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SECOND ORDER OPTIMALITY CONDITIONS FOR OPTIMAL CONTROL OF QUASILINEAR PARABOLIC EQUATIONS

. We discuss an optimal control problem governed by a quasilinear parabolic PDE including mixed boundary conditions and Neumann boundary control, as well as distributed control. Second order necessary and suﬃcient optimality conditions are derived. The latter leads to a quadratic growth condi- tion without two-norm discrepancy. Furthermore, maximal parabolic regularity of the state equation in Bessel-potential spaces H − ζ,pD with uniform bound on the norm of the solution operator is proved and used to derive stability results with respect to perturbations of the nonlinear diﬀerential operator.


1.
Introduction. This article is concerned with optimal control problems governed by quasilinear parabolic partial differential equations (PDEs). Our goal is twofold. First, we aim at establishing second order optimality conditions with minimal gap between necessary and sufficient conditions. Second, we are interested in the precise regularity of the state equation, which is crucial for, e.g., Lipschitz stability estimates. We use the theory to study perturbations of the problem with respect to the nonlinearity, where we rely both on second order optimality conditions and the improved regularity. The prototypical problem with control q and state u is with ξ being a scalar function, and µ is a spatially dependent coefficient function. Boundary conditions are implicitly included in the definition of the differential operator in (1.1b); see Section 2 for the precise assumptions. Optimal control of quasilinear parabolic PDEs of this type has many important applications, for example heat conduction in electrical engineering [44] and semiconductors [57]. As we will see, both distributed control in two and three spatial dimensions as well as Neumann boundary control in two dimensions is included in this setting. Boundary control in three dimensions can be considered for purely time-dependent controls.
In the literature there are many contributions to optimal control of nonlinear parabolic equations that may be distinguished by the differential operator being either monotone or nonmonotone. Existence of solutions to optimal control problems governed by parabolic equations of monotone type has been proved in [46]. Necessary optimality conditions have further been established in [3,53,62]; cf. also the introduction in [31]. Less abstract hypotheses have been used in [14] to show first order optimality conditions. Concerning optimal control of nonmonotone parabolic equations, fewer results have been published. Existence of solutions has been considered in [50,51] for distributed control. The studies impose pointwise control constraints and the control enters nonlinearly in the state equation. More recently, first order necessary conditions for a quasilinear equation subject to integral state constraints have been proved in [31] with distributed controls in L 2 ((0, T ) × Ω). It is worth mentioning, that in the latter study all the coefficients of the elliptic operator may depend on u, ∇u and the control q. However, the derivatives of the coefficients have to satisfy certain growth bounds, so that our prototype problem does not comply with assumption (2.5) in [31]. We also refer to the recent work [12], where the authors prove first and second order optimality conditions for quasilinear parabolic control problems with distributed control with possibly unbounded nonlinearity, yet on smooth domains. Optimal control of semilinear parabolic equations, on the other hand, even with pointwise state constraints, is well-investigated; see for instance [19,20,43,56]. Concerning the case of quasilinear elliptic equations, first-and second order optimality conditions have been established in [13,16].
Recently, uniform Hölder estimates for linear parabolic equations subject to mixed boundary conditions and rough domains have been established in [49], which in turn implies that the state belongs to ). This is the starting point for our investigation. Adapting the ideas of Casas and Tröltzsch from [16,18], we prove second order necessary as well as sufficient optimality conditions for Neumann boundary control in spatial dimension two, and for purely time-dependent control and distributed control in dimensions two and three. The main difficulty is that this requires the first and second derivative of the reduced objective functional to be extended to L 2 (Λ, ), but the linearized state equation contains an additional term involving the gradient of u; cf. Proposition 4.4. We overcome this issue by a careful regularity analysis of the state equation based on the results in [23]; see Proposition 3.3.
