On the regularity of the solutions of Dirichlet optimal control problems in polygonal domains
aa r X i v : . [ m a t h . NA ] M a y On the regularity of the solutions ofDirichlet optimal control problems inpolygonal domains ∗ Thomas Apel † Mariano Mateos ‡ Johannes Pfefferer § Arnd R¨osch ¶ October 3, 2018
Abstract
A linear quadratic Dirichlet control problem posed on a possibly non-convexpolygonal domain is analyzed. Detailed regularity results are provided in classicalSobolev (Slobodetski˘ı) spaces. In particular, it is proved that in the presence of controlconstraints, the optimal control is continuous despite the non-convexity of the domain.
Key Words optimal control, boundary control, Dirichlet control, non-convex polyg-onal domain
AMS subject classification
The investigation of optimal control problems with partial differential equations has beenof increasing interest in the last decades. In this paper we will study the control problem(P) min J ( u ) = 12 Z Ω ( Su ( x ) − y Ω ( x )) dx + ν Z Γ u ( x ) dσ ( x )subject to ( Su, u ) ∈ H / (Ω) × L (Γ) ,u ∈ U ad = { u ∈ L (Γ) : a ≤ u ( x ) ≤ b for a.a. x ∈ Γ } , ∗ The second author was partially supported by the Spanish Ministerio de Ciencia e Innovaci´on underprojects MTM2011-22711. † Universit¨at der Bundeswehr M¨unchen, 85577 Neubiberg, Germany, ([email protected]) ‡ Departamento de Matem´aticas, E.P.I. de Gij´on, Universidad de Oviedo, Campus de Gij´on, 33203Gij´on, Spain ([email protected]). § Universit¨at der Bundeswehr M¨unchen, 85577 Neuibiberg, Germany, (Johannes.Pfeff[email protected]) ¶ Universt¨at Duisburg-Essen, Fakult¨at f¨ur Mathematik, Thea-Leymann-Straße 9, D-45127 Essen, Ger-many ([email protected]) IRICHLET CONTROL IN POLYGONAL DOMAINS where Su is the solution y of the state equation − ∆ y = 0 in Ω , y = u on Γ , (1.1)the domain Ω ⊂ R is bounded and polygonal, Γ is its boundary, a < b and ν > y Ω is a function whose precise regularity will be stated when necessary.Note that, for u ∈ U ad , the state equation does not possess a variational solution ingeneral, so a very weak solution is considered (see Theorem 4). We will discuss here theregularities of the optimal state ¯ y , the optimal control ¯ u and the corresponding adjointstate ¯ ϕ which are limited by singularities due to corners of the domain and due to thepresence of control constraints.The classical Sobolev (Slobodetski˘ı) spaces are denoted by W t,p (Ω) and, in the caseof p = 2, by H t (Ω). As usual, for t > W t,p (Ω) or H t (Ω) will denote the closurerespectively in W t,p (Ω) or H t (Ω) of D (Ω), the space of infinitely differentiable functionswith compact support in Ω, and W − t,q (Ω) with q − + p − = 1 [resp. H − t (Ω)] is the dualspace of W t,p (Ω) [resp. H t (Ω)].The seminal paper on Dirichlet control problems is the work by Casas and Raymond[6]. They investigate the problem even with a semilinear state equation. Assuming a con-vex polygonal domain with maximal interior angle ω < π , they prove ¯ u ∈ W − /p,p (Γ)and ¯ y ∈ W ,p (Ω) with p < p Ω = 2 / (2 − min { λ , } ) and λ = π/ω both for the case withand without control constraints. Note that p Ω > u ∈ H − /p (Γ) and¯ y ∈ H / − /p (Ω) for p < p Ω . Deckelnick, G¨unter and Hinze [9] focus on approximationissues in the case of smooth domains (class C ) in two and three space dimensions. Theregularity is determined by the box constraints since corner (or edge) singularities donot occur. Finally we would like to mention the paper [15] by Of, Phan, and Steinbachwhere the control is searched in H / (Ω) such that the state ¯ y ∈ H (Ω) satisfies theweak formulation; in this case the regularity issues are less severe.Due to the assumptions on the domain, the publications [6], [9], and [14] have in com-mon that the adjoint problem can be solved in H (Ω) ∩ H (Ω). For that reason the veryweak formulation of the state equation is well defined in these publications with testfunctions from H (Ω) ∩ H (Ω). This is not the case when non-convex domains are con-sidered. Instead, the very weak fomulation of the state equation should be defined withtest functions from H (Ω) ∩ H (Ω), where H (Ω) := { v ∈ H (Ω) : ∆ v ∈ L (Ω) } , seethe paper by Casas, Mateos and Raymond [5, (A.16)] and also [1] for further approacheshow to understand the solution of the Poisson equation with non-smooth boundary data.In the paper at hand we prove basic regularity results for the solution of the state andadjoint state equations in Section 2. We extend in Theorem 4 to non-convex domains thewell-known H / (Ω) regularity of the solution of the state equation (1.1) and we provein Theorem 7 that the maximum principle also holds for very weak solutions. Our mainregularity results for the variables in the optimal control problem are proved in Section3. Among many other results, we prove that, in the presence of control constraints and2 IRICHLET CONTROL IN POLYGONAL DOMAINS under minimal regularity assumptions on the data ( y Ω ∈ L s ∗ (Ω), s ∗ > s ∗ = 2. This case has notbeen treated by the cited references for convex domains. In Section 4 we prove that, forregular data, the regularity of the optimal solution is indeed slightly better. We giveconditions for the control to be in H / − ε (Ω) for all ε > − ∆ y = f in Ω , y = u + g on Γ , (1.2)but we can take the unique function y which solves − ∆ y = f in Ω, y = g on Γ, andreplace y := y + y , y Ω := y Ω − y and recover problem (P) with (1.1) for data sufficientlysmooth. Let us denote by M the number of sides of Γ and { x j } Mj =1 its vertexes, ordered coun-terclockwise. For convenience denote also x = x M and x M +1 = x . We will denoteby Γ j the side of Γ connecting x j and x j +1 , and by ω j ∈ (0 , π ) the angle interior to Ωat x j , i.e., the angle defined by Γ j and Γ j − , measured counterclockwise. Notice thatΓ = Γ M . We will use ( r j , θ j ) as local polar coordinates at x j , with r j = | x − x j | and θ j the angle defined by Γ j and the segment [ x j , x ]. In order to describe the regularity ofthe functions near the corners, we will introduce for every j = 1 , . . . , M the infinite cone K j = { x ∈ R : 0 < r j , < θ j < ω j } and a positive number R j such that the sets N j = { x ∈ R : 0 < r j < R j , < θ j < ω j } , satisfy N j ⊂ Ω for all j and N i ∩ N j = ∅ if i = j .3 IRICHLET CONTROL IN POLYGONAL DOMAINS
For every j = 1 , . . . , M we will also consider ξ j : R → [0 , { x ∈ R : r j
The condition in (2.2) is necessary to deduce the W ,p (Ω)-regularity of the regular partof the solution in the lemma below. The meaning of the sets J mp is the following: ξ j r mλ j j W ,p (Ω) for all j ∈ J mp . Notice that, for all p < + ∞ , J mp = ∅ for m ≥ λ j > / − /p < p < + ∞ . We also remark that J p ⊂ J p ⊂ J p . Lemma 1.
