SECOND ORDER ANALYSIS FOR OPTIMAL CONTROL PROBLEMS: IMPROVING RESULTS EXPECTED FROM ABSTRACT THEORY∗

An abstract optimization problem of minimizing a functional on a convex subset of a Banach space is considered. We discuss natural assumptions on the functional that permit establishing sufficient second-order optimality conditions with minimal gap with respect to the associated necessary ones. Though the two-norm discrepancy is taken into account, the obtained results exhibit the same formulation as the classical ones known from finite-dimensional optimization. We demonstrate that these assumptions are fulfilled, in particular, by important optimal control problems for partial differential equations. We prove that, in contrast to a widespread common belief, the standard second-order conditions formulated for these control problems imply strict local optimality of the controls not only in the sense of L∞, but also of L2.

every v ∈ U \ {0} imply that ū is a strict local minimum of J.This second order condition says that the quadratic form v → J (ū)v 2 is positive definite in R n , which is equivalent to the strict positivity of the smallest eigenvalue λ m of the associated symmetric matrix.Moreover, J (ū)v 2 ≥ λ m v 2 for every v ∈ R n .
However, if U is an infinite-dimensional space, then the condition J (ū)v 2 > 0 is not equivalent to J (ū)v 2 ≥ λ m v 2 for some λ m > 0. Is one of the two conditions sufficient for local optimality?The answer has been well known for a long time and it is documented extensively in the literature: the first condition is not sufficient while the second, together with the first order optimality condition, implies strict local optimality of ū in the right setting.Let us discard the first (weaker) condition by an example.
Example 1.1.Consider the optimization problem (Ex 1 ) m i n u∈L ∞ (0,1) The function ū(t) ≡ 0 satisfies the first order necessary condition J (ū) = 0 and However, ū is not a local minimum of (Ex 1 ).Indeed, if we define 0 otherwise, then it holds that J(u k ) = − 1 k 4 < J(ū) and u k − ū L ∞ (0,1) = 2 k .Now the question seems to be answered-the second and stronger condition should be sufficient for optimality.The next example shows, however, that this is not true in general.
The reader will easily confirm that this property holds for any solution û of the problem.What is wrong?
The reason is that J is not of class C 2 in L 2 (0, 1), our fast computations were too careless.Therefore, we cannot apply the abstract theorem on sufficient conditions for local optimality in L 2 (0, 1).On the other hand, J is of class C 2 in L ∞ (0, 1) and the derivatives computed above are correct in L ∞ (0, 1).However, the inequality J (ū)v 2 ≥ δ v 2  L ∞ (0,1) does not hold for any δ > 0. This phenomenon is called the two-norm discrepancy: the functional J is twice differentiable with respect to one norm, but the inequality J (ū)v 2 ≥ δ v 2 holds in a weaker norm in which J is not twice differentiable; see, for instance, [13].This situation arises frequently in infinite-dimensional problems but it does not happen for finite dimensions because all the norms are equivalent in this case.The classical theorem on second order optimality conditions can easily be modified to deal with the two-norm discrepancy.
Theorem 1.3.Let U be a vector space endowed with two norms, ∞ and 2 , such that and assume that the following properties hold: and there exists some ε > 0 such that B∞ (ū; ε) ⊂ A and Then there holds so that ū is a strictly locally optimal with respect to the norm • ∞ .
In the above theorem and hereafter B ∞ (ū; ε) (respectively, B 2 (ū; ε)) will denote the ball of radius ε and centered at ū with respect to the norm • ∞ (respectively, • 2 ).The proof of this theorem is quite elementary.To the best of our knowledge, Ioffe [13] was the first who proved a result of this type by using two norms in the context of optimal control for ordinary differential equations.We also refer to the discussion of the two-norm discrepancy by Malanowski [15] and Maurer [16].In the context of PDE constrained optimization, the proof of Theorem 1.3 can be found, e.g., in [10] or [18,Thm. 4.29].
Theorem 1.3 can be applied to Example 1.2 to deduce that ū is a strict local minimum in the sense of L ∞ (0, 1).Strict local optimality of ū means that J(u) > J(ū) holds for all admissible u out of a certain neighborhood of ū.This does not yet exclude the possibility that ū is an accumulation point of locally optimal solutions.