Moreover, in applications it is often required to guarantee uniform boundedness of the solution operator to certain linearized-type equations, cf. Lemma 5.1. Once existence of a solution to the nonlinear state equation is established in an appropriate function space, improved regularity results can be transferred from the linear D (Ω) allows for distributional objects such as surface charge densities or thermal sources concentrated on hypersurfaces, cf. [39, Theorems 3.6 and 6.9]. We point out that in [39] the authors proved local-in-time existence of solutions in the H −ζ,p D (Ω) setting, but not existence on the whole time interval (0, T ). While the improved regularity is not needed for the second order optimality conditions, in Section 5 we discuss an application to stability analysis that uses both the improved regularity and the second order sufficient optimality conditions. The paper is organized as follows. In Section 2 we state the precise assumptions of the problem and collect specific examples of control settings that are covered by Problem (1.1). Maximal parabolic regularity on L s (I; W −1,p D ) (for second order optimality conditions) and L s (I; H −ζ,p D ) (for stability estimates) is proved in Section 3. Section 4 is devoted to the analysis of the optimal control problem including second order necessary and sufficient optimality conditions. In Section 5 we investigate stability of optimal solutions with respect to perturbations on ξ. Some interesting but technical results are collected in the appendix.
2. Notation and assumptions. We now give the precise assumptions concerning the geometry, the operators, and the problem data.

Notation.
For Ω ⊂ R d a Lipschitz domain, θ ∈ (0, 1], and p ∈ (1, ∞), we define the space H θ,p D (Ω) as the closure of C ∞ D (Ω) = ψ| Ω : ψ ∈ C ∞ (R d ), supp (ψ) ∩ Γ D = ∅ in the Bessel-potential space H θ,p (Ω), i.e. (Ω), where p stands for the conjugate Sobolev exponent, i.e. 1 = 1/p + 1/p . If ambiguity is not to be expected, we drop the spatial domain Ω from the notation of the spaces. The domain of a linear (possibly unbounded) operator A on a Banach space X is denoted by D X (A). As usual R(z, A) = (z − A) −1 denotes the resolvent of an operator A.
By the symbol M d (µ • , µ • ) we denote the set of measurable mappings µ : Ω → R d×d having values in the set of real-valued matrices which satisfy the uniform ellipticity condition (iii) We do not exclude the cases Γ D = ∅ or Γ D = ∂Ω. (iv) The additional requirement of volume-preserving bi-Lipschitz transformations is satisfied in many practical situations. In spatial dimension three, two crossing beams allow for a volume-preserving bi-Lipschitz transformation; see Section 7.3 in [39]. In particular, domains with Lipschitz boundary satisfy Assumption 1; see Remark 3.3 in [39]. (v) Note that the assumption of volume-preserving φ x is only used for the existence result [49] and the characterization of H −ζ,p D to be an interpolation space between L p and W −1,p D ; see (3.8).
Assumption 2. Let ξ be real-valued, twice continuously differentiable, and ξ Lipschitz continuous on bounded sets. However, in many practical situations with d = 3 the isomorphism property holds with p > 3; see [24]. In particular, if Ω is of class C 1 , µ is uniformly continuous and Γ N = ∅, then Assumption 3 holds for all p ∈ (1, ∞). In the example of two crossing beams, cf. Remark 2.1, if µ is constant on each beam, then Assumption 3 holds with homogeneous Dirichlet boundary conditions. Last, we define the setting for the controls. To have one notation for different control situations, we define a measure space (Λ, ) and the control operator B mapping L ∞ (Λ, ) into L s (I; . Concrete examples are given below. Given a measure space (Λ, ), define the control space as Q = L ∞ (Λ, ) and the set of admissible controls as is linear and bounded. Moreover, B can be continuously extended to an operator The desired stateû and the initial value u 0 satisfyû ∈ L ∞ (I; L 2 ), and u 0 ∈ (H −ζ,p D , D H −ζ,p D (−∇ · µ∇)) 1− 1 s ,s , respectively, and λ > 0 is the regularization (or cost) parameter.
The extension property of the control operator stated in Assumption 4 is only needed for second order sufficient conditions. Therein, we require the linearized equation to be solvable for right-hand sides in L 2 (I; W −1,p D ). In order to ease readability, we denote the maximal restriction of −∇ · µ∇, A(u) and B, respectively, to any of the spaces occurring in this article by the same symbol.
For clarity, we summarize the differentiability and integrability exponents used throughout the paper in Table 1 and point out their approximate values. Before continuing the analysis, let us state typical situations covered by the general setting of Problem (1.1).