Consider < p < + ∞ satisfying (2.2) and f ∈ L p (Ω) . Then there existunique real numbers ( c j,m ) j ∈ J mp and a unique solution z f ∈ H (Ω) of problem (2.1) suchthat z f = z reg + X m =1 X j ∈ J mp c j,m ξ j r mλ j j sin( mλ j θ j ) where z reg ∈ W ,p (Ω) and the ξ j are the cut-off functions introduced above.Proof. The result is a direct consequence of [10, Theorem 4.4.3.7]. This result can beapplied since z f ∈ H (Ω) thanks to [10, Lemma 4.4.3.1]. Corollary 2. If f ∈ L (Ω) , then z f ∈ H t (Ω) for all t < λ and ∂ n z f ∈ L (Γ) .Proof. To prove this fact, we apply Lemma 1: the regular part is in H (Ω); on the otherhand, since λ j ≥ λ > / m ≥
1, then ξ j r mλ j j ∈ H t (Ω) for all t < λ . From thiswe obtain that z f ∈ H t (Ω). Since we can choose t > / ∂ n z f ∈ L (Γ).The next result states the regularity of the solution of problems with boundary datain W − /p ∗ ,p ∗ (Γ) with p ∗ ≥
2. Let us recall (cf. [10, Theorem 1.5.2.3]) that the traceof any function z ∈ W ,p ∗ (Ω) is in W − /p ∗ ,p ∗ (Γ), the trace mapping is onto, and, for p ∗ >
2, this space can be characterized as W − /p ∗ ,p ∗ (Γ) = ( g ∈ M Y i =1 W − /p ∗ ,p ∗ (Γ i ) : g is continuous at every corner x j ) . For p ∗ = 2 the continuity requirement in the corners can be weakened to an integralcondition, see [10, Theorem 1.5.2.3(c)]. Lemma 3.
Let g ∈ W − /p ∗ ,p ∗ (Γ) for some p ∗ ≥ . Then there exists a unique solution z ∈ W ,p (Ω) , for all p ≤ p ∗ , p < p D , of the equation − ∆ z = 0 in Ω , z = g on Γ . Proof.
Due to the trace theorem, there exists a function G ∈ W ,p ∗ (Ω) such that itstrace is γG = g . Moreover, we have that ∆ G ∈ W − ,p ∗ (Ω). If we define ζ = z − G , thenit satisfies the boundary value problem − ∆ ζ = − ∆ G in Ω , ζ = 0 on Γ . IRICHLET CONTROL IN POLYGONAL DOMAINS
Let p ≤ p ∗ , p < p D . Then ζ ∈ W ,p (Ω) (see above or cf. Dauge [8, Theorem 1.1(i)]; seealso Jerison and Kenig [12, Thms. 0.5, 1.1, 1.3]) and hence so does z ∈ W ,p (Ω).Since the space of controls is L (Γ), the state equation must be understood in thetransposition sense. Following [2, 5], for u ∈ L (Γ), we will say that y ∈ L (Ω) is asolution of (1.1) if for every f ∈ L (Ω) Z Ω yf dx = − Z Γ u∂ n z f dσ ( x ) , (2.4)where z f is defined in Lemma 1. The definition makes sense thanks to Corollary 2.Existence, uniqueness and regularity of the solution in y ∈ H / (Ω) if Ω is convexdomains is a classical result and can be proved via transposition and interpolation. Letus briefly recall how this result is obtained for a convex domain. Consider the solutionoperator S of (1.1) with Su = y . Due to the Lemma 3 it is S ∈ L ( H / (Γ) , H (Ω)) . (2.5)Using the classical transposition method (cf. [13]) we also have that S ∈ L ( H − / (Γ) , L (Ω)) . (2.6)The final result is obtained by interpolation using that L (Γ) = [ H / (Γ) , H − / (Γ)] / and H / (Ω) = [ H (Ω) , L (Ω)] / , see e.g. [13, Chap. 1, Eq. (2.41)]) for the first result and notice that interpolationresults are valid for spaces posed on Lipschitz domains (cf. [3, Theorem 12.2.7] or [12]).We cannot use this scheme straightforward because (2.6) uses explicitly that for every f ∈ L (Ω), z f ∈ H (Ω), and this is not true for non-convex polygonal domains.For problems posed on non-convex polygonal domains, Berggren [2, Theorem 4.2]proves existence, uniqueness and regularity of the solution y ∈ H t (Ω) for every 0 < t <ǫ ≤ / ǫ depends on the domain). We can also achieve y ∈ H / (Ω) in non-convexdomains. The proof of our following result uses interpolation spaces; a different proofby using integral operators is given in [1]. Theorem 4.
For every u ∈ L (Γ) there exists a unique solution y ∈ H / (Ω) of (1.1) and k y k H / (Ω) ≤ C k u k L (Γ) . Proof.
