SECOND-ORDER NECESSARY AND SUFFICIENT OPTIMALITY CONDITIONS FOR OPTIMIZATION PROBLEMS AND APPLICATIONS TO CONTROL THEORY

This paper deals with a class of nonlinear optimization problems in a function space, where the solution is restricted by pointwise upper and lower bounds and by finitely many equality and inequality constraints of functional type. Second-order necessary and sufficient optimality conditions are established, where the cone of critical directions is arbitrarily close to the form which is expected from the optimization in finite dimensional spaces. The results are applied to some optimal control problems for ordinary and partial differential equations.

1. Introduction.Let (X, S, µ) be a measure space with µ(X) < +∞.In this paper we will study the following optimization problem: where u a , u b ∈ L ∞ (X) and J, G j : L ∞ (X) −→ R are given functions with differentiability properties to be fixed later.We will state necessary and sufficient optimality conditions for a local minimum of (P).Our main goal is to reduce the classical gap between the necessary and sufficient conditions for optimization problems in Banach spaces.We shall prove some optimality conditions very close to the ones for finite dimensional optimization problems.In the case of finite dimensions, strongly active inequality constraints (i.e., with strictly positive Lagrange multipliers) are considered in the critical cone by associated linearized equality constraints.Roughly speaking, this is what we are able to extend to infinite dimensions.Due to the lack of compactness, the classical proof of the sufficiency theorem known for finite dimensions cannot be transferred to the case of general Banach spaces.Our direct method of proof is able to overcome this difficulty.To our best knowledge, this result has not yet been presented in the literature.Of course, the bound constraints u a (x) ≤ u(x) ≤ u b (x) introduce some additional difficulties in the study because they constitute an infinite number of constraints.In section 2 we introduce a slightly stronger regularity assumption than that considered in the Kuhn-Tucker theorem, which allows us to deal with the bound constraints.
In section 4 we discuss the application of our general results to different types of optimal control problems.We consider the control of ODEs as well as that of partial differential equations of elliptic and parabolic type.

Necessary optimality conditions.
In this section we will assume that ū is a local solution of (P), which means that there exists a real number r > 0 such that for every feasible point of (P), with u − ū L ∞ (X) < r, we have that J(ū) ≤ J(u).
For every ε > 0, we denote the set of points at which the bound constraints are ε-inactive by We make the following regularity assumption: ∃ε ū > 0 and {h j } j∈I0 ⊂ L ∞ (X), with supp h j ⊂ X εū , such that G i (ū)h j = δ ij , i,j ∈ I 0 , (2.1) where I 0 is the set of indices corresponding to active constraints.We also denote the set of nonactive constraints by I − I − = {j ≤ m|G j (ū) < 0}.
Obviously (2.1) is equivalent to the independence of the derivatives {G j (ū)} j∈I0 in L ∞ (X εū ).Under this assumption we can derive the first-order necessary conditions for optimality satisfied by ū.For the proof, the reader is referred to Bonnans and Casas [3] or Clarke [10].
Theorem 2.1.Let us assume that (2.1) holds and that J and {G j } m j=1 are of class C 1 in a neighborhood of ū.Then there exist real numbers { λj } m j=1 ⊂ R such that λj ≥ 0, m 1 + 1 ≤ j ≤ m, λj = 0 if j ∈ I − , (2.2) Since we want to establish some optimality conditions useful for the study of control problems, we need to take into account the two-norm discrepancy; for this question, see, for instance, Ioffe [17] and Maurer [19].Then we have to impose some additional assumptions on the functions J and G j .
(2.8) Associated with d, we set Given { λj } m j=1 by Theorem 2.1, we define the cone of critical directions (2.11) In the following theorem we state the necessary second-order optimality conditions.
Theorem 2.2.Assume that (2.1), (A1), and (A2) hold; { λj } m j=1 are the Lagrange multipliers satisfying (2.2) and (2.3); and J and {G j } m j=1 are of class C 2 in a neighborhood of ū.Then the following inequality is satisfied: (2.12) To prove this theorem we will make use of the following lemma.Lemma 2.3.Let us assume that (2.1) holds and that J and {G j } m j=1 are of class C 2 in a neighborhood of ū.Let h ∈ L ∞ (X) satisfy G j (ū)h = 0 for every j ∈ I, where I is an arbitrary subset of I 0 .Then there exist a number ε h > 0 and C 2 -functions with u t = ū + th + j∈I γ j (t)h j , {h j } j∈I given by (2.1).