Example 2.4 (Neumann boundary control; d = 2). Given q a , q b ∈ L ∞ (I × Γ N ) define the control space and set of admissible controls as According to [39,Theorem 6.9] the adjoint of the trace operator Tr * is continuous from ). Note that for d = 3, we have to require 3/p − 1/2 ≥ θ and θ > 1/2, but 3/p − 1/2 < 1/2. This motivates the analysis of purely time dependent controls which are also interesting in practice, since distributed controls are usually difficult to implement; see [20] and references therein for applications.
The control space and the space of admissible controls, respectively, are given by where q a , q b ∈ L ∞ (0, T ; R m ). The inequality above is understood componentwise. We note that Assumption 4 is satisfied due to the continuous embedding H −ζ,p   3. The quasilinear parabolic state equation. We start with the discussion of existence and regularity for the state equation (1.1b). First we introduce the concept of maximal parabolic regularity for nonautonomous operators from [25,Definition 2.1]. Note that if the operator is time-independent, then the definition coincides with the usual notion of maximal parabolic regularity for autonomous equations; cf. [7, Section III. 1.5]. Thereafter, in Section 3.1 we summarize regularity results for right-hand sides in L s ((0, T ); W −1,p D ). We emphasize that this regularity is sufficient for the discussion of first and second order optimality conditions. However, the improved regularity of the state discussed in Section 3.2 is of independent interest and can be used in, e.g., the stability analysis with respect to perturbations in ξ as performed in Section 5.
Definition 3.1. Let X, Y be Banach spaces such that Y → d X, and [0, T ] t → A(t) ∈ L(Y, X) be a bounded and measurable map. Moreover, s ∈ (1, ∞) and A(t) is a closed operator in X for each t ∈ [0, T ]. Then A is said to satisfy maximal parabolic regularity on X, if for every f ∈ L s (I; X) and w 0 ∈ (X, Y ) 1−1/s,s there exists a unique solution w ∈ W 1,s (I; X) ∩ L s (I; Y ) satisfying where the time derivative is taken in the sense of X-valued distributions on I; see Chapter III.1 in [7].
Note that we will sometimes use the notion of maximal parabolic regularity on e.g. L s (I; X) whenever dealing with nonautonomous operators.  [23,Lemma 3.4]). Let X, Y be Banach spaces such that Y → X and s ∈ (1, ∞). If τ ∈ (1 − 1 s , 1), then If in addition Y → c X, then the embeddings above are compact as well. ). This subsection is devoted to maximal parabolic regularity of the nonautonomous operator A(u) on L s (I; W −1,p D ). To this end, we first consider the time-independent operator −∇ · µ∇ for an arbitrary coefficient function µ ∈ M d (µ • , µ • ).

Maximal parabolic regularity on
. Given any solution u to the state equation, we study maximal parabolic regularity of the nonautonomous operator ). This will particularly be useful for the stability analysis of (1.1) in Section 5. However, it is not needed for the second order optimality conditions in Section 4. To prove maximal parabolic regularity of A(u) on L s (I; H −ζ,p D ), we establish two ingredients: First, we show that −∇ · µ∇ is uniformly R-sectorial with respect to µ; see (3.12). Section 3.2.1 contains this result on L p , whereas Section 3.2.2 is concerned with R-sectoriality on W −1,p D . Second, we verify that A(u) satisfies the Acquistapace-Terreni condition; see (3.19) in Section 3.
, and the linear mapping is continuous.
Combining Proposition 3.2 and (3.9) yields the following embedding, where compactness is due to Corollary 3.7 ([39, Corollary 6.16]). If s > 2 1−ζ , then there is α > 0 such that For the following considerations, let Σ θ denote the open sector in the complex plane with vertex 0 and opening angle 2θ, which is symmetric with respect to the positive real half-axis, i.e.
Proof. Due to [11, Proposition 4.6 (ii), Theorem 11.5 (ii)], the operator −∇ · µ∇ + 1 admits bounded H ∞ -calculi on L p and W −1,p where µ • (µ) is the coercivity constant of µ. In particular, the spectra are contained in a sector Σ θ . Using (3.8) holds. The smallest such C is called R-bound of T and denoted by R(T ).