Notice that for 0 < ε < /
2, we also have that, for θ = 1 / (1 + 2 ε ), L (Γ) = [ H / (Γ) , H − ε (Γ)] θ and H / (Ω) = [ H (Ω) , H / − ε (Ω)] θ and therefore the result will be true if we can prove that S ∈ L ( H − ε (Γ) , H / − ε (Ω)) forsome ε >
0. 6
IRICHLET CONTROL IN POLYGONAL DOMAINS
Fix 0 < ε < min { λ − / , / } . For any u ∈ H − ε (Γ), we will say that y = Su if forevery f ∈ L (Ω) Z Ω yf dx = −h u, ∂ n z f i H − ε (Γ) ,H ε (Γ) . (2.7)Notice that since ε < λ − / z f ∈ H / ε (Ω) and that ∂ n z f ∈ Q Mj =1 H ε (Γ j ) = H ε (Γ) because ε < / k ∂ n z f k H ε (Γ) ≤ C k f k H ε − / (Ω) . (2.8)Notice also that if u ∈ H / (Γ) then the unique variational solution y ∈ H (Ω) of − ∆ y = 0 in Ω , y = u on Γis a solution of (2.7) and if u ∈ L (Γ), then (2.7) is the same as (2.4).Let us prove uniqueness of the solution of (2.7) in L (Ω) first. If u = 0 and y = Su ∈ L (Ω), then, taking f = y as test function in (2.7) we get R Ω y dx = 0, and therefore y ≡
0. Since the problem is linear, the solution is unique.We next prove existence of a solution y ∈ H / − ε (Ω) of (2.7). We know (cf. [10,Theorem 1.4.2.4]) that H / − ε (Ω) = H / − ε (Ω) and hence (cid:0) H / − ε (Ω) (cid:1) ′ = H ε − / (Ω).Denote F = { f ∈ L (Ω) such that k f k H ε − / (Ω) = 1 } . For any u ∈ H / (Γ), y = Su , we use that L (Ω) is dense in H ε − / (Ω) and (2.8) toobtain k y k H / − ε (Ω) = sup f ∈F h f, y i H ε − / (Ω) ,H / − ε (Ω) = sup f ∈F Z Ω yf dx = sup f ∈F −h u, ∂ n z f i H − ε (Γ) ,H ε (Γ) ≤ sup f ∈F k ∂ n z f k H ε (Γ) k u k H − ε (Γ) ≤ C sup f ∈F k f k H ε − / (Ω) k u k H − ε (Γ) = C k u k H − ε (Γ) . (2.9)Take a sequence u k ∈ H / (Γ), u k → u ∈ H − ε (Γ), and let y k = Su k . We have justproved that k y k − y m k H / − ε (Ω) ≤ k u k − u m k H − ε (Γ) and therefore y k converges in H / − ε (Ω) to some y ∈ H / − ε (Ω) that is (the unique)solution of the equation: Z Ω yf dx = Z Ω lim k y k f dx = lim k Z Ω y k f dx = lim k −h u k , ∂ n z f i H − ε (Γ) ,H ε (Γ) = −h lim k u k , ∂ n z f i H − ε (Γ) ,H ε (Γ) = −h u, ∂ n z f i H − ε (Γ) ,H ε (Γ) . Finally, (2.9) implies that S ∈ L ( H − ε (Γ) , H / − ε (Ω)) and the proof is complete.7 IRICHLET CONTROL IN POLYGONAL DOMAINS
The next result is rather technical. It will be used to describe precisely the structureof the optimal state in the proof of Theorem 13 below. In that result we will be able towrite the control as the sum of a regular part and a singular part. We show in Lemma2.5 how to solve the state equation for singular boundary data. Besides the usual regularand singular parts that we described in Lemma 1, a new singular part of the solutionarises from the boundary data.Define the jump functions at the corners χ j = (cid:26) { x ∈ ∂K j : θ j = 0 }− { x ∈ ∂K j : θ j = ω j } . (2.10)Notice that χ j = 1 on Γ j and χ j = − j − . Lemma 5.
Consider any pair of subsets H , H ⊂ { , . . . , M } and real numbers − / <η j, and a j, for all j ∈ H and < η j, and a j, for all j ∈ H such that η j,n /λ j Z forany j ∈ H n , n = 1 , . Define u = X j ∈ H a j, ξ j r η j, j + X j ∈ H χ j a j, ξ j r η j, j on Γ , take p such that < p < inf (cid:26) − η j,n : j ∈ H n , n = 1 , and η j,n < (cid:27) , where we consider inf ∅ = + ∞ and define J mp as in (2.3) . Then there exist unique realnumbers ( c j,m ) j ∈ J mp , m = 1 , , and a unique solution y ∈ H / (Ω) of equation (1.1) such that y = y reg + X n =1 X j ∈ H n a j,n ξ j r η j,n j s j,n ( θ j ) + X m =1 X j ∈ J mp c j,m ξ j r mλ j j sin( mλ j θ j ) , (2.11) where s j,n ( θ j ) = ( − n +1 − cos( η j ω j )sin( η j ω j ) sin ( η j θ j ) + cos ( η j θ j ) (2.12) and y reg ∈ W ,p (Ω) . If, further, η j, > for all j ∈ H , then y ∈ H (Ω) .Proof. Since η j, > − /
2, we have that u ∈ L (Γ) and y ∈ H / (Ω) thanks to Theorem4. If also η j, >
0, then u ∈ H / (Γ) and hence Lemma 3 gives us that y ∈ H (Ω).Notice next that η j,n > − / / (1 − η j,n ) > / p is well defined.A direct computation shows that for n = 1 , y j,n = r η j,n j s j,n ( θ ) are respectively thesolutions of the problems − ∆ y j, = 0 in K j , y j, = r η j, j on ∂K j , IRICHLET CONTROL IN POLYGONAL DOMAINS − ∆ y j, = 0 in K j , y j, = χ j r η j, j on ∂K j . Since ∆ y j, = 0, we have that∆ X n =1 X j ∈ J n ξ j y j,n = n X j =1 X j ∈ J n ( y j,n ∆ ξ j + 2 ∇ y j,n ∇ ξ j ) . Since |∇ y j,n | ≤ Cr η j,n − j , the condition imposed on p implies that f = ∆ X n =1 X j ∈ J n a j,n ξ j y j,n ∈ L p (Ω) , and we can write y = y f + P n =1 P j ∈ H n a j,n ξ j y j,n , where − ∆ y f = f in Ω , y f = 0 on Γ . Applying Lemma 1 we obtain that y f = y reg + P m =1 P j ∈ J mp c j,m ξ j r mλ j j sin( mλ j θ j ) andthe proof is complete.Although the maximum principle is a well known result for weak solutions of equation(1.1) (see the celebrated paper by Stampacchia [16]), we have not been able to find areference of its validity for solutions defined in the transposition sense (2.4). For thesake of completeness, we include such a result. First, we prove the following technicallemma. Lemma 6.
