Second Order Optimality Conditions for Weak and Strong Local Solutions of Parabolic Optimal Control Problems

Second order sufficient optimality conditions are considered for a simplified class of semilinear parabolic equations with quadratic objective functional including distributed and terminal observation. Main emphasis is laid on problems where the objective functional does not include a Tikhonov regularization term. Here, standard second order conditions cannot be expected to hold. For this case, new second order conditions are established that are based on different types of critical cone. Depending on the choice of this cone, the second order conditions are sufficient for local minima that are weak or strong in the sense of calculus of variations.


Introduction
In this paper, we survey second-order optimality conditions for the following slightly simplified class of optimal control problems: subject to the parabolic state equation ∂y ∂t (x, t) − ∆y(x, t) + R(x, t, y(x, t)) = u(x, t) in Q y(x, t) = 0 on Σ y(x, 0) = y 0 (x) in Ω (1.2) and to the pointwise control constraints a ≤ u(x, t) ≤ b for almost all (x, t) ∈ Q. (1.3) Here, Ω ⊂ R N , N ∈ N, is a bounded Lipschitz domain with boundary Γ ; T > 0 is a fixed terminal time and we set Q = Ω × (0, T ), and Σ = Γ × (0, T ).
The function y, the state of the control system, is defined as weak solution of (1.2), while the control function u is assumed to be bounded and measurable. The nonlinearity R : Q × R → R ("reaction term") is assumed to be a Carathéodory function of class C 2 with respect to the second variable and satisfies the following assumptions R(·, ·, 0) ∈ L p (Q), p > N/2 + 1, and ∃C R ∈ R such that    ∀ρ > 0 ∃ε > 0 such that ∂ 2 R ∂y 2 (x, t, y 2 ) − ∂ 2 R ∂y 2 (x, t, y 1 ) < ρ ∀(x, t) ∈ Q and |y 2 − y 1 | < ε.
(1. 6) These assumptions include in particular functions R : R −→ R of class C 2 that are monotone non-decreasing, a standard case in the optimal control of semilinear parabolic equations. Moreover, polynomials of odd order, R(y) = a k y k + . . . + a 1 y + a 0 with k = 2n−1, n ∈ N, and a k > 0 are allowed. Then R is an even order polynomial and hence bounded from below. Highly nonlinear functions, as R(y) = exp(y), can be also considered. Further, y 0 is a given initial state belonging to L ∞ (Ω).
We discuss second-order conditions that are sufficient for local optimality of stationary solutions of the problem. Stationary solutions are pairs (ȳ,ū) that satisfy the first-order necessary optimality conditions of the control problem (1.1)-(1.3). For non-vanishing Tikhonov regularization parameter ν > 0, associated results are known since years, we refer to our recent survey paper [12] and the references therein.
Our main interest, however, is the case ν = 0, where the objective functional in (1.1) does not contain the last integral that is often called a Tikhonov regularization term; see Corollaries 3 and 4. We became interested in this degenerate case by numerical observations: The numerical solution method of optimal control problems for the more general FitzHugh-Nagumo system, a well known system of mathematical physics that includes a second linear partial differential equation, turned out to be surprisingly stable for very small regularization parameters ν > 0. Eventually, this observation brought us to the ques tion, whether we can prove stability of locally optimal solutions as ν 0.
To answer this question, we developed second-order sufficient optimality conditions for local solutions that are strong in the sense of calculus of variations and also work for the degenerate case ν = 0. Our first associated result is presented in [10]. It is valid for the FitzHugh-Nagumo system and can even be applied to sparse optimal controls, where the objective functional includes in addition the L 1 (Q)-norm of the control function.
To our best knowledge, the first result on strong local minima in PDE control was recently obtained in [1] for the case of semilinear elliptic equations. Moreover, an unknown referee called our attention to the preprint [2] on a parabolic control problem.
Although our results are similar to the ones of [1], [2], they are more general than those of [1], [2]. In particular, we admit the case of a vanishing Tikhonov regularization parameter ν, while ν > 0 is required in [1], [2].
In our paper, we introduce a new extended cone of critical directions E τ u prior to Theorem 9. This cone can be used to deal with the issue of stability for vanishing Tikhonov regularization parameter ν, cf. more detailed remarks in the introduction to Section 3.
On the one hand, the results of [10] are more general than the results we will present in this paper, because they include a nondifferentiable sparsity term. On the other hand, the reaction term R was assumed to be a special third-order polynomial and the domain Ω was assumed to have a dimension not larger than three. Moreover, due to the more complicated form of the FitzHugh-Nagumo system, the analysis in [10] is very technical. Therefore, we present our theory of second-order conditions here for a simpler equation but with more general nonlinearity R and for all dimensions N ∈ N.