Remark 3.10. The R-bound has the following properties.
To prove uniform R-sectoriality of −∇·µ∇ on L p , we first consider Gaussian bounds of the heat kernels associated with the respective semigroups. Using an argument due to Davies, the Gaussian bound may be extended to hold on a sector Σ θ . Since R-boundedness is inherited by domination, we obtain R-boundedness of the semigroup operators and, thus, R-boundedness of the resolvents employing the Laplace transformation. This is a well-established idea originating from [65, Section 4e].
Proposition 3.11. Let 0 < µ • < µ • and let S −∇·µ∇+1 denote the semigroup generated by Proof. This is a special case of [8,Theorem 4.4], cf. also [60,Theorem 7.5]. The assumptions on the space W 1,2 D are verified in the proof of Theorem 3.1 in [61]. The constants in [8] are constructive and can be chosen uniformly with respect to µ due to µ ∈ M d (µ • , µ • ). This yields the Gaussian bound . Let S be a holomorphic semigroup on L 2 with holomorphy sector Σ θ0 and uniform bound S(z) L 2 ≤ C for all z ∈ Σ θ0 . Suppose S satisfies Gaussian bounds as in Proposition 3.11. Then the semigroup S is holomorphic on L p for any p ∈ [1, ∞] with holomorphy sector Σ θ0 . Moreover, S(z) has a kernel K z satisfying upper Gaussian estimates. More precisely, for all ε ∈ (0, 1] and θ ∈ (0, ε θ 0 ) there exists c > 0 such that uniformly in z ∈ Σ θ . The constant c depends exclusively on ε, C, the domain Ω, and the constant c of Proposition 3.11.
We remark that the doubling property and the uniformity in growth condition required for [26, Proposition 3.3, cf. also text after proof] are satisfied since Ω is a d-set; see Proposition A.2 in the appendix. Furthermore, the notation for kernel bounds used in [26] is equivalent to ours up to positive constants due to the uniformity condition.
for almost all x ∈ Ω and all z ∈ Σ θ . Due to [26,Proposition 2.4], the latter can be bounded by the Hardy-Littlewood maximal function which defines a bounded linear operator M on L p ; see [26, p. 97]. Thus, where the bound follows from the proof of [66, Theorem 2.10].
Clearly, from the estimates (3.13) and (3.14) and using Remark 3.10, we infer the resolvent estimate (Ω). We next establish uniform R-sectoriality on W −1,p D . Since (−∇ · µ∇ + 1) 1/2 provides an isomorphism from L p onto W −1,p D and commutes with the resolvent of −∇ · µ∇ + 1, the result on W −1,p D follows from the result on L p , as the square root operators are uniformly bounded.
We emphasize that the regularity requirements of [11] are considerably weaker than Gröger regular; see Propositions A.2 and A.3 in the appendix and for Kato's square root property [28,Theorem 4 The following lemma is a direct consequence of Proposition 3.15 and the corresponding results (3.13) and (3.16) on L p . and

Acquistapace-Terreni condition.
As the next step towards our regularity result Theorem 3.20 we verify the so-called Acquistapace-Terreni (AT) condition. A family of operators { A(t) : t ∈ [0, T ] } on a Banach space X satisfies the (AT) condition if there are constants 0 ≤ β < α < 1, θ ∈ (0, π/2) and c > 0 such that for all t, s ∈ [0, T ] and z ∈ C \ Σ θ . We verify this condition for To this end, we first consider the differential operators on L p , and thereafter on W −1,p D .
Proposition 3.18. There exist θ ∈ (0, π/2) and c > 0 such that Proof. Since the operator A(t) is an isomorphism from W 1,p D onto W −1,p D , in particular A(t) has constant domain with respect to t. The resolvent identity yields Using Hölder continuity of u and Lipschitz continuity of ξ, we have  Proof. We will verify the supposition of Lemma D.1.