Consider f ∈ L (Ω) , f ≥ and let z f ∈ H (Ω) be the solution of equation (2.1) . Then ∂ n z f ≤ a.e. on Γ .Proof. Take u ∈ C ∞ (Γ), u ≥
0. Thanks to Lemma 3, the solution of equation (1.1) satis-fies y ∈ H (Ω) and the maximum principle for weak solutions, as proved by Stampacchia[16], holds. Therefore y ≥
0. Integration by parts shows then that0 ≤ Z Ω yf dx = − Z Γ u∂ n z f dσ ( x )and the result follows by the usual density argument. Theorem 7.
Consider u ∈ L ∞ (Γ) . Then the solution y of equation (1.1) belongs to L ∞ (Ω) and k y k L ∞ (Ω) ≤ k u k L ∞ (Γ) . Proof.
Define K = k u k L ∞ (Γ) . We will prove that y ≤ K a.e. on Ω, the proof for − y ≤ K being analogous. 9 IRICHLET CONTROL IN POLYGONAL DOMAINS
We already know that y ∈ H / (Ω) ֒ → L (Ω) (cf. Theorem 4). Define y K = y − K and y + K = max( y K , ∈ L (Ω). For every f ∈ L (Ω) we deduce from (2.1) that Z Ω y K f dx = − Z Γ ( u − K ) ∂ n z f dσ ( x )We take f = y + K ≥
0. For this choice of f , we know from Lemma 6 that ∂ n z f ≤ ≤ Z Ω ( y + K ) dx = Z Ω y K y + K dx = − Z Γ ( u − K ) ∂ n z f dσ ( x ) ≤ . So y + K ≡ y ≤ K . The following result is standard and the proof can be found in [6]. Though in thatreference only convex domains are taken into account, this result is independent of theconvexity of the domain, once we have proved Corollary 2 and Theorem 4. Here and inthe rest of the paper, Proj [ a,b ] ( c ) = min { b, max { a, c }} for any real numbers a, b, c . Lemma 8.
Suppose y Ω ∈ L (Ω) . Then problem (P) has a unique solution ¯ u ∈ L (Γ) with related state ¯ y ∈ H / (Ω) and adjoint state ¯ ϕ ∈ H (Ω) . The following optimalitysystem is satisfied: ¯ u ( x ) = Proj [ a,b ] (cid:18) ν ∂ n ¯ ϕ ( x ) (cid:19) for a.e. x ∈ Γ , (3.1) − ∆¯ y = 0 in Ω , ¯ y = ¯ u on Γ , (3.2) − ∆ ¯ ϕ = ¯ y − y Ω in Ω , ¯ ϕ = 0 on Γ . (3.3)As in the proof of Theorem 4, the solution of the state equation (3.2) must be under-stood in the transposition sense, whereas the adjoint state equation (3.3) has a variationalsolution.Next we state a regularity result for the adjoint state in the framework of classicalSobolev–Slobodetski˘ı spaces. In the rest of this section we will suppose that y Ω ∈ L s ∗ (Ω)where, 2 ≤ s ∗ < + ∞ satisfies2( s ∗ − λ j s ∗ Z ∀ j ∈ { , . . . , M } . (3.4) Theorem 9.
There exist a unique function ¯ ϕ reg ∈ W ,s ∗ (Ω) and unique real numbers (ˆ c j,m ) j ∈ J ms ∗ , where J ms ∗ is defined in (2.3) , such that ¯ ϕ = ¯ ϕ reg + X m =1 X j ∈ J ms ∗ ˆ c j,m ξ j r mλ j j sin( mλ j θ j ) . (3.5)10 IRICHLET CONTROL IN POLYGONAL DOMAINS
Proof.
Since the problem is control constrained, ¯ u ∈ L ∞ (Γ), and by the maximumprinciple proved in Theorem 7, ¯ y ∈ L ∞ (Ω). Therefore, ¯ y − y Ω ∈ L s ∗ (Ω), and we can useLemma 1: since the related adjoint state ¯ ϕ is the solution of (3.3) we have that thereexist unique ¯ ϕ reg ∈ W ,s ∗ (Ω) and (ˆ c j,m ) j ∈ J ms such that relation (3.5) holds.For any s ≥ H s = { j ∈ J s : λ j > } and for s ≥ m = 2 , H ms = { j ∈ J ms : ˆ c j, = 0 } , where the coefficients ˆ c j,m are the coefficients obtained in Theorem 9. Notice that theindexes in H s correspond to convex corners. The indexes in H ms correspond to thosenon-convex corners where the main part of the singularity of the adjoint state vanishes,and hence the behavior of the solution at those non-convex corners can be somehowcompared to the behavior of the solution at the convex corners. Notice also that ∪ m =1 J ms = ∪ m =1 H ms [ { j ∈ J s : λ j < c j, = 0 } Consider also p ≥ m = 1 , , p ≤ s ∗ , p < − mλ j if j ∈ H ms ∗ . (3.6)This condition on p appears in a natural way in the proof of Theorem 11, see (3.10). If s ∗ >
2, then we can choose p >
2. With this choice we have that
Lemma 10.
Let p satisfy (3.6) and for m = 1 , , consider j ∈ H ms ∗ . Then ξ j r mλ j − j ∈ W − /p,p (Γ) . (3.7) Proof.
Take j ∈ ∪ m =1 H ms ∗ and consider N j the bounded cone of radius 2 R j defined inSection 2. We first prove that u j = r mλ j − j ∈ W ,p ( N j ).Since j ∈ ∪ m =1 H ms ∗ , then mλ j >
1, so mλ j − > r mλ j − j ∈ C ( ¯ N j ) ⊂ L p ( N j ).On the other hand |∇ u j | = ( mλ j − r mλ j − j , and making the usual change of variablesto polar coordinates, we have Z Z N j |∇ u j | p dx = ( mλ j − p ω j Z R j r j r ( mλ j − pj dr, the last integral being convergent if and only if ( mλ j − p +1 > −
1. Taking into accountthat j ∈ J ms ∗ implies mλ j <
2, the previous condition is fulfilled if and only if p < − mλ j ,which is the assumption.Relation (3.7) now follows from the smoothness of the cut-off function, the tracetheorem and the continuity of ξ j r mλ j − j . 11 IRICHLET CONTROL IN POLYGONAL DOMAINS
Theorem 11.