If the two-norm discrepancy occurs in an optimal control problem, we consider two norms, namely the L ∞ -norm for differentiation and the L 2 -norm for expressing the coercivity of J .Then local optimality should hold only in the stronger L ∞ sense.
However, we will prove in this paper that for standard optimal control problems for distributed parameter systems, where the control appears linearly in the state Downloaded 05/23/13 to 193.144.185.28.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.phpequation, the sufficient second order condition also implies strict local optimality in the L 2 sense.Even more, we can find an L 2 -neighborhood of this local minimum where local uniqueness holds.This means that there does not exist any other stationary point of (P) in that neighborhood.Let us underline this even more: in many cases with two-norm discrepancy, results expected to hold only in an L ∞ -neighborhood around the local solution are even true in an L 2 -neighborhood.
The plan of this work is as follows.In section 2 we will formulate an abstract optimization problem and fix the assumptions that lead to the results mentioned above.In sections 3 and 4 we will apply the abstract results to elliptic control problems with Neumann and Dirichlet boundary controls, respectively.Finally, in section 5 a distributed parabolic control problem is considered.We do not need the restrictions on the dimension of the spatial domain, which are usually required in these cases.
2. An abstract optimization problem in Banach spaces.Let U ∞ and U 2 be Banach and Hilbert spaces, respectively, endowed with the norms • ∞ and • 2 .We assume that U ∞ ⊂ U 2 with continuous embedding; in particular, the choice Moreover, an objective function J : A −→ R is given.We consider the abstract optimization problem (P) min u∈K J(u).
The next well known result expresses the first order optimality conditions in the form of a variational inequality.Theorem 2.1.If ū is a local solution of (P) and J is differentiable at ū, both in the sense of U ∞ , then We say that ū is a local solution of (P) in the sense of U ∞ , if J(ū) ≤ J(u) holds for all u ∈ K ∩ {u ∈ U ∞ : u − ū ∞ < ε} with some ε > 0. If the strong inequality J(ū) < J(u) is satisfied in this set for all u = ū, then this solution is called a strict local solution.Notice that any local solution of (P) in the U 2 sense is also a local solution in the U ∞ sense.Therefore, (2.1) holds also for local solutions of (P) in the U 2 sense.
The rest of this section is devoted to the study of the necessary and sufficient second order optimality conditions for problem (P).Throughout the section all notions of differentiability of J are to be understood in the sense of U ∞ .We fix an element ū of K and require the following assumptions on (P).
(A1) The functional J : A −→ R is of class C 2 .Furthermore, for every u ∈ K there exist continuous extensions The reader might have the impression that assumptions (A1) and (A2), mainly (A2), are too strong.However, we will see in the next sections that they are fulfilled by many optimal control problems.
Associated with ū, we define the sets (2.6) where cl 2 (S ū) denotes the closure of S ū in U 2 .The set S ū is called the cone of feasible directions and C ū is said to be the critical cone.Now, we formulate the necessary second order optimality conditions under a regularity assumption stated in the next theorem; we refer to [2, section 3.2] or [9] for the proof.
Theorem 2.2.Let ū be a local solution of (P) in U ∞ .Assume that (A1) and the regularity condition Let us mention that the regularity assumption of the above theorem is equivalent to the notion of polyhedricity of K; see [1] or [2, section 3.2].
Finally, we prove a theorem on sufficient second order optimality conditions.Its novelty is that the obtained quadratic growth condition holds in a U 2 -neighborhood of ū rather than only in a U ∞ -neighborhood.
Theorem 2.3.Suppose that assumptions (A1) and (A2) hold.Let ū ∈ K satisfy (2.1) and Then, there exist ε > 0 and δ > 0 such that Proof.We argue by contradiction and assume that for any positive integer k there exists u k ∈ K such that (2.9) From assumption (A1) and (2.1) we deduce We also derive the converse inequality.Due to the definition of v k and (2.3), we have for some θ k ∈ (0, 1) Hence, (2.9) leads to Invoking again (2.9) and (2.1) we get by a Taylor expansion Therefore, it holds that Using first (2.7) and then (2.4), the above inequality leads to ), it follows that v = 0. Finally, using (2.5) and the fact that v k 2 = 1, we get the contradiction as follows: Remark 2.4.The main novelty in the proof is the use of assumptions (2.4) and (2.5).They generalize the requirement of other papers that J (ū) is a so-called Legendre form.In former contributions to the subject it was not known that assumption (2.4) can be deduced in the context of control theory by an application of Egorov's theorem.Therefore, they needed the U ∞ -convergence of the sequence {u k } ∞ k=1 .In our approach, a generalization of the Legendre quadratic form hypothesis was necessary to achieve the final contradiction in the precedent proof; (2.5) was developed in this way.We refer to the first author's paper [5].For the use of Legendre forms, the reader is referred to Hestenes [12] and Ioffe and Tihomirov [14].