Step 2. It remains to argue uniformity of the (AT) condition. Suppose for the moment that u 0 = 0. According to [49,Theorem 2.13 ii)], there is α > 0 such that the mappings are equicontinuous for all u ∈ C(I × Ω), due to the lower and upper bound on ξ and µ; see Assumption 2. Whence, Now we are in the situation with homogeneous initial conditions as before, and since 4. Optimal control problem. After the detailed discussion of the state equation we now return to the optimal control problem. By means of Proposition 3.5 it is justified to introduce the control-to-state mapping S : Q → W 1,s (I; W −1,p D ) ∩ L s (I; W 1,p D )−∇ · µ∇, S(q) = u, where u denotes the solution of (1.1b) for any control q ∈ Q = L ∞ (Λ, ). Recall that p ∈ (d, 4) from Assumption 3 is close to the spatial dimension d and the numbers ζ and s are chosen according to Assumption 4 with ζ very close to 1 and s > 2 ζ−d/p , 2 1−ζ approaching +∞; see also Table 1. The control-to-state mapping S leads to the reduced objective function j : Q → R + 0 , q → J(q, S(q)). Here and in the following we omit trivial embedding operators to improve readability. Then, the optimal control problem (1.1) is equivalent to Minimize j(q) subject to q ∈ Q ad .
(P ) Since the set of admissible controls Q ad is not empty due to Assumption 4, we obtain by standard arguments, see, e.g., [64], the following existence result for optimal controls. In particular, we use compactness of the embedding Lemma 4.1. The optimal control problem (P ) admits at least one globally optimal controlq ∈ Q ad with associated optimal stateū = S(q).
We point out that the reduced objective function is not necessarily convex due to the nonlinear state equation and introduce the notation of local solutions.

Definition 4.2.
A controlq ∈ Q ad is called a local solution of (P ) in the sense of L 2 (Λ, ) if there exists a constant ε > 0 such that the inequality is satisfied for all q ∈ Q ad with q − q L 2 (Λ, ) ≤ ε.

4.1.
Differentiability of the control-to-state mapping. We first prove differentiability of the control-to-state mapping S and thereafter derive first and second order optimality conditions. According to Assumption 2, the nonlinearity ξ is assumed to be twice continuously differentiable with locally Lipschitz continuous second derivative. Hence, the Nemytskii operator associated with ξ is twice continuously Fréchet-differentiable in L ∞ (I × Ω); see, e.g., [64,Lemma 4.11]. Since u belongs to W 1,s (I; W −1,p D ) ∩ L s (I; W 1,p D ) → L ∞ (I × Ω) due to Proposition 3.5 with Proposition 3.3, the preceding remarks allow us to introduce the following notation. We set for v, v 1 , v 2 ∈ W 1,r (I; W −1,p D ) ∩ L r (I; W 1,p D ) and r ∈ (1, ∞). Proof. We consider Moreover, if f n f in L 2 (I; W −1,p D ), then v n → v in L r1 (I; C κ ) for r 1 ∈ (1, 2p/d), where v n denotes the corresponding solution to (4.2) with right-hand side f n .
Proof. First, for all v ∈ L ∞ and ϕ ∈ W 1,p D , we observe that a.e. in I, due to boundedness of µ in L ∞ (Ω) and ξ (u) in L ∞ (I × Ω). Hence with u = S(q), and w = S (q)(δq 1 , δq 2 ) is the unique solution of where v i = S (q)δq i , i ∈ {1, 2}, and u = S(q). Furthermore, S (q) can be uniquely extended to a continuous mapping from L 2 (Λ, ) to W 1,2 (I; W −1,p D ) ∩ L 2 (I; W 1,p D ). Proof. This follows from the implicit function theorem and Proposition 4.4.

4.2.