Let p satisfy (3.6) . Then, the optimal control ¯ u belongs to W − /p,p (Γ) ,the optimal state ¯ y belongs to W ,q (Ω) for all q ≤ p , q < p D . In particular, if s ∗ > ,both are continuous functions.Proof. We will exploit the projection relation (3.1) and the expression for the adjointstate obtained in (3.5).Notice first that ¯ ϕ reg ∈ W ,s ∗ (Ω) and ¯ ϕ reg = 0 on Γ, so ∂ n ¯ ϕ reg ∈ W − /s ∗ ,s ∗ (Γ) (cf.[5, Lemma A.2] for the case s ∗ = 2 or [4] for the case s ∗ > s ∗ >
2, then ∂ n ¯ ϕ reg ( x j ) = 0 on every corner the normal derivative of the regular part is a continuousfunction on Γ.We are going to compute now the normal derivative of the singular part. For any m ∈ { , , } and j ∈ J ms ∗ , we have ∂ n ξ j r mλ j j sin( mλ j θ j ) ∈ C ∞ (Γ \ { x j } ), so we have thatfor every compact set K ⊂ Γ \ { x j : j ∈ J s ∗ } ∂ n ¯ ϕ ∈ W − /s ∗ ,s ∗ ( K ) . Near the corners, for r j < R j we have on Γ j (where θ j = 0) that ∂ n r mλ j j sin( mλ j θ j ) ξ j = − r j ∂ θ r mλ j j sin( mλ j θ j )= − mλ j r mλ j − j cos( mλ j
0) = − mλ j r mλ j − j (3.8)and on Γ j − (where θ j = ω j ) that ∂ n r mλ j j sin( mλ j θ j ) ξ j = 1 r j ∂ θ r mλ j j sin( mλ j θ j )= mλ j r mλ j − j cos (cid:18) m πω j ω j (cid:19) = ( − m mλ j r mλ j − j . (3.9)Next we will distinguish two cases.Case 1: if j ∈ ∪ m =1 H ms ∗ then mλ j > r j →
0. Noticing (3.7), we have that the choice of the exponent p made in (3.6) givesus ∂ n ¯ ϕ reg + X m =1 X j ∈ H ms ∗ ˆ c j,m ξ j r mλ j j sin( mλ j θ j ) ∈ W − /p,p (Γ) . (3.10)So far, we can deduce that, for every compact set K ⊂ Γ \ { x j : λ j < , ˆ c j, = 0 } ∂ n ¯ ϕ ∈ W − /p,p ( K ) . (3.11)Case 2: Now j ∈ J s ∗ , λ j <
1, and ˆ c j, = 0. We have ( − m = − mλ j − < x → x j ∂ n ˆ c j, r λ j j sin( λ j θ j ) ξ j = − sign(ˆ c j, ) ∞ . IRICHLET CONTROL IN POLYGONAL DOMAINS
If it happens that also j ∈ J s ∗ ∪ J s ∗ , we have that for m = 2 and m = 3, mλ j − > x → x j ∂ n X m =1 ˆ c j,m r mλ j j sin( mλ j θ j ) ξ j = − sign(ˆ c j, ) ∞ . If s ∗ >
2, also on this corner lim x → x j ∂ n ¯ ϕ reg ( x ) = 0 , and, triviallylim x → x j ∂ n ¯ ϕ ( x ) = lim x → x j ∂ n ¯ ϕ reg ( x ) + ∂ n X m =1 ˆ c j,m r mλ j j sin( mλ j θ j ) ξ j ! = − sign(ˆ c j, ) ∞ . (3.12)If s ∗ = 2, as we said at the beginning of the proof, ∂ n ¯ ϕ reg ∈ H / (Γ), so it needs not beeven a bounded function. Nevertheless, since the singular part behaves like a negativepower of r j , this term dominates and we also have that (3.12) holds.As a consequence, there exists ρ j > x ∈ Γ with | x − x j | < ρ j eitherProj [ a,b ] ∂ n ¯ ϕ ≡ a or Proj [ a,b ] ∂ n ¯ ϕ ≡ b depending on the sign of ˆ c j, . So the control is flatnear non-convex corners. This, together with the projection formula (3.1) and (3.11)implies that the optimal control belongs to W − /p,p (Γ). Finally, the regularity of theoptimal state ¯ y follows from Lemma 3. Remark 12.
We would like to remark that the case of having ˆ c j, = 0 can be seen asa “rare” case in practice (although this can happen; see Example 14 below). So the“normal” case is that H ms = ∅ for m = 2 ,
3. In this case, in the choice of p made in (3.6)the indexes m = 2 , p will only depend max { ω j : ω j < π } , sowe get the same regularity for the control as that obtained in [6] for convex domains.To describe more accurately the regularity of the state and the control, we have tointroduce some further notation. Consider the coefficients ˆ c j,m obtained in Theorem 9and define the coefficients a j,m = − mλ j ˆ c j,m ν if 0 ∈ [ a, b ]0 if 0 [ a, b ] (3.13)and the functions s j,m ( θ j ) = ( − m +1 − cos(( mλ j − ω j )sin(( mλ j − ω j ) sin (( mλ j − θ j ) + cos (( mλ j − θ j ) . (3.14)The sets H ms will be used now to describe the singular part of the state and the control:if j ∈ H ms , then ξ j r mλ j − j W ,s (Ω) and ξ j r mλ j − j W − /s,s (Γ). Regarding (3.16), wealso mention that if λ j ≤ − /s , then ξ j r λ j j W ,s (Ω).13 IRICHLET CONTROL IN POLYGONAL DOMAINS
Theorem 13.