To explain more specific difficulties related to second order conditions, we consider also a modified version of example (Ex 2 ).
Even more, though there exists a continuous extension J (u) ∈ B(L 2 (0, 1), R) for every u ∈ L ∞ (0, 1), the continuity property J (u k ) → J (ū) in B(L 2 (0, 1), R) does not hold for every sequence {u k } ∞ k=1 that is bounded in L ∞ (0, 1) and converges strongly in L 2 (0, 1) to ū.Indeed, it suffices to consider u k and v k defined by We have This proves the lack of the continuity property J (u k ) → J (ū) in B(L 2 (0, 1), R).However, if this property would be satisfied, then the assumptions (2.2)-(2.5)can be simplified, as one of the referees suggested.We can substitute them as follows: (A2 ) (i) For any sequence Theorem 2.3 holds under assumptions (A1) and (A2 ).Indeed, it is obvious that (A1) and (A2 ) imply (2.3) and (2.4).Therefore, the proof is the same except for the final contradiction that can be obtained as follows.First, using the notation of the former proof, we observe that (A2 )-(i) implies Since J (ū) is a Legendre form, we deduce that v k → 0 strongly in U 2 .This contradicts the fact that v k U2 = 1 holds for all k.
Unfortunately, the assumption (A2 )-(i) is too restrictive.It does not hold for the simple problem (Ex 3 ) and it also fails for optimal control problems with highly nonlinear terms in the cost functional or in the state equation.The abstract framework given by assumptions (A1) and (A2) has a wider range of applications.
As an important consequence of Theorem 2.3, we are able to show local uniqueness of stationary points in the sense of L 2 .Recall that ũ ∈ K is said to be a stationary point if Proof.We prove the assertion by contradiction.Assume that there exists a sequence {u k } ∞ k=1 ⊂ K such that u k → ū in U 2 , u k = ū for all k, and J (u k )(u−u k ) ≥ 0 for every u ∈ K.Then, using the quadratic growth condition (2.8) and performing a Taylor expansion of J(ū) around u k , we get we deduce from the inequality above that If we are able to show v ∈ C ū, then the above inequality contradicts (2.7) and the proof is complete.Let us prove this.Obviously, v k belongs to S ū for every k, hence we have v ∈ cl 2 (S ū), since cl 2 (S ū) is convex and closed in U 2 .Let us check that J (ū)v = 0. From (2.1) we get J (ū)v k ≥ 0. Therefore, the inequality J (ū)v ≥ 0 follows from (2.3).On the other hand, J (u k )v k ≤ 0 follows from the definition of u k .Invoking again (2.3), we obtain J (ū)v ≤ 0, which completes the proof.
Assumption (2.7) has another consequence that was known up to now only in an L ∞ -neighborhood of ū.The result expresses some alternative formulation of secondorder sufficient conditions that is useful for applications in the numerical analysis.
Theorem 2.7.Under the assumptions of Theorem 2.3, there exist a ball B 2 (ū; ε) in U 2 and numbers ν > 0 and τ > 0 such that where We argue again by contradiction.Assume the existence of a sequence , and k hold for all k.Therefore, (2.2) and (2.4) imply J (ū)v = 0 and J (ū)v 2 ≤ 0. It is also clear that v ∈ cl 2 (S ū).It follows that v ∈ C ū and, as a consequence of (2.7), v = 0. Finally, taking into account that v k 2 = 1, we obtain the contradiction from (2.5): Remark 2.8.Suppose that assumptions (A1) and (A2) are changed as follows: Downloaded 05/23/13 to 193.144.185.28.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1.In (A1), the relations (2.2) are required to hold for every u ∈ A. 2. In (A2), all properties are required for all sequences {u k } ∞ k=1 converging to ū and belonging to A instead of K. Then (2.10) holds for every element u ∈ A ∩ B 2 (ū; ε).The same proof remains valid just by changing K by A.
