Second-order optimality conditions for an optimal control problem governed by a regularized phase-field fracture propagation model

We prove second-order optimality conditions for an optimal control problem of tracking-type for a time-discrete regularized phase-field fracture or damage propagation model. The energy minimization problem describing the fracture process contains a penalization term for violation of the irreversibility condition in the fracture growth process, as well as a viscous regularization corresponding to a time-step restriction in a temporal discretization of the problem. In the control problem, the energy minimization problem is replaced by its Euler–Lagrange equations. While the energy minimization functional is convex due to the viscous approximation, the associated Euler–Lagrange equations are of quasilinear type, making the control problem nonconvex. We prove second-order necessary as well as second-order sufficient optimality conditions without two-norm discrepancy.


Introduction
In this paper, we are interested in second-order optimality conditions for an optimal control problem for regularized fracture propagation. More precisely, we consider an optimal control problem of tracking-type of the following form: Find a control q in an admissible control set Q ad , with associated state pair u := (u, ϕ), that satisfies min q∈Q ad , u∈V J(q, u) : subject to : A(u) + R(ϕ; γ ) = B (q).
The precise functional analytic setting along with a concrete mathematical definition of the operators A, R, and B will be introduced in the next subsection. Let us give a brief introduction to the model problem, which stems from a bi-level optimization problem with an upper-level tracking-type functional and CONTACT Ira Neitzel neitzel@ins.uni-bonn.de a lower-level variational fracture propagation problem. The latter is an energy minimization problem, which is eventually replaced by its Euler-Lagrange equations. The lower-level fracture propagation problem behind this formulation was considered in [1][2][3]. An Ambrosio-Tortorelli regularization (cf. [4]) is used to avoid the irregular fracture set. This means that in addition to the displacement u, a phase-field variable ϕ is introduced. The latter has values 0 ≤ ϕ ≤ 1 and describes the condition of the material at every point in the domain, with ϕ = 1 where the material is completely sound, and ϕ = 0 where the material is fully broken, guaranteeing a smooth transition between those two states. The control q acts as a boundary force. We consider one time-step of a time-discrete but spatially continuous problem formulation. In the energy minimization problem describing the fracture process, a viscous regularization corresponding to a timestep restriction in a temporal discretization of the problem (cf. [5,6]) is used that guarantees strict convexity of the lower-level minimization problem. Nevertheless, the Euler-Lagrange equations are of quasilinear type, making the overall control problem nonconvex. Moreover, a violation of the irreversibility condition in the fracture growth process is penalized using the regularization from [7]. The corresponding terms appear in the operator R in the differential equations, whereas the differential operator A stems from the actual fracture propagation process. We refer to, e.g. the introduction of [8] for a more detailed description of the mathematical model. Compared to previous works on optimal control of fractures (cf. [9,10]) where paths of fixed length or prescribed fracture paths could be treated, this variational fracture approach is more flexible and also allows the treatment of arbitrary fracture paths and branching (cf. [5]).
To the best of our knowledge, Neitzel et al. [8] were the first to add an outer optimization problem to a variational phase-field fracture model, leading to an optimal control problem in the form of (NLP γ ,η ) as described above. The chosen tracking-type cost functional to steer the displacement u towards a desired displacement u d with an L 2 -cost term for the boundary control is a standard-setting in the field of optimal control of partial differential equations (PDEs).
Our work complements the studies of this model problem that were conducted in [6,8,11,12]. The control problem, without viscous regularization but with an additional trivial kernel assumption, has been analyzed with respect to first-order necessary optimality conditions in [8]. In [6], the convergence of regularized solutions (with respect to the penalization parameter γ ) has been proven, along with estimates on the constraint violations for fixed γ . In [11], this convergence result was extended to the dual variables, and it was shown that in the limit the functions satisfy an optimality system of an MPCC. In the same publication, the sequential quadratic programming (SQP) method was introduced for both the regularized and nonregularized problem, the involved sub-problems were investigated, and it was proven that the limit point of the SQP method, in case of convergence, satisfies the first-order optimality system of the regularized original problem. A convergence result of the FEM discretization of a linearized fracture control problem was shown in [12]. Further, algorithmic concepts and some examples, where a desired phase-field ϕ d in the tracking-type functional is traced instead of a desired displacement u d , can be found in [13].
Let us mention a few closely related results dealing with the numerical challenges of the lower-level problem. In [14], a computational framework for phasefield fracture models in porous media, which have an application in, e.g. the fracking industry, has been presented in detail. For a recent overview about modelling and numerical analysis of phase-field fracture models in multiphysics, we refer the interested reader to [15].