Assume further that ab = 0 . Then there exist a unique ¯ u reg ∈ W − /s ∗ ,s ∗ (Γ) ,a unique ¯ y reg ∈ W ,s (Ω) , for all s ≤ s ∗ , s < p D , and unique real numbers ( c j ) λ j ≤ − /s such that ¯ u ( x ) = ¯ u reg + X m =1 , X j ∈ H ms ∗ a j,m ξ j r mλ j − + X j ∈ H s ∗ χ j a j, ξ j r λ j − (3.15)¯ y = ¯ y reg + X m =1 X j ∈ H ms a j,m ξ j r mλ j − j s j,m ( θ j ) + X λ j ≤ − /s c j ξ j r λ j j sin( λ j θ j ) (3.16) where the χ j are the jump functions at the corners defined in (2.10) , the s j,m ( θ ) aredefined in (3.14) and the ξ j are the cut-off functions.Proof. From the considerations in the proof of Theorem 11 we have that far from thecorners with index j ∈ ∪ m =1 H ms ∗ , the optimal control is the projection of a functionthat is either regular enough or tends to a signed ∞ at one point, so it is clear that¯ u reg ∈ W − /s ∗ ,s ∗ (Γ).Next we will check what happens in the neighborhoods of the corners with index j ∈ ∪ m =1 H ms ∗ .If 0 [ a, b ] then the control would also be flat in neighborhoods of the corners withindex j ∈ ∪ m =1 H ms ∗ again because the normal derivative of the adjoint state is continuousnear the corner and 0 at the corner. Then (3.15) holds with a j,m = 0.If a < < b , then the optimal control will coincide with ∂ n ¯ ϕ/ν . Using formulas (3.8)and (3.9) and taking into account the definition of the jump functions on the corners χ j (2.10), we have that there exists ρ j > x ∈ Γ such that | x − x j | < ρ j ¯ u ( x ) = 1 ν ∂ n ¯ ϕ ( x ) = ν ∂ n ¯ ϕ reg ( x ) − ˆ c j, λ j ν r λ j − j if j ∈ H s ∗ , ν ∂ n ¯ ϕ reg ( x ) − χ j ˆ c j, λ j ν r λ j − j − ˆ c j, λ j ν r λ j − j if j ∈ H s ∗ ∪ H s ∗ , (3.17)and (3.15) holds for a j,m = − m ˆ c j,m λ j /ν .Let us finally check (3.16). We will write ¯ y = y + y , where − ∆ y = 0 in Ω , y = ¯ u reg on Γand − ∆ y = 0 in Ω , y = X m =1 , X j ∈ H ms ∗ a j,m ξ j r mλ j − + X j ∈ H s ∗ χ j a j, ξ j r λ j − on Γ . IRICHLET CONTROL IN POLYGONAL DOMAINS
Using Lemma 3 we have that y ∈ W ,s (Ω) for s ≤ s ∗ , s < p D . From Lemma 5 we havethat there exist a unique y ,reg ∈ W ,p (Ω), p defined in (3.6), and unique real numbers c j,m such that y = y ,reg + X m =1 X j ∈ H ms ∗ a j,m ξ j r mλ j − j s j,m ( θ j )+ X m =1 X j ∈ J mp c j,m ξ j r mλ j j sin( mθ j )Since p ≥ s ≤ s ∗ < ∞ , in dimension 2 we have thanks to usual Sobolev’s imbedding W ,p (Ω) ֒ → W ,s (Ω) that y ,reg ∈ W ,s (Ω).As we already mentioned, among the terms of the second addend, those which corre-spond to H ms are not in W ,s (Ω). Notice that if s ∗ < p D , and hence s < p D , then therecould be some terms in W ,s (Ω), which we gather in a function y a,reg ∈ W ,s (Ω).For the last addend, we have that ξ j r mλ j j sin( mθ j ) W ,s (Ω) iff mλ j ≤ − /s .Since s ≥
2, this excludes the case m >
1. We gather all the other terms in a function y b,reg ∈ W ,s (Ω).So finally we have that the (3.16) holds for ¯ y reg = y + y ,reg + y a,reg + y b,reg ∈ W ,s (Ω)and c j = c j, .Let us present now the example announced in Remark 12. For the example, we wantto remark that our results are also applicable in curvilinear polygons without manychanges (see [10, Th. 5.2.7]). The only thing to take into account is that if the angle ω j between two curved arcs is grater than π , then we must impose also s ∗ < ω j / ( ω j − π )(this is not the case in the following example). Example 14.
Let ω = 3 π/ { x ∈ R :0 < r < , < θ < ω } , where ( r, θ ) are the usual polar coordinates. We have that ω = ω = π/
2, and hence λ = 2 / λ = λ = 2. Suppose y Ω ∈ L ∞ (Ω), so we maychoose any s ∗ < + ∞ . We have J s ∗ = J s ∗ = { } and J s ∗ = ∅ . We also have p <
3. Theadjoint state can be written as¯ ϕ = ¯ ϕ reg + ˆ c , ξ r / sin (cid:18) θ (cid:19) + χ ˆ c , ξ r / sin (cid:18) θ (cid:19) where ¯ ϕ reg ∈ W ,s ∗ (Ω) for all s ∗ < + ∞ . Take − a = b = 1 for instance. If ˆ c , = 0, then H ms ∗ = ∅ for m = 1 , , u = ¯ u reg ∈ W − /s ∗ ,s ∗ (Γ).Define now y Ω ( x ) = (cid:26) θ < ω / − θ > ω / θ = ω /
2. Theskew-symmetry of the data suggests that the solution is skew-symmetric, i. e. that15
IRICHLET CONTROL IN POLYGONAL DOMAINS the symmetric contribution with r / sin (cid:0) θ (cid:1) vanishes, ˆ c , = 0 (result which we haveconfirmed numerically), and hence¯ u = ¯ u reg + χ a , ξ r / so ¯ u ∈ W − /p,p (Γ) for all p < a = 0 or b = 0. These cases can be treatedwith the same techniques. Nevertheless, many cases may appear depending on whichof the bounds is zero and the sign of the coefficients of the singular part ˆ c j,m . As anexample, we will show how to treat some of these cases. We will discuss first what wethink is the “generic” case, and then a seemingly more “rare” case. Without loss ofgenerality suppose a = 0, b > Case 1
Take j ∈ H s ∗ and suppose ˆ c j, = 0. Then in the expression for the normalderivative of the adjoint state (3.17), the term ξ j r λ j − j dominates the term ∂ n ϕ reg , since ξ j r λ j − W − /s ∗ ,s ∗ (Γ) and ∂ n ¯ ϕ reg ∈ W − /s ∗ ,s ∗ (Γ). Therefore, if ˆ c j, <