First order optimality conditions. We first verify that the derivative of the reduced objective functional can be expressed in terms of an adjoint state that is defined in an abstract fashion by taking the adjoint of the solution operator to the linearized state equation; see Proposition 4.4. Thereafter, we show improved regularity of the adjoint state and end this section with first order optimality conditions in form of a variational inequality and further properties of the adjoint state needed for the second order optimality conditions. Lemma 4.6. Let u = S(q) be the state corresponding to q ∈ L ∞ (Λ, ). There is a unique adjoint state z = z(q) ∈ L 2 (I; W 1,p D ) such that j (q)(δq) = (λq + B * z, δq) L 2 (Λ, ) , δq ∈ L 2 (Λ, ).   for any z T ∈ (W −1,p D , W 1,p D ) 1−1/r ,r , and, in particular, for z T = 0. Using the dense embedding (due to W 1,p Note that the initial condition is implicitly contained in the latter space. The isomorphism (4.5) yields the existence of z ∈ L r (I; [6,Section 6]. In summary, since z ∈ L r (I; W 1,p D ), the expression B * z is well-defined and identity (4.4) with proves (4.3) taking r = r = 2.
Note that in the L s (I; W −1,p D ) setting we are not able to show that the adjoint state possesses a time derivative, even though the right-hand side is smooth. However, employing the regularity result Theorem 3.20 for the state, we can derive improved regularity for the adjoint state. In view of the projection formula for the optimal control, see (4.7), this directly transfers to the optimal controlq, which might be used in, e.g., the numerical analysis of finite element discretizations. We emphasize that the result is not needed for the second order optimality conditions in Section 4.3.
Lemma 4.8. Letq ∈ Q ad be a local solution of (P ). Then it holds We refer to, e.g., Lemma 2.21 in [64], for a proof of the variational inequality. Employing the adjoint state z associated withq of Lemma 4.6 the first order necessary condition (4.6) can be expressed as Using the pointwise projection P Q ad on the admissible set Q ad , defined by then as in, e.g., Theorem 3.20 of [64], the optimality condition simplifies further tō For the discussion of second order optimality conditions, we will need the following continuity result concerning the linearized and adjoint state. ). Furthermore, the associated adjoint states satisfy z n → z in L r (I; W 1,p D ) for any r ∈ (1, ∞). Proof. Set S(q n ) = u n . Clearly, continuity of the control-to-state mapping implies u n → u in W 1,s (I; W −1,p D ) ∩ L s (I; W 1,p D ). According to Proposition 4.4 the mapping is a topological isomorphism for each u n . Whence, due to smoothness of the inversion mapping, for the first assertion it suffices to show The second assertion follows from the first one as z n = S (q n ) * (u n −û) for r ≥ 2p/(p + d), hence for any r ∈ (1, ∞).

4.3.
Second order optimality conditions. We now discuss second order necessary and thereafter sufficient optimality conditions using a cone of critical directions. The analysis substantially relies on the following expression for the second derivative of the reduced objective functional employing again the adjoint state from the preceding subsection.
where u = S(q), v i = S (q)η i , i = 1, 2, and z(q) denotes the adjoint state associated with the state u = S(q). The mapping η → j (q)η 2 is continuous and weakly lower semicontinuous on L 2 (Λ, ). Moreover, if q n → q in L s (Λ, ) and η n η in In particular, if η = 0, then Since for u = S(q) it holds j(q) = L(u, q, z) for all z ∈ L r (I; W 1,p D ), differentiating in (4.12) twice with respect to q in direction η i ∈ L r (Λ, ), and using Lemma 4.5 yields where v i = S (q)η i and w = S (q)[η 1 , η 2 ]. Defining z to be the adjoint state as in Lemma 4.6, all terms involving w vanish and we obtain (4.9) for η i ∈ L r (Λ, ).
We would like to extend j to L 2 (Λ, ), i.e. we have to argue that the expression in (4.9) is well-defined for η i ∈ L 2 (Λ, ). The only critical terms are those involving the adjoint state. According to Lemma 3.4 we have u ∈ L s (I; W 1,p D ) with s from Assumption 4. Moreover, z ∈ L r (I; W 1,p D ) holds for any r ∈ (1, ∞) due to Lemma 4.6.
To prove (4.10), we first note that S (q n )η n S (q)η n in W 1,2 (I; W −1,p D ) ∩ L 2 (I; W 1,p D ) and z(q n ) → z(q) in L r (I; W 1,p D ) with r > 2p/(p−d) as above, by means of Proposition 4.9. Hence, the compact embedding W 1,2 (I; where we have used weak lower semicontinuity in the last step. If η = 0, then the particular structure of the second derivative (4.9) with S (q)η n → 0 in L r1 (I; L ∞ ) for r 1 ∈ (1, 2p/d) due to Proposition 4.4 yields where the last inequality is due to (4.13). This proves (4.11).