Then ω is of class C 2 in a neighborhood of (0, 0), Therefore we can apply the implicit function theorem and deduce the existence of ε > 0 and functions where γ(t) = (γ j (t)) j∈I .Furthermore, by differentiation in the previous identity we get Taking into account the continuity of γ and G j and that γ(0) = 0, we deduce the existence of ε h ≤ ε such that (2.13) holds for every t ∈ (−ε h , +ε h ).

15)
I includes all equality constraints, all strongly active inequality constraints (i.e., λj > 0), and, depending on h, possibly some of the weakly active inequality constraints (i.e., λj = 0).Then we are under the assumptions of Lemma 2.3.Let us set (2.11) we deduce that G j (ū) = 0 and G j (ū)h < 0 for j ∈ I 0 \ I. Therefore we have that G j (u t ) < 0 for every j / ∈ I and t ∈ (0, ε 0 ), for some ε 0 > 0 small.On the other hand, the assumptions on h, along with the additional condition (2.14) and the fact that supp h j ⊂ X εū , imply that u a (x) ≤ u t (x) ≤ u b (x) for t ≥ 0 small enough.Consequently, by taking ε 0 > 0 sufficiently small, we get that u t is a feasible control for (P) for every t ∈ [0, ε 0 ).Now we know G j (u t ) = 0 for j ∈ I and λj = 0 for j / ∈ I 0 (cf.(2.2)).According to (2.11) we require G j (ū)h = 0 for active inequalities with λj > 0; hence if i belongs to I 0 \ I, then λj = 0 must hold.This leads to has a local minimum at 0 and, taking into account that γ j (0) = 0, The last identity follows from the fact that h vanishes on X 0 .Since the first derivative of φ is zero, the following second-order necessary optimality condition must hold: Here we have used (A1).Now let us consider h ∈ L ∞ (X) satisfying (2.11) but not (2.14), i.e., h is any critical direction.The main idea in this case is to approach h by functions h ε , which belong to the critical cone C 0 ū and satisfy (2.14) as well.Then for every ε > 0, we define A ε = X ε ∪ {x ∈ X : ū(x) = u a (x) or ū(x) = u b (x)}.This is the complement of the set of points x satisfying (2.14).Set where χ Aε is the characteristic function of A ε and I is given by (2.15).We verify that h ε belongs to C 0 ū, while hχ Aε is possibly not contained in this cone.Thus for every j ∈ I, using (2.1) and taking 0 < ε < ε ū, we have In the case of j ∈ I 0 \ I, we have G j (ū)h < 0. Then it is enough to take ε sufficiently small to get G j (ū)h ε < 0.
Finally, it is clear that h ε (x) → h(x) a.e. in X as ε → 0. Therefore, assumption (A2) allows us to pass to the limit in the second-order optimality conditions satisfied for every h ε and to conclude (2.12).
3. Sufficient optimality conditions.Whenever nonlinear optimal control problems are solved, second-order sufficient conditions play an essential role in the numerical analysis.For instance, they ensure local convergence of Lagrange-Newton-SQP methods; see Alt and Malanowski [2], Dontchev et al. [11], Ito and Kunisch [18], or Schulz [23], and the references cited therein.Such conditions are important for error estimates as well.We refer, for instance, to Arada, Casas, and Tröltzsch [1] and Hager [15].Finally, we mention that second-order conditions should be checked numerically to verify local optimality of computed solutions; see Mittelmann [21].
In this section, ū is a given feasible element for the problem (P).Motivated again by the considerations on the two-norm discrepancy, we have to make some assumptions involving the L ∞ (X) and L 2 (X) norms, as follows.