0, we wouldhave that ∂ n ¯ ϕ ( x ) ∈ [0 , b ] in a neighborhood of x j , and hence ¯ u ( x ) can be computed asin (3.17). On the other hand, if ˆ c j, >
0, then ∂ n ¯ ϕ ( x ) ≤ x j , so¯ u ( x ) ≡ ω = 3 π/ a = 0, b = 1 and either y Ω ≡ y Ω ≡ −
1, which wouldgive sign( c j, ) = − sign( y Ω ). Case 2 If j ∈ H s ∗ and ˆ c j, <
0, then ∂ n ¯ ϕ ( x ) ∈ [0 , b ] in a neighborhood of x j on the sideΓ j , but ∂ n ¯ ϕ ( x ) ≤ x j on the side Γ j − , so on Γ j , ¯ u ( x ) would havethe same expression as in (3.17), but, on Γ j − , ¯ u ( x ) would be flat near the corner x j .This would be the case of taking Example 14 for a = 0, b = 1 and y Ω defined in (3.18). Taking advantage of the regularity of the optimal state, we can obtain several resultsfor more regular data. We will write some results that we think will be useful for thenumerical analysis of problem (P). To be specific, to obtain error estimates for problemswith regular data, we will need that for y Ω ∈ H (Ω), ¯ u ∈ H / − ε (Γ) (cf. Corollary 16)and W ,p (Ω) regularity of the regular part of the adjoint state if y Ω ∈ W ,p (Ω), p ≥ y Ω ∈ H t ∗ (Ω), with 0 < t ∗ ≤ t ∗ ) /λ j Z ∀ j ∈ { , . . . , M } . For t > − m ∈ Z define˜ J mt = { j ∈ { , . . . , M } such that 0 < mλ j < t } IRICHLET CONTROL IN POLYGONAL DOMAINS
Notice again that due to our choice of t ∗ , we only will deal with the cases m = 1 , , Corollary 15.
There exist a unique function ¯ ϕ reg ∈ H t ∗ (Ω) and unique real numbers ˆ c j,m such that ¯ ϕ = ¯ ϕ reg + X m =1 X j ∈ ˜ J mt ∗ ˆ c j,m ξ j r mλ j j sin( mλ j θ j ) . Proof.
Since t ∗ >
0, there exists s ∗ > y Ω ∈ L s ∗ (Ω), and hencewe can apply Theorem 11 and we have that also ¯ y ∈ W ,q (Ω) ֒ → H (Ω) ֒ → H t ∗ (Ω),where q > § § t ∗ , logarithmic terms do not appearin the development of the singular part.To describe the regularity of the optimal control and state, we first introduce the sets˜ H mt in an analogous way as we did for the sets H ms , the indexes being taken now in thesets ˜ J mt defined above instead of the sets J ms and considering the coefficients ˆ c j, obtainedin Corollary 15. We next define the exponents t >
0, related to the regularity of thecontrol, and ˜ t >
0, related to the regularity of the state such that t ≤ t ∗ , t < , t < mλ j − j ∈ ˜ H mt , m = 1 , , t ≤ t, ˜ t < λ . The meaning of these bounds is the following. The regularity of the optimal control willbe limited by the regularity of the data, the impossibility of having a control globally in H / (Γ) due to the corners and the bound constraints, and the singular behavior of thecontrol at the convex corners or the “special” nonconvex corners that may lay in ˜ H mt for m = 2 ,
3. The regularity of the optimal state will be limited by the regularity of thecontrol and the singular behavior at the nonconvex corners of the solution.
Corollary 16.
Suppose that ♯∂ Γ { x ∈ Γ : ¯ u ( x ) = a or ¯ u ( x ) = b } < + ∞ (4.1) (the number of points on the boundary in the topology of Γ of the active set is finite).Then the optimal control ¯ u belongs to H / t (Γ) and ¯ y ∈ H t (Ω) .Proof. The proof follows the lines of that of Theorem 11. Since ¯ ϕ reg ∈ H t ∗ (Ω) and¯ ϕ reg = 0 on Γ its normal derivative will be in H / t (Γ) provided t ≤ t ∗ and t < ϕ reg = 0 on Γ is needed for t ∗ ≥ / t ∗ = 1 this continuityis not enough to have that the normal derivative is in H / (Γ).)This H / t (Γ) regularity is not affected by the projection formula (3.1) because t < c j, = 017 IRICHLET CONTROL IN POLYGONAL DOMAINS and λ j <
1. The rest of the singular terms in the expression for the normal derivative ofthe adjoint state will be in H / t (Γ) since t < mλ j − y ∈ H t (Ω). Since ¯ u ∈ H / t (Γ), there exists some U ∈ H t (Ω)such that U = u on Γ. Moreover, ∆ U ∈ H − t (Ω). So z = ¯ y − U is the solution of theboundary value problem − ∆ z = ∆ U in Ω , z = 0 on Γ . Using the regularity results in [7, § § z reg ∈ H t (Ω) and unique coefficients c j,m such that z = z reg + X m =1 X j ∈ ˜ J m − t c j,m ξ j r mλ j sin( mλ j θ j ) . Since ˜ t < λ , the singular part is in H t (Ω), and so is the optimal state.Finally, we will describe the adjoint state for even more regular data. In the rest ofthis section we will suppose y Ω ∈ W ,p ∗ (Ω), p ∗ ≥ p > m ∈ Z define J m ,p = (cid:26) j ∈ { , . . . , M } such that 0 < mλ j < − p and mλ j Z (cid:27) . Now we have that J m ,p = ∅ if m >
5. We have to add the condition mλ j Z otherwiselogarithmic terms may appear. Define also: L m ,p = (cid:26) j ∈ { , . . . , M } such that 0 < mλ j < − p and mλ j ∈ Z (cid:27) . A direct calculation gives us that L m ,p = ∅ if m = 2 or m ≥ L ,p ⊂ { j : ω j = π/ } and L ,p ⊂ { j : ω j = 3 π/ } , and hence mλ j = 2 if j ∈ L m ,p .Consider now p ≥ m = 1 , , p ≤ p ∗ , p < p D , p < − mλ j if j ∈ H ms ∗ ∀ s ∗ < ∞ . (4.2)In addition, we need to to assume3 p − λ j p Z ∀ j ∈ { , . . . , M } . With this notation, we have the following result.18
IRICHLET CONTROL IN POLYGONAL DOMAINS
Corollary 17.