Some further related results regarding analysis and numerics of the optimization of fractures include [16,17], where the authors limit the propagation of a crack in a material by controlling the release of the associated energy, they do, however, not aim at achieving a desired configuration as in the case of (NLP γ ,η ). Shape optimization was used in [18,19]. We also want to mention [20,21], where control of a viscous-damage model was considered in a continuous setting, and [22,23], where the author investigated necessary conditions of an optimal control problem of a two-field damage model, and strong stationarity of a nonsmooth (viscous damage) coupled system, respectively.
An open question for (NLP γ ,η ) is the investigation of second-order necessary and sufficient optimality conditions, which is the goal of the present work. The complicated lower-level fracture problem introduces a fair amount of mathematical challenges, and we will put our focus on establishing the required differentiability and Lipschitz continuity results for the control-to-state operator associated with it. We will then combine them with well-established techniques for the objective functional J, relying on its L 2 -tracking-type structure. We are interested, in particular, in second-order sufficient conditions (SSC). Due to the nonconvex structure of (NLP γ ,η ), only then we can ensure that critical points satisfying the first-order necessary conditions are in fact (isolated) local minima. SSC are also essential as a theoretical tool when looking at the stability of perturbed problems, for instance, when analysing a priori error estimates of discretization schemes, or when looking at convergence results of numerical methods such as the SQP method.
In general, SSC are an area of active research in the optimal control community. To our knowledge, the first PDE-related publication for second-order sufficient optimality conditions was [24]. For an overview about many aspects considered since then, we refer to the work [25] and the references therein. One of the central difficulties in infinite dimensions is the two-norm discrepancy first described in [26], i.e. if differentiability and coercivity of the second derivative hold in different spaces. In particular, for many optimal control problems, coercivity of the second derivative holds in L q , typically with q = 2, but differentiability can only be ensured in a stronger space, often L ∞ . This introduces additional challenges also in the application of SSC results as mentioned above.
One of the main results of this work is to show that we can avoid this two-norm discrepancy through careful choice of the introduced spaces and exploiting the regularity results we can establish for the solution operator corresponding to the phase-field fracture model (EL γ ,η ). Avoiding the two-norm discrepancy not only directly ensures local optimality in the Hilbert space L 2 , but might also be helpful for research in, e.g. the stability analysis required in the context of numerical methods that should simplify due to the lack of the L ∞ -norm gap in calculations.
Let us mention some previous results in the context of SSC for PDE constrained optimal control, restricting ourselves mainly to the elliptic setting. SSC for the semi-linear case with state constraints have been established in [27]. In [28], for the same setting, SSC that are closest to the associated necessary ones have been presented. SSC for a more abstract optimization in Banach spaces have been analyzed in [29]. For the quasilinear case, SSC in the control-constrained case can be found in [30,31]. At this point, we also want to mention [32], where SSC of an optimal control problem that is governed by a nonsmooth quasilinear PDE were investigated. In a more general context, no-gap SSC for bang-bang optimal control problems have been analyzed in [33], and for a very recent contribution concerning second-order optimality conditions for a nonsmooth optimal control problem, we refer to [34].
Finally, we also want to give a short overview about related results in the field of MPCCs, since as mentioned above, the model problem converges to an MPCC in the penalization limit. For MPCCs, SSC are a challenging task due to the lack of smoothness in the control-to-state operator. A lot of the recent research thus focused on regularization methods to establish different (M, B, C, strong) stationary concepts (cf. [35,36]). In [37] a comparison of those concepts for the obstacle problem has been made. The authors of [38,39] tackled the lack of smoothness of the control-to-state operator by investigating generalized derivatives of this operator for the obstacle problem. SSC for the obstacle problem have recently been under investigation in [40]. For a control problem governed by variational inequalities, SSC could be established in [41,42]. In [43], the authors established SSC for a regularization of an optimal control problem that is governed by an evolution variational inequality.
We finish this section with an outline of the present paper. In Section 2, we provide all details and assumptions for the model problem along with an overview about the notation used. Then, in Section 3, we collect and provide the necessary existence and regularity results for (EL γ ,η ). The most important preliminary result is the Lipschitz continuity of the associated control-to-state operator G and its derivatives G and G in Lemma 3.6, Lemma 3.8, and Lemma 3.9. After collecting solvability results and first-order necessary conditions for (NLP γ ,η ), our main results, the second-order necessary conditions of Theorem 4.4 and the secondorder sufficient conditions of Theorem 4.6 in Section 4, follow from applying optimal results for an abstract setting from [44]. In particular, we arrive at a result on sufficient conditions without two-norm discrepancy, following from plain positivity of the second derivative of the reduced functional in a critical point for directions from a cone of critical directions. This means that the gap between necessary and sufficient conditions is minimal, which is a result of the structure of our problem, that ensures that for (NLP γ ,η ), positivity and coercivity of the second derivative of the reduced functional are equivalent.