(A3) There exists a positive number r > 0 such that J and {G j } m j=1 are of class C 2 in the L ∞ (X)-ball B r (ū), and for every η > 0 there exists ε ∈ (0, r) such that for Analogously to (2.9) and (2.10), we define for every τ > 0 The next theorem provides the second-order sufficient optimality conditions of (P).Although they seem to be different from the classical ones, we will prove later that they are equivalent; see Theorem 3.2 and Corollary 3.3.
(ii) Some technical definitions.Let us set Finally, we take where (iii) Approximation of u − ū by elements of the critical cone.Let u be a feasible point for problem (P), with u − ū L ∞ (X) < .Then u − ū will not, in general, belong to the critical cone.Therefore, we use the representation u − ū = h + h 0 , where h is in the critical cone and h 0 is some small correction.
Let us introduce the set of indices This is the set of indices for which we need to correct G j (ū)(u− ū), since the conditions of the critical cone are not met.We need to carry out this correction for equality constraints if G j (ū)(u − ū) = 0. We also need to apply this correction for an active inequality constraint satisfying G j (ū)(u − ū) > 0 or for a strongly active inequality constraint if G j (ū)(u − ū) < 0 holds.We define for all j ∈ I u where the elements h j are introduced in assumption (2.1).Then h satisfies (2.11).This is seen as follows: Downloaded 04/23/13 to 193.144.185.28.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php (the last inequality follows from j / ∈ I u ).Thus G j (ū)h fulfills the conditions of the critical cone.If j ∈ I u , then and G j (ū)h also fulfills the conditions of the critical cone.
Let us now estimate h 0 in L 2 (X).For every j ∈ I u there exists If α j ≥ 0, we deduce from (3.11) and (3.1) that Let us define This is the set of all indices, where we do not obtain an estimate of α j having the order u − ū 2 L 2 (x) .We should notice at this point that λj > 0 holds for all j ∈ I − u .(Since u must be feasible, j stands for an inequality constraint.Therefore, 0 > α j = G j (ū)(u − ū), and j ∈ I u implies j ∈ I + .)Then we have Downloaded 04/23/13 to 193.144.185.28.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.phpholds, since ρ < 1.Therefore, Now from (2.8), (2.11), (3.1), (3.5), and (3.6) it follows that Using the definition of ε from (3.9), we have On the other hand, From the definitions of C 1 and ρ given in (3.7) and (3.6) along with (3.15), (3.16), and (3.17), we get The definition of α j given by (3.10) along with assumption (3.1) imply From (3.8) and the above inequality, we deduce Definition (3.9) and (3.21) lead to Finally, combining (3.18), (3.19), and (3.22), we conclude the desired result: Now we prove the equivalence between the sufficient optimality conditions stated in Theorem 3.1 and the classical ones.
Theorem 3.2.Let ū be a feasible point of (P) satisfying (2.2) and (2.3).Let C ū be the set of elements h ∈ L ∞ (X) satisfying (2.11), and C τ ū be given by (3.3).Let us suppose that assumptions (2.1), (A1), and (A3) hold.Let τ > 0 be given.Then the following statements are equivalent: Therefore, it is enough to take δ = δ 1 .Let us prove the opposite implication.Let h ∈ C ū.We set h τ = hχ X τ , where χ X τ is the characteristic function of X τ and where the functions h j are given by (2.1).
Let us see that h 0 ∈ C τ ū .Since supph j ⊂ X εū and h − h τ = h(1 − χ X τ ), we have that h 0 (x) = 0 for x ∈ X τ .Now we distinguish between the cases j ∈ I h and j ∈ I 0 \ I h .
If j ∈ I h , then If this inequality reduces to an equality G j (ū)(h−h τ ) = 0, then h 0 verifies that the condition is in C τ ū .In the remaining case in which j ∈ I 0 \ I h but G j (ū)(h − h τ ) < 0, using again the definition of I h , we deduce that G j (ū)h < 0. (G j (ū)h = 0 and G j (ū)(h − h τ ) < 0 would give j ∈ I h .)Consequently, since h ∈ C ū, we have that j > m 1 and λj = 0 (otherwise, h ∈ C τ ū and λj > 0 would imply G j (ū)h = 0).Then the inequality G j (ū)h 0 < 0 also means that h 0 shows the condition to be in C τ ū .We now prove that ĥ where g j being given in (2.4).Indeed, if α j > 0, then If α j < 0, then from the definition of I h we have that G j (ū)h = 0; therefore Combining the previous two inequalities and the definition of ĥ, we get (3.25).