There exist a unique function ¯ ϕ reg ∈ W ,p (Ω) and unique real numbers ˆ c j,m and ˆ d j,m such that ¯ ϕ = ¯ ϕ reg + X m =1 X j ∈ J m ,p ˆ c j,m ξ j r mλ j j sin( mλ j θ j )+ X m =1 , X j ∈ L m ,p ˆ d j,m ξ j r j (log( r j ) sin(2 θ j ) + θ j cos(2 θ j )) . Proof.
Since p ∗ ≥ y Ω ∈ L s ∗ (Ω) for any s ∗ < ∞ , and hence we can apply Theorem 11and we have that also ¯ y ∈ W ,p (Ω). The result follows directly from the adjoint stateequation (3.3) thanks to the regularity result [10, Th. 5.1.3.5]¯ ϕ = ¯ ϕ reg + X m =1 X j ∈ J m ,p ˆ ξ j r mλ j j sin( mλ j θ j )+ X m =1 , X j ∈ L m ,p ˆ d j,m ξ j r mλ j j (log( r j ) sin( mλ j θ j ) + θ j cos( mλ j θ j )) . and using that mλ j = 2 if j ∈ L m ,p . For problems without control constraints, we obtain similar results. Indeed, for convexdomains and data y Ω ∈ L s ∗ (Ω), s ∗ >
2, it is obvious from Theorem 18 below that theoptimal control is a bounded function and hence all the results stated before apply.Nevertheless, for nonconvex domains, we will not obtain a continuous control and thesingularities must be taken into account near all the corners. Therefore, the indexes forthe expansion of the singular parts must be taken running through all the sets J ms , andnot only through the sets H ms . Theorem 18.
Suppose now that − a = b = ∞ and y Ω ∈ L s ∗ (Ω) , s ∗ ≥ . Then thereexists a unique ¯ ϕ reg ∈ W ,s (Ω) , ¯ u reg ∈ W − /s,s (Γ) , ¯ y reg ∈ W ,s (Ω) , for all s ≤ s ∗ , s < p D , and unique real numbers (ˆ c j,m ) j ∈ J ms and ( c j ) λ j < − /s such that ¯ ϕ = ¯ ϕ reg + X m =1 X j ∈ J ms ˆ c j,m r mλ j j sin( mλ j θ j ) ξ j (5.1)¯ u ( x ) = ¯ u reg + X m =1 , X j ∈ J ms a j,m ξ j r mλ j − + X j ∈ J s χ j a j,m ξ j r λ j − (5.2)¯ y = ¯ y reg + X m =1 X j ∈ J ms a j,m ξ j r mλ j − j s j,m ( θ j ) + X λ j < − /s c j ξ j r λ j j sin( λ j θ j ) (5.3)19 IRICHLET CONTROL IN POLYGONAL DOMAINS where a j,m and s j,m ( θ ) are given by the formulas (3.13) and (3.14) .Proof. The proof is very similar to those of theorems 9, 11 and 13. We will only empha-size on the main difference: at the beginning of the proof of Theorem 9 we used that theoptimal control was bounded to obtain that the optimal state was a function in L s ∗ (Ω).Now the optimal control is not bounded, so we use a bootstrapping argument to showthat ¯ y ∈ L s (Ω) for s ≤ s ∗ , s < p D . The result follows then using the same techniques asbefore. Notice that now we do not need the sets H ms , since we do not have to exclude inthe expression of the singular part of the control the corners where the normal derivativeof the adjoint state is not bounded.In a first step we have that ¯ u ∈ L (Γ), and hence ¯ y ∈ H / (Ω) ⊂ L (Ω). If s ∗ ≤ < p D for any polygonal domain.Suppose s ∗ >
4. The normal derivative of the regular part of the adjoint state isin W − / , (Γ), but now, since we have no control constraints, we have to take intoaccount the normal derivative of the singular part near the non-convex corners. Wehave so far that the optimal control can be written as the sum of a regular part, which isin W − / , (Γ) plus a singular part, that behaves as r λ − . For the regular part we applyLemma 3 and for the singular part we apply Lemma 5, and we have that the optimal statecan be written as the sum of a regular part which is in W , (Ω) ⊂ L s ∗ (Ω) plus a singularpart that behaves at worst as r λ − ξ ∈ L s (Ω) for all s < / (1 − min { , λ } ) = p D . Sowe have that ¯ y ∈ L s (Ω) for all s ≤ s ∗ , s < p D .We will finish this section stating some regularity results of the optimal solution inthe unconstrained case in some special situations. Corollary 19.
Suppose the assumptions of Theorem 18 are satisfied. Then, for all p ≤ s ∗ , p < p Ω we have that ¯ ϕ ∈ W ,p (Ω) , ¯ u ∈ W − /p,p (Γ) and ¯ y ∈ W ,p (Ω) .Proof. Since p ≤ s ∗ and p < p Ω < p D , the regular parts of the involved functions satisfy¯ u reg ∈ W − /p,p (Γ), ¯ y reg ∈ W ,p (Ω) and ¯ ϕ reg ∈ W ,s (Ω) due to Theorem 18.On the other hand, the assumption p < p Ω implies ξ r λ − ∈ W ,p (Ω), and henceobviously ξ r λ ∈ W ,p (Ω) and ξ r λ − ∈ W − /p,p (Γ). Since these are the worst termswe may find in the singular parts, the proof is complete.With the same techniques of Section 4, (Corollary 15 and 16) we can obtain thefollowing result. Corollary 20.
Suppose the assumptions of Theorem 18 are satisfied and y Ω ∈ H t ∗ (Ω) for some t ∗ ≤ . Define t > such that t ≤ t ∗ , t < , t < λ − . Then ¯ u ∈ H / t (Γ) and ¯ y ∈ H t (Ω) . IRICHLET CONTROL IN POLYGONAL DOMAINS
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