

18 REGULARITY OF OBSTACLE CONTROL 49 EXAMPLE 3. For simplicity, we suppose n. Let B Ž.. Choose Ž. C such that Ž x. for x 4 8. Let x Ž x, x., f x f x, x x, x x x ln ln x x.,, Then it is easy to chec that f C and f W p for any. p,. On the other hand, f is C smooth except at the origin. Near the origin, for x Ž x, x. B Ž., we have 8 x x xx x x ln x x f x, x ln ln x x Ž. Ž. Ž. Ž. 4 x x x 4 x x x, x x ln x x x x ln x x x x xx x x ln x x f x, x ln ln x x Ž. Ž. Ž. Ž. 4 x x x 4 x x x, x x ln x x x x ln x x Ž. Ž. Ž. Ž. 8 x x 4 x x fž x, x.. x x ln x x x x ln x x So fcž.. Now let z M with M a large number such that M inf Ž r. ž M sup Ž x./. 3 x e ž / By Corollary 3.3, we have y z y M y. Therefore y, M y f. Since y C for any Ž,., we have,,, M y C. Since f C, y C Ž.. In the proof of Theorem 3. Ž ii., we have seen that if y is the optimal state of Problem Ž C., then it is a solution of variational inequality Ž.3. for some. The following theorem shows the converse is also true in some sense., p TEOREM 3.4. Suppose W Ž., p, y TŽ. is the solution of ariational inequality Ž.3.. Then y is an optimal state of Problem Ž C. with z y. In particular, we hae y, a.e.. y 4 x

19 5 ONGWEI LOU, Proof. If we use the fact y W p Žcf 3, 4, 5., we can get the result easily by Lemma. and Theorem.5. But we would lie to give a, proof here without using the fact y W p Ž.., Now, let z y. Since y and W p Ž., Ž we have z L. By Theorem.5, there exists a y,. Ž. satisfying Ž.5.. Thus we need only prove y y. By the definition of y, we have y Ž y. dx, Ž.. Ž 3.. Modifying the proof of Theorem.5 and noting that z y, z y y, and, we have ence y y. 4 y y y y dx 4 y y y y y y dx 4 y y y y y y dx 4 y y y y y y y y dx 4 y y y y y dx 4 y y y y y y dx 4 y y y y y dx Ž y. y y Ž y. 4 dx. ACKNOWLEDGMENTS This paper is written under the guidance of Professor Jiongmin Yong. Many valuable suggestions were offered by Professor Xunjing Li. The author thans both of them for their help.

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