The following corollary is an immediate consequence of Theorems 3.1 and 3.2.Corollary 3.3.Let ū be a feasible point for problem (P) satisfying (2.2) and (2.3), and suppose that assumptions (2.1), (A1), and (A3) hold.Assume also that for some δ > 0 and τ > 0 given.Then there exist ε > 0 and α > 0 such that J(ū)+α u− ū 2 L 2 (X) ≤ J(u) for every feasible point u for (P), with u− ū L ∞ (X) < ε.Remark 3.4.Comparing the sufficient optimality condition (3.4) with the necessary condition (2.12), we notice the existence of a gap between the two, arising from two facts.First, the constant δ 1 is strictly positive in (3.4), and it can be zero in (2.12), which is the classical situation even in finite dimensions.Second, we cannot substitute, in general, C τ ū , with τ > 0, for C 0 ū in (3.26), as is done in (2.12), because of the presence of an infinite number of constraints.Quite similar strategies are employed by Maurer and Zowe [20], Maurer [19], Donchev et al. [11], and Dunn [12].

An abstract control problem.
Let, in addition to the measure space (X, S, µ), Y and Z be real Banach spaces; let A : Y → Z be a linear continuous operator; and let where the control u is taken from L ∞ (X).We assume that for all u ∈ L ∞ (X) the equation Ay + B(y, u) = 0 admits a unique solution y ∈ Y , so that a control-state mapping G : u → y is defined.Moreover, the inverse operator (A + ∂B ∂y (y, u)) −1 : Z → Y is assumed to exist for all (y, u) ∈ Y × L ∞ (X) as a linear continuous operator.Then the implicit function theorem yields that G is of class C 2 from L ∞ (X) to Y .The first-and second-order derivatives G (u) and G (u) are given as follows: Define y = G(u), z h = G (u)h, and while z h1h2 is uniquely determined by We omit the proof, which can easily be transferred from that of Theorem 2.3 in [7].

Optimal control of ODEs.
In this section we discuss an optimal control problem governed by an ODE.We concentrate on a very simplified setting to give the reader an easy insight into the application of the theory.For further problems, we refer to the book by Hestenes [16].Define y(t), u(t))dt, j = 1, . . ., m, and consider the optimal control problem (ODE) Here, T is a fixed time.To reduce the number of technicalities, let us discuss only real-valued functions y and u.The vector-valued case can be handled analogously.
For the same reason, we assume that the functions ψ, f j , and b are of class C 2 on R and [0, T ] × R × [min u a , max u b ], respectively, although weaker Carathéodory-type conditions would suffice.We introduce the state space Y = {y ∈ W 1,∞ (0, T )|y(0) = 0} and set (Ay

y(t), u(t)).
A is continuous from Y to Z = L ∞ (0, T ), and B is of class C 2 from Y × L ∞ (0, T ) to Z.In this way, (ODE) is related to (OC) as a particular case, where X = [0, T ], and µ Downloaded 04/23/13 to 193.144.185.28.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php is the Lebesgue measure, dµ = dt.For convenience, the variable t ∈ X is substituted for the variable x, which was used in the former sections.
In an obvious way this ϕ generates a linear functional belonging to Z * , but it has more regularity than arbitrary functionals of this space.
Remark 4.4.Given the inhomogeneous initial condition y(0) = y 0 , we have to work with the space Y = W 1,∞ (0, T ) and must include the initial condition in the definition of A. Then the additional term ϕ 0 (y(0) − y 0 ) would appear in (4.19).This requires some more notational effort.However, the optimality conditions are not changed.Therefore, without loss of generality we confine ourselves to a homogeneous initial condition.