Well posedness of state equation and control problem
We begin our analysis by recalling known results on the solvability of the state equation (1.2). We discuss existence and uniqueness of the state y for given u and prove differentiability properties of the control-to-state mapping G : u → y. Adopting a standard notation we set For the convenience of the reader we recall that a function y ∈ W (0, T ) ∩ L ∞ (Q) is said to be a weak solution of (1.2), if Q −y ∂v ∂t holds for all v ∈ H 1 (0, T ; H 1 0 (Ω)) such that v(·, T ) = 0. The following theorem can be shown in the same way as [4, Theorem 1].
Theorem 1 Under our assumptions, for all u ∈ L p (Q) with p > N/2+1, the equation (1.2) has a unique weak solution yu ∈ W (0, T ) ∩ L ∞ (Q). There exists a constant C independent of u such that and we have yu ∈ C(Ω × (0, T ]). If y 0 is in addition continuous inΩ, then yu belongs to C(Ω × [0, T ]).
The main idea of the proof is the use of the substitution y(x, t) = e λt v(x, t) with sufficiently large parameter λ, cf. [4]. In this way, an equation with monotone nonlinearity is obtained where the known results on existence, uniqueness, and regularity [16,Theorem 5.5] or in [5] can be applied.
Following [11], cf. also [9], we introduce the control-to-state mapping G : Then the following differentiability properties can be proved, see [4] or the more general version [9, Theorem 2.2].
Theorem 2 (Differentiability of the control-to-state mapping) The mapping G is of class C 2 . The derivative ηv := G (u)v is equal to the function η that solves the linearized equation Remark 1 The operator G is not differentiable from L 2 (Q) to W (0, T ). However, G (u) and G (u) can be extended to continuous linear and bilinear mappings from L 2 (Q) to W (0, T ), for any u ∈ L p (Q) with p > N/2 + 1.
Since yu is uniquely determined by u, we can formulate the optimal control problem (1.2)-(1.3) in the following control reduced form: where the (control reduced) objective functional is defined by and the set of admissible controls U ad is Thanks to Theorems 1 and 2, it is easy to prove that for all ν ≥ 0 the control problem (Pν ) has at least one solutionūν ; see [9,Theorem 3.1]. In view of the differentiability properties of G, also Jν is of class C 2 . In a standard way, the derivatives can be expressed by using an adjoint equation. For given u ∈ L p (Q), the equation is called the adjoint equation. Its solution ϕu is the adjoint state associated with u.
Theorem 3 The functional Jν : L p (Q) −→ R, p > N/2 + 1, is of class C 2 . Its first order derivative is given by

5)
where ϕu is the solution in W (0, T ) ∩ L ∞ (Q) of the adjoint equation (2.4). For directions v 1 , v 2 ∈ L p (Q), the second order derivative of Jν is The proof of the existence and uniqueness of the solution ϕu ∈ W (0, T ) ∩ L ∞ (Q) of the adjoint system and the formula (2.5) can be found in [9, §3.2]. The expression of the second derivative follows from the chain rule, (2.2), and (2.4).