Then we obtain the usual second order necessary optimality condition.
Proof. We will use [18,Theorem 2.2]. The delicate point is the extension of the derivatives j and j to L 2 (Λ, ). However, this is guaranteed due to Lemma 4.5 and Proposition 4.10. Hence, the regularity condition from Proposition 4.11 allows to apply [18,Theorem 2.2].
Second order sufficient optimality conditions are typically formulated using coercivity of j . Indeed, for the given objective functional this is equivalent to the seemingly weaker positivity condition of j , as already observed for semilinear parabolic PDEs in [19]. Proof. The proof is identical to the one of Theorem 4.11 in [19] except for the different structure of j , where we use the formula given in Proposition 4.10.
Proof. To prove this result, we apply [18,Theorem 2.3]. The delicate point is to verify assumption (A1), which is the continuous extension of j and j to L 2 (Λ, ). However, in our setting this is guaranteed due to Propositions 4.9 and 4.10. To see that assumption (A2) is also satisfied, let q n ∈ Q ad with q n →q in L 2 (Λ, ) and η n ∈ L 2 (Λ, ) with η n η in L 2 (Λ, ). Employing the adjoint representation (4.3) with the continuity result of Proposition 4.9 we infer lim n→∞ j (q n )η n = lim n→∞ (λq n + B * z n , η n ) L 2 (Λ, ) = (λq + B * z, η) L 2 (Λ, ) = j (q)η, where z n denotes the adjoint state associated with q n . The remaining properties (4.10) and (4.11) required for [18,Theorem 2.3] have been proven in Proposition 4.10.
5. Application to stability analysis. As an application of the second order optimality conditions of Section 4 and the improved regularity in L s (I; H −ζ,p D ) of Section 3 (in particular Theorem 3.20), we investigate the dependence of the optimal solution on certain perturbations. The stability analysis of optimal control problems is of independent interest, e.g., if the nonlinearity is not known exactly, cf. [57], for the convergence of optimization algorithms, or in the context of finite element discretizations. Since the novelty and challenge of this paper is due to the nonlinear operator A, this section is restricted to perturbations in ξ. To this end, consider a family of perturbed nonlinearities ξ ε ∈ Ξ defined in Theorem 3.20 satisfying ξ − ξ ε ∞ ≤ c ε, ε > 0. (5.1) Note that due to uniform boundedness of the states in C(I × Ω) Assumption (5.1) might be weakened to hold on compact subsets of R. For ease of readability we rely on the stronger supposition. A similar problem subject to perturbations on the desired stateû has been studied in [19,Section 4.4] for a semilinear heat equation. For any perturbed nonlinearity ξ ε ∈ Ξ fulfilling (5.1) let S ε denote the associated control-to-state mapping. Define ). The perturbed optimal control problem reads as Minimize j ε (q) subject to q ∈ Q ad .
(P ε ) We first prove a general Lipschitz stability result of the control-to-state mapping.
Lemma 5.1. There is a constant c > 0 independent of ξ ε ∈ Ξ such that for all q ∈ Q ad and η ∈ L 2 (Λ, ) it holds If in addition ξ ε ∈ C 1 (R), then Proof. We denote in short u = S(q) and u ε = S ε (q). According to Theorem 3.20 all solutions u ε are uniformly bounded in As in Proposition 4.4 we see that for each u ε the left-hand side of (5.3) defines an isomorphism. Furthermore, using Lipschitz continuity of ξ on bounded sets, we immediately infer that u ε → b ε is continuous from C(I × Ω) into itself. Whence, the map- ). Compactness of u ε in C([0, T ]; W 1,p D ) yields uniformity of the norm of the solution operators to (5.3). Hence, we obtain the first assertion by estimating the right-hand side of (5.3) by , as well as uniform boundedness of u ε due to Theorem 3.20 and boundedness of Q ad .