This extension to the open set A can be important in cases where the sequence {u k } cannot be required to be in K.For instance, this might be interesting for numerical discretizations.
In what follows, we demonstrate the applicability of our results to PDE constrained optimal control problems.

Application I. An elliptic Neumann control problem.
In this section we study the optimal control problem where where −∞ < α < β < +∞, and y u is the solution of the following Neumann problem: Hereafter, ν(x) denotes the unit outward normal vector to Γ at the point x and ∂ ν y is the normal derivative of y.We impose the following assumptions on the functions and parameters appearing in the control problem (P 1 ).Assumption (N1).Ω is an open, bounded, and connected subset of R n , n ≥ 2, with Lipschitz boundary Γ and f : R −→ R is a function of class C 2 such that f (t) ≥ c o > 0 for all t ∈ R. The reader is referred to [7] for more general nonlinear terms in the state equation.
Assumption (N2).We assume that L : Ω × R −→ R and l : Γ × R × R −→ R are Carathéodory functions of class C 2 with respect to the second variable for L and with respect to the second and third variables for For every M > 0 there exist functions ψ M ∈ L p(Ω), p > n/2, and φ M ∈ L q(Γ), q > n − 1, and a constant are satisfied for a.a.x ∈ Ω and every u, y ∈ R, with |y| ≤ M and |u| ≤ M .Downloaded 05/23/13 to 193.144.185.28.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.phpMoreover, for every ε > 0 there exists η > 0 such that for a.a.x ∈ Ω and all Here D 2 (y,u) l(x, y, u) denotes the Hessian matrix of l with respect to the variables (y, u).
We also assume the Legendre-Clebsch type condition We should mention that the frequently used function On the state equation (2.1), the following result is known.Theorem 3.1.Under assumption (N1), for every u ∈ L q(Γ) (3.2) has a unique solution

respectively.
The proof of the existence and uniqueness of a solution y u in H 1 (Ω) ∩ L ∞ (Ω) is standard; see, for instance, [3].For the continuity of y u , the reader is referred to [11] or [17].Let us show for convenience the differentiability of G.We set It is known that, given y ∈ W 1,r (Ω) such that Δy ∈ L r (Ω), 1 < r < +∞, one can define ∂ ν y ∈ W −1/r,r (Γ); see [6].Therefore, V is well defined for r = min{p, 2}.Endowed with the graph norm, V is a Banach space.Moreover, we deduce from [11] or [17] that V is embedded in C( Ω).Now, we consider It is easy to check that F is C 2 , F (y u , u) = (0, 0) for every u ∈ L q(Γ) and that defines an isomorphism.Now the implicit function theorem yields that G is of class C 2 and (3.4) and (3.5) are fulfilled.In view of this theorem, the chain rule applies to show the following result.Downloaded 05/23/13 to 193.144.185.28.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.phpTheorem 3.2.If the assumptions (N1) and (N2) are satisfied, then the mapping J : L ∞ (Γ) −→ R, defined by (3.1), is of class C 2 .For all u, v, v 1 and v 2 of L ∞ (Γ) we have where Remark 3.3.From the above expressions for J (u) and J (u) and assumption (N2) we deduce that J (u) and J (u) can be extended to linear and bilinear forms, respectively, on L 2 (Γ).
Indeed, since u ∈ L ∞ (Γ), then set y u ∈ H 1 (Ω) ∩ C( Ω).In particular, there exists M > 0 such that ||u|| ∞ ≤ M and y u ∞ ≤ M holds.Moreover, (3.8) yields where C is independent of M , y, and u.We also know All these estimates ensure the existence of two constants, M 1 > 0 and M 2 > 0, such that for every Furthermore, the constants M i can be taken the same for every u belonging to a bounded set of L ∞ (Γ).