First-Order Necessary Optimality Conditions
Since the control problem (Pν ) is not convex, we consider local minima. In this section we set up the associated first-order necessary optimality conditions and draw some conclusions from the optimality system.
We say thatūν is a local minimum of problem (Pν ) in the sense of L p (Q), 1 ≤ p ≤ +∞, if there exists ε > 0 such that Here, Bε(ūν ) denotes the L p (Q)-ball centered atūν with radius ε.
The boundedness of U ad in L ∞ (Q) implies thatūν is a local minimum in the L 2 (Q) sense if and only if it is a local minimum in the L p (Q) sense for any 1 ≤ p < +∞. On the other hand, ifūν is a local minimum in the L ∞ (Q) sense, then it is a local minimum in the L p (Q) sense for any 1 ≤ p ≤ +∞. Hereafter, local minima will be always understood as local minima in the L 2 (Q) sense. Notice that the control u is the image of yu of the differential operator y → ∂y/∂t − ∆y + R(·, ·, y). Therefore u plays the role of the derivative y of the unknown function x → y(x) in the classical calculus of variations. In this sense, the neighborhood Bε(ūν ) is a neighborhood that accounts also for the derivatives of y.
Later, we will also investigate conditions for local minima that are strong in the sense of calculus of variations. This means that Jν (ūν ) ≤ Jν (u) ∀u ∈ U ad such that yu − yū ν L ∞ (Q) < ε, no matter how far u is fromūν .
For the definition and discussion of weak and strong local minima in the classical calculus of variations, we refer to [15,Chpt. 2] and [18,Sect. 37.4e].
The following first order necessary optimality conditions have to be satisfied by any local minimum of (Pν ).
Theorem 4 Letūν be a local minimum of (Pν ), letȳν be the associated state, and let ϕν := ϕū ν ∈ W (0, T ) ∩ L ∞ (Q) be the associated adjoint state defined as the unique solution of (2.4) for u :=ūν . Thenūν obeys the variational inequality The well-known projection formula below follows by a standard discussion of the variational inequality (3.2).
Corollary 2 Letūν andφν be as in Theorem 4 and assume that ν = 0. Then the following implications hold, From the above relations we deduce that the optimal controlūν is bang-bang if the set of points of Q whereφν vanishes has a zero Lebesgue measure.
For these well known results, we refer the reader to [8, Corollary 3.2] and [6, Theorem 3.1] that deal with the cases ν > 0 and ν = 0, respectively, in a similar situation. Moreover, these implications are extensively explained in [16, p.70] for the elliptic case.

Second Order Optimality Conditions
Next, we develop the second order analysis for (Pν ) and begin with second order necessary conditions. Let a controlūν ∈ U ad fulfill the optimality conditions of Theorem 4. This means thatūν obeys, along with the adjoint stateφν , the variational inequality (3.2). Associated withūν , we introduce the standard cone of critical directions The next theorem is well known; the reader is referred to [3] or [11] for a general result.
Theorem 5 (Second order necessary condition) The set Cū ν is a convex and closed cone in L 2 (Q). Ifūν is locally optimal for (Pν ), then Now we turn over to second order sufficient conditions. Depending on different situations, besides Cū ν we will introduce two more cones of critical directions, namely the cones C τ u and E τ u . The cone Cū, considered for the necessary second order conditions, will also appear in the formulation of second order sufficient conditions for ν > 0 in Thm. 6. This is a result on weak local optimality. However, as a corollary, we obtain the surprising result that, in this case,ūν yields even a strong local minimum.
The two other cones are introduced for the case ν = 0. The second order conditions based on E τ u guarantee thatȳ affords a minimum to the objective functional that is strong in the sense of calculus of variations, cf. Theorem 9 and relation (4.35). This property of strong local optimality cannot be deduced, if we substitute C τ u for E τ u . In Section 5, we shall apply these different second order conditions to proving results on stability of optimal solution with respect to certain perturbations of the data in the optimal control problem.

The case ν > 0
The following properties of Cū ν will be needed.
Our next auxiliary result is important. The equivalence that is stated in the following lemma is not in general true for infinite-dimensional optimization problems. However, it holds true if the Tikhonov regularization term ν 2 u 2 L 2 (Q) is included in the cost functional with ν > 0; see [12] for the proof.