Having in mind the particular form of ϕ, we see that here (4.5) is nothing more than the definition of the adjoint equation It is obvious that (4.20) admits a unique solution φ ∈ W 1,∞ (0, T ).In section 5 we show that (A1) is satisfied for (ODE).We obtain the following derivatives of the Lagrange function: (all derivatives taken at (ȳ, ū)); hence ∂L/∂u can be identified with d ∈ L ∞ (0, T ), where f 0 , b , f j stand for 2 × 2 Hessian matrices taken at (t, ȳ(t), ū(t)).It is easy to verify that (A2) is satisfied.
The first-order necessary optimality conditions are stated in Corollary 4.1.In particular, the following variational inequality has to be satisfied: Downloaded 04/23/13 to 193.144.185.28.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.phpfor all u a ≤ u(t) ≤ u b ; hence ū(t) = u a , where d(t) > 0, and ū(t) = u b , where d(t) < 0. (These points form the set X 0 .)No information is obtained where d is zero.Roughly speaking, this is the set for which higher-order conditions are needed.
The second-order necessary conditions are formulated in Corollary 4.2.We have to specify the linearized equation (4.12) and the form of the derivatives in the relations (4.14).The linearized equation is while

Optimal boundary control of an elliptic equation.
As a further application, we consider an elliptic control problem.For convenience, we discuss a simplified version and refer for further reading to [9].
Let Ω ⊂ R N be a bounded domain with boundary Γ of class C 0,1 .Let ν denote the outward unit normal vector at Γ, and ∂ ν be the associated normal derivative.Define We assume that the functions γ j = γ j (x, y), ψ j = ψ j (x, y), and f j = f j (x, y, u) are of class C 2 on Ω × R and Ω × R 2 , respectively.Moreover, real Borel measures µ j are given on Ω.Here, µ is the Lebesgue surface measure induced on Γ, dµ = dS.The appearance of the measures µ j in the functionals will heavily influence the verification of assumptions (A1)-(A3).Therefore, the easier case ψ j = 0, j = 1, . . ., m, is of interest as well.
Consider the optimal control problem (ELL) In this setting, the boundary control u is looked upon in the space L ∞ (Γ), hence X = Γ, while the state y belongs to (Here q > N/2 and p > N − 1 are given fixed.)Endowing Y with the graph norm, it is known that Y ⊂ C( Ω), the embedding being continuous.Assume that b = b(x, y, u) satisfies the same conditions as the f j .Additionally, we require that (∂b/∂y)(x, y, u) .
The equation Ay + B(y, u) = 0, which is equivalent to our elliptic boundary value problem, admits for each u ∈ L ∞ (Γ) exactly one solution y ∈ Y .The mapping u → y is of class C 2 from L ∞ (Γ) to Y .Now we proceed in the same way as in the preceding section.The Lagrange function is where ϕ ∈ W 1,s (Ω) for all s < N N −1 is the adjoint state.The adjoint state ϕ together with its trace ϕ |Γ forms a Lagrange multiplier of Z * = L q (Ω) × L p (Γ) having higher regularity.Here (4.5) reduces to the adjoint equation (all partial derivatives taken at (x, ȳ(x), ū(x))).This equation has a unique solution φ ∈ W 1,s (Ω) associated with (ȳ, ū, λ).Notice that for N = 2 the Sobolev imbedding theorem yields ϕ ∈ L σ (Ω) for all σ < ∞, but not in general ϕ ∈ L ∞ (Ω).For N ≥ 3 the regularity of ϕ is even lower.This indicates that we have to discuss assumptions (A1)-(A3) with more care.We shall do this in the last section.
The situation is easier in the case ψ j = 0, j = 0, . . ., m.Then all data given in the adjoint equation are bounded and measurable, and the regularity theory of elliptic equations yields φ ∈ C( Ω) (see [5]).
Let us establish the first-and second-order derivatives of L. We get We observe that, due to our notation, there is almost no difference in the expressions derived for the case of (ODE) in (4.21), (4.23).The first-and second-order conditions for our elliptic problem (ELL) admit the following form: Set Then d has the same form as in (4.22).The first-and second-order optimality conditions are given by Corollaries  In this way, we have obtained the second-order sufficient condition for a simplified elliptic control problem.For the discussion of more general problems, we refer to [7], [9].We should underline again that so far we have stated the optimality condition in a formal way.It remains to verify (A1)-(A3) to make our theory work.Low regularity of the adjoint state ϕ can be an essential obstacle for this.We refer to section 5.