We now demonstrate that Theorem 2.3 can be applied to the problem (P 1 ).To this end, we set , along with the boundedness in L ∞ (Γ), implies that u k → u in L q (Γ) for every 1 ≤ q < +∞.In particular this is true for q.Therefore, invoking Theorem 3.1 we get Using this fact in (3.8), we deduce, with the help of assumption (N1), that ϕ u k → ϕ u in H 1 (Ω) ∩ C( Ω).From (3.4) we also know that In view of these convergence properties Downloaded 05/23/13 to 193.144.185.28.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.phpand assumptions (N1)-(N2), we easily obtain (2.3).Now we consider the expression for J (u k )v 2  k and observe that it is easy to pass to the limit in all the integral terms except in the last.To confirm (2.4) we apply Lemma 3.5 stated below for X = Γ, μ = σ, and Then we deduce from (3.9) that lim inf Together with the previous comments, the above equality confirms (2.4).Let us prove (2.5).Since v = 0, then all the integral terms of J (u k )v 2 k tend to zero, except the last.To get (2.5), we use (3.3) and find Lemma 3.5.Let (X, Σ, μ) be a measure space with μ(X) < +∞.Suppose that satisfy the following assumptions: , it holds that g ∈ L ∞ (X).Denote the lower limit in (3.9) by λ.Then there exists a subsequence of functions, denoted in the same way, such that the integrals of the right-hand side of (3.9) converge to λ.Again, we can select a new subsequence of this such that g k (x) → g(x) a.e. in X.Let ε > 0 be arbitrary.By Egorov's theorem there exists a measurable set Finally, passing to the limit ε → 0 we get (3.9)After having verified all the necessary assumptions, we are justified to apply Theorems 2.2 and 2.3 to the problem (P 1 ).Given ū ∈ K, we see that the cone of critical directions C ū defined in section 2 can be expressed for the problem (P 1 ) in the form and ȳ = y ū and φ = ϕ ū denote the state and adjoint state associated to ū, respectively.Let us check this claim.We have to prove that the defined cone C ū coincides with the set {v ∈ cl 2 (S ū) : J (ū)v = 0} denoted by E ū for a while.We recall the following well-known property of the optimal control (see, e.g., [18,Lemma 2.26]): The set of elements of L 2 (Γ) enjoying this property is closed; consequently v also has this property.Therefore, from the above property of ū we get Now, we prove the converse inclusion.We will even get more, namely This implies that the regularity assumption of Theorem 2.2 is satisfied: 0 a.e. in Γ.Thus, we have that every v k belongs to D ū, which leads to v ∈ cl 2 (D ū), as desired.
Corollary 3.6.Let assumption (N1) be satisfied and suppose that ū is a local minimum of (P 1 ) in the L ∞ (Γ) sense.Then there holds J (ū)(u− ū) ≥ 0 for all u ∈ K and then there exist ε > 0 and δ > 0 such that Under assumption (N1) and (N2), there exists a ball B 2 (ū; ε) in L 2 (Γ) such that there is no other stationary point in B 2 (ū; ε)∩K than ū.Moreover, there exist numbers ν > 0 and τ > 0 such that Observe that the above cone C τ ū is not equal to the cone E τ ū defined in Theorem 2.7.However, if v ∈ C τ ū , then Thus, we have that C τ ū ⊂ E τΓ ū , with τ Γ = τ |Γ|.Hence, Theorem 2.7 can be applied.Let us underline that the mapping G is differentiable only in L q (Γ) for q > n − 1.For all n ≥ 3, G is not differentiable in L 2 (Γ).Even if the objective functional J were quadratic, the classical theory of second order conditions would assure only local optimality in the sense of L ∞ (Γ).The general nonlinear cost functional J is differentiable only in L ∞ (Γ).Hence, for any dimension n, the classical theory of second order conditions would assure only the local optimality of ū in the L ∞ (Γ) sense.In contrast to this, our result guarantees local optimality in the sense of L 2 (Γ).Let us recall a well-known fact.Since K is bounded in L ∞ (Γ), then ū is a (strict) local solution of (P 1 ) in the sense of L 2 (Γ) if and only if it is a (strict) local solution of (P 1 ) in the sense of L r (Γ) for all 1 ≤ r ≤ ∞.
where −∞ < α < β < +∞ and y u is the solution of the state equation The following hypotheses are assumed about the functions involved in the control problem (P 2 ).Assumption (D1).We assume that y d ∈ L p(Ω), with p ≥ 2 and p > n/2, and Λ > 0.
Assumption (D2).The function f : R −→ R is of class C 2 and f (t) ≥ 0 for all t ∈ R.