Lemma 2
The following statements are equivalent In the case of unconstrained optimization, the coercivity condition (4.5) can be required in the whole space and is sufficient for a local minimum. For a comparison with this case, we refer to [19,Sect. 2.2].
Finally, we recall the following result.
Compared with the classical calculus of variations, this result means thatū affords to Jν a weak local minimum. However, we are able to prove the surprising fact that the local optimality ensured by Theorem 6 is even strong.
Theorem 7 (ν > 0; Strong local optimality) Letūν satisfy all assumptions of Theorem 6. Then δ > 0 and ε > 0 exist such that Proof Assume the contrary, i.e. that (4.7) does not hold for any δ and ε . Then, for any integer k ≥ 1, we can find a control u k ∈ U ad with yu k −ȳν L ∞ (Q) < 1/k such that (4.8) We can select a subsequence, denoted in the same way, such that {u k } k≥1 is weakly * convergent to someũ ∈ L ∞ (Q). From the parabolic equation, we have The right-hand side converges weakly in L p (Q) to −R(x, t,ȳν ) +ũ. Therefore, by the weak continuity of the solution mapping for linear parabolic equations, the sequence {y k } converges weakly in W (0, T ) to a solutionỹ that satisfies From y k ỹ in W (0, T ) and y k →ȳν in L ∞ (Q), we findỹ =ȳν and henceũ =ū. In this way, u k * ūν in L ∞ (Q) is obtained. In particular, we also have that u k ūν in L 2 (Q).
By our notation and the convergence y k →ȳν in L ∞ (Q), we have J 0 (u k ) → J 0 (ūν ); hence we deduce from (4.8) Passing to the limit k → ∞, we arrive at This implies that u k 2 L 2 (Q) → ūν 2 L 2 (Q) holds in addition to the weak convergence. Therefore, the convergence of {u k } toūν is strong in L 2 (Q). Consequently, given > 0 such that (4.6) holds, we have that u k −ūν L 2 (Q) < ε for all k sufficiently large. Then (4.8) contradicts (4.6).
In the framework of optimal control of partial differential equations, the first result about strong local optimality was proved in [1] for the elliptic case.