4.4.
Optimal distributed control of a parabolic equation.We confine ourselves to a distributed parabolic control problem.A more general class, including boundary control and boundary observation, is considered in a separate paper by Raymond and Tröltzsch [22].Let Ω be defined as in the last section, and set Q = Ω × (0, T ), Σ = Γ × (0, T ).Define t, y(x, t), u(x, t))dxdt, j = 1, . . ., m.We assume again that the functions ψ j , f j , and γ j are of class C 2 on Q × R and Q × R 2 , respectively.Moreover, real Borel measures µ j , j = 0, . . ., m, are given on Ω and Q, respectively.Now µ is the Lebesgue measure on Q, dµ = dxdt.Downloaded 04/23/13 to 193.144.185.28.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.phpConsider the optimal control problem (PAR) In this setting, the distributed control u is looked upon in the space L ∞ (Q); hence we set X = Q.The state y belongs to Y = {y ∈ W (0, T )|y(0) = 0, y t − ∆y ∈ L q (Q), ∂ ν y ∈ L p (Σ)}, where q > N/2 + 1 and p > N + 1 are given fixed.It is known that Y ⊂ C( Q), the embedding being continuous for the graph norm.Assume that b = b(x, t, y, u) satisfies the same conditions as the f j .Additionally, we require that ∂b/∂y(x, t, y, u) The equation Ay + B(y, u) = 0, which is equivalent to our parabolic initial-boundary value problem, admits for each u ∈ L ∞ (Q) exactly one solution y ∈ Y .We refer to [5].The mapping u → y is of class C 2 from L ∞ (Q) to Y .Here, the Lagrange function is where ϕ is the adjoint state and dS again denotes the Lebesgue surface measure induced on Γ. Equation (4.5) turns out to be the adjoint equation ))µ 0 in Ω (all partial derivatives taken at (x, ȳ, ū)).This equation has a unique solution φ ∈ W 1,s (Ω) associated with (ȳ, ū, φ, λ).If, however, ψ j = 0, j = 1, . . ., m, then φ is more regular, φ ∈ W (0, T ) ∩ C( Q).
The first-and second-order conditions for the parabolic case are covered by Corollaries We leave the calculations of the derivatives in (4.14) to the reader; they are obtained by an obvious modification of (4.28).We should mention again that these optimality conditions are meaningful only if the assumptions (A1)-(A3) are satisfied.

Verification of the assumptions.
Our theory relies on the general assumptions (A1)-(A3).We shall see that (A1)-(A3) are naturally satisfied for the problem (ODE), while the situation is more complicated in the case of the elliptic or parabolic PDE.
(i) Problem (ODE).(A1).It is obviously sufficient to look at one of the functionals G j (u) = F j (G(u), u) to assess the situation.We have where y = G (ū)h.Here, ∂f j /∂y, ∂f j /∂u are bounded and measurable functions.Moreover, the estimate must hold with some g j ∈ L 2 (0, T ); hence (A1) is fulfilled.
(A2).Here, the derivative is characteristic for the discussion.All entries of f j are bounded and measurable.If . This shows (A2).
(A3).First, we must estimate differences of the type G j (ũ) − G j (ū) for ũ in a L ∞ -neighborhood of ū.We get where ỹ = G(ũ), ȳ = G(ū), y = G (ū)h.Due to our assumptions, we find that 2 is the coupling of the nonlinearity b with φ.It is the essential advantage of our simplified case (ODE) that φ ∈ L ∞ (0, T ).Therefore, we are justified to estimate ≤ c h 2 L 2 (0,T ) .
(5.5) Discussing all second-order terms in this way, we easily verify that (A3) is also satisfied.
(ii) Elliptic problem (ELL).We repeat the discussion of (A1)-(A3) along the lines of (i) but concentrating on the essential differences with the case of (ODE).Here, it holds that G j (ū)h = where y = G (ū)h.In contrast to (5.2), now the mapping G (ū) is not in general continuous from L 2 (Γ) to C( Ω).This property only holds for N = dim Ω = 2 (see [9]).