As usual, we will say that an element y u ∈ L ∞ (Ω) is a solution of ( Now, given an optimal control ū with associated adjoint state φ, we define d = Λū − ∂ ν φ.Then, the critical cone C ū is defined as for problem (P 1 ) and the analogous versions of Corollaries 3.6 and 3.7 hold for problem (P 2 ).The reader should notice that the mapping G(u) = y u is not differentiable, probably even not well defined in L q (Γ) for any q < ∞.Therefore, the use of L ∞ (Γ) as control space is crucial.Once again, the classical theory of second order conditions is improved by assuring the strict local optimality of ū in the sense of L 2 (Γ) under the standard second order optimality conditions.L(x, t, y u (x, t), u(x, t)) dxdt,

Application
where Ω T = Ω × (0, T ) and y u is the solution of the state equation Here, Σ T = Γ × (0, T ).We impose the following assumptions on the functions and parameters appearing in the control problem (P 3 ).Assumption (P1).y 0 ∈ C 0 (Ω) and the function f : Ω T ×R −→ R is a Carathéodory function of class C 2 with respect to the second variable and satisfies the conditions where q, p ∈ [1, +∞] and respect to the last two variables and L(•, •, 0, 0) belongs to L 1 (Ω T ).For every M > 0, there exist a function For every ε > 0 there exists η > 0 such that for a.a.(x, t) ∈ Ω T and all u i , y i ∈ R, with i = 1, 2.Here D 2 (y,u) L(x, t, y, u) denotes the Hessian matrix of L with respect to the variables (y, u).

respectively.
The reader is referred to [4] for the proof of the existence of a unique solution in L 2 ([0, T ], H 1 0 (Ω)) ∩ C( ΩT ); see also [18,Theorem 5.5].For the proof of the differentiability we can proceed analogously to the proof of Theorem 3.1.We set endowed with the graph norm.Defining we can apply again the implicit function theorem to deduce (5.4) and (5.5).Theorem 5.2.If the assumptions (P1) and (P2) are satisfied, then the functional J : L ∞ (Ω T ) −→ R, defined by (5.1), is of class C 2 .For all u, v, v 1 , and v 2 of L ∞ (Ω T ) we have v weakly in L 2 (Ω T ).Then (2.3)-(2.5)are satisfied.Proof.We follow the steps of the proof of Proposition 3.4.First, we mention that the convergence of {u k } ∞ k=1 in L 2 (Ω T ) along with the boundedness in L ∞ (Ω T ) imply that u k → u in L p([0, T ], L q(Ω)).Applying Theorem 5.1 we get that y u k = G(u k ) → G(u) = y u strongly in L 2 ([0, T ], H 1 0 (Ω)) ∩ C( ΩT ).Invoking assumption (P1), we obtain from (5.5) that ϕ u k → ϕ u in L 2 ([0, T ], H 1 0 (Ω)) ∩ C( ΩT ).Now (5.4) implies that z v k = G (u k )v k → G (u)v strongly in L 2 ([0, T ], H 1 0 (Ω)) ∩ C( ΩT ).These convergence properties and assumptions (P1) and (P2) yield (2.3).We now proceed as in the proof of Proposition 3.4.The only delicate term for passing to the limit is the last in the expression for J (u k )v 2 k .Inequality (2.4) follows from Lemma 3.5, where we set X = Ω T , μ is the Lebesgue measure in Ω T , and 0 < Λ ≤ g k (x, t) → g(x, t) in L 1 (Ω T ), with Now we can apply Theorem 2.3 to problem (P 3 ).For given ū ∈ K, the cone of critical directions C ū defined in section 2 admits for (P 1 ) the form Here, ȳ = y ū and φ = ϕ ū are the state and adjoint state associated with ū.As for the elliptic control problem (P 1 ), the cone of critical directions C ū coincides with the one defined in section 2 and the regularity condition of Theorem 2.2 holds.Therefore, analogous Corollaries 3.6 and 3.7 hold for the control problem (P 3 ).Once again G is not differentiable in L 2 (Ω T ) for all n > 1 and, because of its general form, the cost functional J is only differentiable in L ∞ (Ω T ).Therefore, the classical theory of second order conditions would assure only the local optimality of ū in the L ∞ (Ω T ) sense.However, our result ensures local optimality in the sense of L 2 (Ω T ) in all cases.

4 .
Application II.An elliptic Dirichlet control problem.In this section, we assume that Ω ⊂ R n is an open domain whose boundary Γ is of class C 1,1 .In this domain we formulate the following control problem:
III.A parabolic distributed control problem.Now we consider the distributed control problem (