The case ν = 0
Our main results on second order conditions for ν > 0 (except Theorem 7) were known since for some years, cf. [11]. In the degenerate case ν = 0, we found our results just recently in [10] for the FitzHugh-Nagumo system. Here, we adapt the ideas of [10] to our semilinear parabolic state equation with slightly more general nonlinearity R.
For general infinite-dimensional optimization problems, the strict positivity of the second derivative of the objective functional on the critical cone is not sufficient for local optimality. An associated example is known from [14]. (The situation is different for our objective functional Jν , if ν > 0. Then the second derivative generates a Legendre form (cf. [15, Chpt. 6.2.1]) and we were able to argue as in the preceding section). Therefore, we have to consider the well-known extended cone C τ u , cf. [13] for the elliptic case or [16, (5.42)] for a parabolic state equation. For given (small) threshold τ > 0, we define C τ u as the set of elements (4.9) The extended cone C τ u covers Cū, i.e. Cū ⊂ C τ u holds for all τ > 0. For every v ∈ L 2 (Q) satisfying the sign conditions (4.1), the inequality is fulfilled, where Qv denotes the set of points (x, t) ∈ Q such that the first condition of (4.9) is not satisfied by v(x, t).
Proof The inclusion Cū ⊂ C τ u is obvious. Let us prove the second claim. Ifφ(x, t) ≥ τ , then Corollary 2 implies thatū(x, t) = a and therefore v( The nonnegative of the first integral in the inequalities above follows from Lemma 1,(i). Now one might be tempted to formulate the second order sufficient conditions as ∃τ > 0 and ∃σ > 0 such that J (ū)v 2 ≥ σ v 2 L 2 (Q) ∀v ∈ C τ u . (4.11) Unfortunately, this condition cannot be expected for ν = 0, since the Tikhonov regularization term is missing. This term is needed to fulfill the condition above. Actually, the inequality (4.11) holds in a few very exceptional cases; see [6] and [10].
The following second order sufficient condition is adequate: Theorem 8 Letū ∈ U ad , along with the adjoint stateφ = ϕū, satisfy the variational inequality (3.2). Assume that where ηv = G (ū) v. Then, there exists ε > 0 such that where Bε(ū) is the ball of L 2 (Q) centered atū with radius ε.
The inequality (4.12) of the second order condition in Theorem 4.13 can be motivated by the form of the second order derivative J (ū) [v, v] in Theorem 3: If ν = 0, then the L 2 (Q)-norm of v 2 is missing, while the associated L 2 -norms of ηv and ηv(T ) are still present.
Before proving this theorem, we derive some auxiliary results.
Lemma 3 Assume that p > N/2 + 1. Then constants C a,b , C 1 , C 2 , C 3 , and C∞ exist such that, for all u ∈ U ad , the following estimates are satisfied: Proof The first estimate follows from Theorem 1 and the estimates for the solution of (2.4). To prove the second and third estimates, we we substitute (yu −ȳ)(x, t) = e µt w(x, t) with some µ ≥ −c R , see (1.4), and subtract the parabolic equations satisfied by yu andȳ. We obtain whereŷu(x, t) =ȳ(x, t) + θ(x, t)(yu(x, t) −ȳ(x, t)) and 0 ≤ θ(x, t) ≤ 1. In view of (1.4), e −µt ∂R ∂y (x, t,ŷu) + µ is a.e. nonnegative. Now a standard estimate for linear parabolic equations leads to an estimate of the type (4.15) for w . Transforming back by yu −ȳ = e µt w delivers the W (0, T )-estimate for yu −ȳ in (4.15).
To confirm the associated estimate for ϕu −φ, we consider the difference of the adjoint equations for ϕu andφ,
From the estimate (4.14) we know that all states yu are uniformly bounded. Moreover, we already have shown the W (0, T )-estimate for yu −ȳ in (4.15). In this way, we are able to bound the right hand sides of (4.19) against u −ū L 2 (Q) and to verify the W (0, T )-estimate for ϕu −φ in (4.15).
For the L ∞ -estimate (4.16), we recall that the L ∞ -norm of the solution of a linear parabolic equation with bounded coefficients can be estimated against the L p -norm of the right-hand side and to the L ∞ -norm of the initial data, provided that p > N/2+1, see [16,Thm. 5.5]. (This result has to be applied to the equations for e −µt (yu −ȳ) and e −µt (ϕu −φ), respectively.) In the terminal condition for ϕu−φ of (4.19), we use that yu(T )−ȳ(T ) L ∞ (Ω) ≤ yu −ȳ L ∞ (Q) . The L ∞ estimation of ϕu −φ in (4.16) is now straightforward.
Finally, we verify (4.17). To this aim, we set analogously w(x, t) = e −µt ηv(x, t). Then, w satisfies (4.18) with e −µt v in the right hand side andȳ substituted forŷu. Next, we fix µ large enough such that e −µt ∂R ∂y (x, t,ȳ) + µ ≥ 0 in Q. Multiplying the equation by w and integrating in Q we infer From here we get Transforming back by ηv(x, t) = e µt w(x, t), we obtain (4.17).
Lemma 4 There exist a constant Cη such that
For convenience of the reader we recall that, by the definition in Lemma 4, we have ηu,v = G (u)v while ηv = G (ū)v belongs to the fixed reference controlū.
Lemma 5 There exists a constant M a,b such that, for all u ∈ U ad and for all v 1 , v 2 ∈ L 2 (Q), the following estimate holds Proof This estimate follows easily from the expression (2.6) of J and (4.14).
We also need the following preparatory result: holds for all v ∈ L 2 (Q) and for all u ∈ U ad such that yu −ȳ L ∞ (Q) < ε.
Proof From (2.6), it follows Now we discuss the estimation of I 1 , I 2 , and I 3 . First, (4.20) yields that (4.26) From here, we also get (4.28) By (4.25) and (4.27), we estimate I 1 , I 3 is handled by (4.26) and (4.28) as follows Finally, we deal with I 2 by The first term is handled with the assumption (1.6) and (4.14). Thanks to (4.16), the second term can be estimated by C yu−ȳ L ∞ (Q) ηv 2 L 2 (Q) with some constant C . The third term is handled as I 1 . The statement of the lemma is a straightforward consequence of the obtained estimates.
Corollary 3 (ν = 0; Sufficient condition for a weak local solution) Under the assumptions of Theorem 8, there exist ε > 0 and δ > 0 such that
The growth condition (4.33) is valid in a ball aroundū, hence we obtained a result on local optimality in weak sense. We were not able to prove that, under these assumptions based on the cone C τ u , the solution is locally optimal in strong sense. To deal with this problem, we introduce another extended cone by E τ u = v ∈ L 2 (Q) satisfying (4.1) and From Lemma 1,(i) we infer that Cū ⊂ E τ u for every τ > 0. Thus the cone E τ u is a small extension of Cū. We are able to prove the following result on second order sufficiency that is based on E τ u : Theorem 9 Letū ∈ U ad , along with the adjoint stateφ, satisfy the variational inequality (3.2). Assume also that τ > 0 and σ > 0 exist such that ∀v ∈ E τ u .