For N > 2 we assume that Ω j , the support of µ j , satisfies Ωj ⊂ Ω.Then the mapping h → G (ū)h is continuous from L 2 (Γ) to C( Ωj ); hence h → G j (ū)h is a linear and continuous functional on L 2 (Γ).The Riesz theorem yields a representation analogous to (5.3).Hence (A1) is shown under additional assumptions on the subdomains Ω j .(A2) then holds true in the same way.Notice that the restriction to Ω j is not needed if all ψ j vanish.Downloaded 04/23/13 to 193.144.185.28.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php To verify (A3) we need even more restrictions on the data.The situation is easy if ψ j = 0, j = 1, . . ., m.Then all given data in the adjoint equation are bounded and measurable, and the regularity theory of elliptic equations yields φ ∈ C( Ω).In this case, (A3) is obviously satisfied.
Let us now assume that at least one of the ψ j is not zero.Then the best regularity of the trace φ|Γ is φ|Γ ∈ L r (Γ) for all r < (N − 1)/(N − 2).For instance, ϕ ∈ L r (Γ) for all r < ∞ is obtained in the case N = 2.We therefore cannot assume that φ ∈ L ∞ (Ω).Regard the elliptic counterpart to (5. (5.6) This expression has to be estimated for h ∈ L 2 (Γ).If φ|Γ / ∈ L ∞ (Γ), which is the normal case, then we must exclude the third term from (5.6).This means that ∂ 2 b/∂u 2 has to disappear-u must appear linearly.Next we consider the second term, where φ|Γ y L 2 (Γ) is estimated against h L 2 (Γ) .The mapping h → y is continuous from L 2 (Γ) to C(Γ) (N = 2), to L r (Γ) for all r < ∞ (N = 3), and to L r (Γ) for all r < 2(N − 1)/(N − 3) (N > 3).Therefore, the second term can be estimated iff N = 2, while it must be cancelled for N > 2. The latter means ∂ 2 b/∂u∂y = 0-here b = b 1 (x, y) + b 2 (x)u must hold.In the same way we arrive at the surprising fact that for N > 3 the first term in (5.6) must vanish, too.In other words, in the case of elliptic boundary control with pointwise functionals F j , we cannot admit nonlinear equations for N > 3.
Remark 5.1.We should underline again that these restrictions are not needed if the functionals F j are sufficiently regular (ψ j = 0, j = 1, . . ., m).Moreover, the case of distributed controls permits us to slightly relax the restrictions on the dimension N .
In the opposite case, the problem of regularity is even more delicate than in the elliptic problem.We cannot discuss the general case in detail and refer to the recent paper [22].Instead of this, let us explain the point for a very particular constraint: Suppose that only one (pointwise) state constraint of the form g 1 (y, u) = T 0 y(x 1 , t)dt = 0 is given, where x 1 ∈ Ω is a fixed position of observation.To make the theory work, we need some strong restrictions: We assume N = dim Ω = 1, i.e., Ω = (a, b), and require that ∂ 2 b/∂u 2 = 0 (the control appears linearly).Then the mapping h → y = G (ū)h is continuous from L 2 (Q) to C( Q), and the functional h → g 1 (y, h) is continuous on L 2 (Q).We know that φ ∈ L s (Q) for all s < 3. (This follows from Theorem 4.3 in [22] for N = 1 and α = α.)Hence φ / ∈ L ∞ (Q), and that is the reason why we cannot admit a control appearing nonlinearly.The estimate of the parabolic counterpart of (5.6) is and h k (x) → h(x) a.e. in X, then  J (ū) + m j=1 λj G j (ū)   h 2 k →   J (ū) + m j=1 λj G j (ū)   h 2 .(2.5) Downloaded 04/23/13 to 193.144.185.28.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.phpIf we define L(u, λ) = J(u) + m j=1 λ j G j (u) and d(x) = f (x) + )h(x)dµ(x) ∀h ∈ L ∞ ( 7), (3.6), (3.10), and (3.11), for some