Applying a meanwhile standard localization argument, cf. [15], we introduce the auxiliary problem Minimize j ε (q) subject to q ∈ Q ad ∩ B ρ (q), (5.4) for ρ > 0 sufficiently small such that the second order sufficient optimality condition (4.15) holds. Existence of at least one solution follows by standard arguments.
Theorem 5.2. Letq ∈ Q ad be a locally optimal control of (P ) satisfying the second order sufficient optimality conditions (4.15). There exist a sequence (q ε ) ε of local solutions to (P ε ) and a constant c > 0 such that Proof. We set To begin with, let (q ε ) ε denote a sequence of global solutions to (5.4). By optimality ofq ε for (5.4) and the quadratic growth condition (4.16) we obtain Thus, using the definition of j and the Cauchy-Schwarz inequality we arrive at Now, applying Lemma 5.1 and (5.1), we obtain where we have used that S(q), respectively S ε (q), can be estimated independently of q due to Theorem 3.20 and boundedness of Q ad . For ε small enough it is clear thatq ε is in the interior of B ρ (q) and hence a local solution of (P ε ).
Assuming differentiability of the nonlinearity ξ ε , we are able to improve the estimate of Theorem 5.2. Precisely suppose that From the Lipschitz stability result of Lemma 5.1 we immediately infer Corollary 5.3. There is c > 0 such that for all q ∈ Q ad it holds |[j (q) − j ε (q)] η| ≤ cε η L 2 (Λ, ) , η ∈ L 2 (Λ, ).
Appendix A. Regularity of domains. For the geometric setting, we introduce: In the paper, we require further regularity properties of the domain Ω. We give short proofs or references of these well-known results for convenience. Here for all r ∈ (0, 1] and y ∈ R d−1 such that (y, 0) ∈ φ x (∂Γ N ∩ U x ) with φ x and U x as in (i), (iii) Γ D is a (d − 1)-set.
Appendix B. Exponential stability of the semigroups on L p (Ω).
Proof. This is essentially the result of [54,Corollary 14] (cf. also [59,Satz 4.2.6]) except for the uniformity. We take a step back and consider the problem A ι (t)S Aι(t) (t − s)f (s) ds.
Step 2. Boundedness of S ι . For maximal parabolic regularity we have to show that S ι is bounded on L s (I 0 ; X) which is done in [54,Corollary 14] based on the operator-valued symbol associated with the resolvent R(z, −A ι (t)). Note that due to the supposition [54,Conditions (4), (5)] are uniform with respect to ι ∈ I.
We first consider the regular version [54,Theorem 6]. Its proof is based on [54,Proposition 11] stating that every symbol a has a Coifman-Meyer type decomposition. Concerning the constants, using Remark 3.10, we infer that C on page 813 [54] depends on the properties of the symbol [54, Definition 3], only. The definition b k (x, ξ) = a(x, 2 k ξ)φ k (2 k ξ) and Remark 3.10 immediately yield C with the same dependence. Employing [9, Lemma 2.3] we see that the estimate of [54, Proposition 11, (ii)] exclusively depends on C . The remaining estimate with C β essentially uses [54,Condition (3)]. The decomposition is then used to define a bounded operator on L p (R n ; X) by means of [54,Proposition 10]. In its proof we first use [54,Theorem 7] yielding constants that are independent of T j . Then we apply Kahane's inequality (exclusively depending on p and X) and the R-bound of D k , but D k depends on the dyadic partition of unity, only. Thereafter we use the definition of T j f (x). The R-bound of a j justifies the next inequality and we are left with terms that are independent of the symbol a. For the second part of the proof, the only point where the symbol enters is in the middle of page 811. There we use the estimate |(I − ∆ z ) m a j (z)| ≤ C2 2jδm due to [54,Proposition 10,(ii)].
Second, we consider the general version [54, Theorem 5] using Nagase's reduction to the smooth case. The symbol a is decomposed into a = b + c [54, Proposition 13], where b is regular and c is treated by [54,Lemma 12]. In the second last estimates on pages 815 and 816 we use [54,Condition (4)], the third last estimate on page 817 uses [54,Condition (5)]. The remaining estimates are independent of a. Last, in the proof of [54, Lemma 12] the constant C exclusively depends on C 0 and C α of the supposition and χ from the proof.