(4.34)
Then, there exists ε > 0 such that Proof Proceeding completely analogous to [10], we obtain a constant M > 0 such that where M a,b was introduced in Lemma 5. From Lemma 6, we deduce the existence of ε 2 > 0 such that ∀u ∈ U ad with yu −ȳ L ∞ (Q) < ε 2 With these prerequisites, we are able to verify (4.35) with ε = min{ε 1 , ε 2 }. To this aim, we select u ∈ U ad such that yu −ȳ L ∞ (Q) < ε and distinguish between two cases.
u . This is the case, where J (ū)(u −ū) is sufficiently big to ensure local optimality without the coercivity assumption (4.34).
Here we estimate as follows: Here, the term J (ū)(u −ū) is so small that the coercivity condition (4.34) has to be invoked.
This time, we use (3.2), (4.34) and (4.37) to get Corollary 4 (ν = 0; Sufficient condition for a strong local minimum) Ifū satisfies the assumptions of Theorem 9, then there exist δ > 0 and ε > 0 such that The proof of this corollary is completely analogous to that of Corollary 3. We take ε 0 and C 3 as in the proof of Corollary 3, and δ and ε as in the statement of Theorem 9. Then the inequality (4.38) follows by substituting ε for min{ε 0 , ε} and δ for δ/C 3 .

Applications to the Stability Analysis with Respect to Perturbations
In this section we explain how our second order sufficient conditions for strong local solutions can be applied to proving stability estimates for the local solution with respect to certain perturbations in the data of our optimal control problem. Here, we concentrate on the case ν = 0, where the second-order sufficient optimality conditions known from earlier papers cannot be applied.
First, we consider perturbations of the desired state y Q and second the behavior of the local solution for the Tikhonov regularization parameter ν tending to zero. The second problem is also an issue of stability analysis, because a small Tikhonov parameter can be viewed as a perturbation of the reference parameter ν = 0.

Perturbation of y Q and y T
Assume that, for all ε > 0 a perturbed desired state y ε Q ∈ L p (Q), p > N/2 + 1, and a perturbed desired final state y ε T ∈ L ∞ (Ω) are given such that holds with some constant C > 0.
Associated with these perturbed target functions, we define the family of perturbed objective functionals and consider the family of perturbed optimal control problems Notice that the perturbed functionals do not include a Tikhonov regularization term, i.e. we have ν = 0.
It follows by the same arguments as for the unperturbed problem that for each ε > 0 at least one optimal controlū ε exists. We denote byȳ ε the associated optimal state and investigate the behaviorȳ ε for ε 0.
First, we quote a result on convergence from [9] that does not yet contain information on the rate of convergence. Notice that the set U ad is bounded. Therefore, a subsequence ofū ε can be selected that converges weakly in L 2 (Q). By selecting a subsequence if necessary, we can assume weak convergence of {ū ε }ε.
Theorem 10 (ν = 0; Convergence for ε 0) If {ū ε }ε is any sequence of optimal controls of the problems (P ε ) that converges weakly in L 2 (Q) to someū, thenū is optimal for (P) and lim ε 0 Conversely, letū be a strict locally optimal control of (P). Then there exists a sequence {ū ε }ε of locally optimal controls of (P ε ) converging weakly toū. For this sequence, (5.3) holds as well. Furthermore, a radius ρ > 0 exists such that everyū ε affords a global minimum to J ε with respect to all elements u ∈ U ad such that yu −ȳ ∞ ≤ ρ.
We refer to the proof in [10]. This theorem ensures the convergence ofȳ ε toȳ as ε 0. The associated rate of convergence is the subject of the next result.
Theorem 11 (ν = 0; Lipschitz stability for ε 0) Assume thatū is a locally optimal control of (P) that satisfies the second order sufficient optimality condition (4.34). Let {ūε} be a sequence of locally optimal controls of (P ε ) that converges weakly toū in L p (Q) as ε 0 and has the properties stated in Theorem 10; denote the associated states byȳ andȳε, respectively. Then a number C > 0 exists such that Proof In view of the assumed properties of the sequence {ū ε } we have ȳε − y L ∞ (Q) → 0 as ε 0. Consequently, for all sufficiently small ε > 0,ȳε is so close toȳ that the quadratic growth condition (4.38) is satisfied by Corollary 4. Invoking this growth condition, we proceed as follows: By shifting the the first and the last two terms to the left-hand side, this inequality becomes The left side is an integral of various differences of squares. Expanding the squares, it can be considerably simplified. For instance, the tracking terms on Q are handled as follows: in view of (5.1). In the same way, the integrals on Ω are simplified to finally obtain Now we invoke (5.5) to conclude The desired result follows immediately.
Remark 3 Assume that the locally optimal controlū considered in Theorem 10 is bang-bang. Then the second order sufficient optimality condition (4.34) -as one of the assumptions of Theorem 11 -can be replaced by the condition (4.12) and the result of the theorem remains true.
Indeed, ifū is bang-bang andūε ū in L 2 (Q), thenūε →ū strongly in every space L p (Q) for p < +∞; see [7,Theorem 4.4]. Therefore,ūε belongs to any L pneighborhood Br(ū) provided that ε is sufficiently small. Then the assumptions of Corollary 3 are satisfied so that the quadratic growth condition (4.33) applies (substitute Br(ū) for Bε(ū) there). In view of that, the estimate (5.5) in the proof above holds for sufficiently small ε > 0.

Tikhonov parameter tending to zero
As a further application of second order optimality conditions for strong local minima, we investigate the behavior of a sequence of optimal controls {ūν } ν>0 of (Pν ) and the corresponding states {ȳν } ν>0 as ν 0.
Again, by the boundedness of U ad , any sequence of solutions of (Pν ) contains subsequences converging weakly in L 2 (Q).
Again, this is a fairly standard result. We refer the reader to [10].
We do not know the rate of convergence of global (local) solutionsūν of (Pν ) to global (local) solutionsū of (P). Notice that the Tikhonov parameter ν vanishes in the limit, hence it should be difficult to find such a rate.
Instead, we are able to estimate the rate of convergence for the associated state functions, if certain second order sufficient optimality conditions are satisfied atū.
It might be surprising that both of the coercivity conditions (4.12) or (4.34) can be used this time. The reason is that, sinceū andūν are requested to be as in Theorem 13, here the strong convergenceūν →ū is assumed while Theorem 11 was based only on weak convergence of {ūν }.
In [17] the reader can find a deeper analysis of the stability when the Tikhonov parameter ν goes to zero in the case of a control problem of a linear elliptic partial differential equations. The extension of their results to the case of nonlinear partial differential equations is an open issue.