Problem formulation, general assumptions and notation
Let us now introduce the precise assumptions on the model problem along with the general functional analytic setting of this paper. For convenience, we recall the formulation.
Find a control q in a set of admissible controls Q ad , a subset of the control space Q, with associated state pair u := (u, ϕ), that satisfies Here, the domain is a polygonal subset of R 2 with boundary ∂ = ∪ D , i.e. the boundary of is split into a Neumann part where we apply the control q, and a part D denoting the remaining part of ∂ with homogeneous Dirichlet boundary conditions. As in [6, Section 2], we additionally assume throughout that is W 2,q -regular for the homogeneous Neumann-problem −ε ϕ + 1 ε ϕ = f , as well as Gröger regularity (cf. [45]) of ∪ .
The given function u d ∈ L 2 ( , R 2 ) in (NLP γ ,η ) denotes a desired displacement, and the Tikhonov parameter α is a fixed positive real number, the cost parameter. The control space is Q = L 2 ( ), and the set of admissible controls is defined by simple box constraints as For a precise definition of (EL γ ,η ), let us first fix some general notation for function spaces. For p > 2, q := p/2 > 1, we define the spaces For the choice of p and q and in N = 2 spatial dimensions, note that W u → V u and W ϕ → V ϕ by the Sobolev/Kondrachov embedding theorem. Further W ϕ → L ∞ , which we will frequently use without further mentioning. Further, choose s ∈ (0, 1/2) and assume that p and s are chosen such that H 1+s ⊂ W 1,p .
We want to point out that the definition of the space W u (thus also of W) and W × differs from [11], where W u was defined as W Here, we will only use these improved regularities when looking at the regularity of the solutions of (EL γ ,η ) and its linearization, and will specifically point out all instances in which we include these spaces. We will denote the respective dual spaces with a superscript * , e.g. V * . Here and throughout, we understand all spaces to be defined on the domain unless otherwise stated and omit the dependence on for readability.
We will further use the following notation for the scalar product/norm: (·, ·) denotes the usual L 2 inner product with corresponding norm · , and (·, ·) corresponds to the inner product of the control space Q = L 2 ( ). In addition, ·, · stands for a duality pairing where we omit the spaces wherever obvious from the context. The operators involved in (EL γ ,η ) are what we will call the nonlinear phasefield operator A : V ⊃ W → V * , the penalization operator R : V ϕ → V * ϕ , and the control-action operator on the Neumann boundary , B : Q → V * . For a displacement/phase-field pair u := (u, ϕ) ∈ W, they are defined by for any v = (v u , v ϕ ) ∈ V, and given phase-field ϕ − ∈ W ϕ with 0 ≤ ϕ − ≤ 1. Note that the operators A and R will later also be considered as a mapping into the more regular spaces W × and L q , but in both cases we can use test functions v ϕ from V, since W × → V * . In a temporal discrete multi-step model, ϕ − would be the phase-field from the previous time-step, and we would make this assumption on an initial phase-field ϕ 0 . Moreover, let the parameters κ, ε, γ > 0, and η ≥ 0 be given. We will refer to γ as the penalization parameter, since the operator R stems from a penalization of the violation of the irreversibility condition for the fracture growth, and to η as (viscosity) regularization parameter (cf. the comments in the introduction). The parameter ε > 0 stems from the phase-field modelling of the fracture growth problem and will be considered fixed here. In short, on each side of the fracture, there is a transition zone with width ε, cf. [8]. Finally, κ appears in g(x) := (1 − κ)x 2 + κ, and C denotes the (symmetric) rank-4 elasticity tensor (cf. [46,Section 3]). For further explanation of the forward problem, we also refer to the exposition in [8].
For further use, we define the operators appearing in the linearized equations. Let d u := (d u , d ϕ ) ∈ V be a pair of displacement and phase-field functions. For Again, we can use A and R as mappings from W into W × , and W ϕ into L q , respectively. In fact, in those spaces the operators A and R will turn out to be the derivatives of the operators A and R. Again, in both cases, we can use test and In the remainder of this paper, we will tacitly assume that η ≥ 0 is large enough such that all results below hold true. We collect this in the following standing assumption:

Assumption 2.1 (Viscous approximation):
Let η ≥ 0 be chosen large enough for all following calculations and results.

The control-to-state operator and the objective functional
We start with the analysis of (EL γ ,η ) and the objective functional J. We will collect and extend known results for the PDE, and eventually introduce a well-defined control-to-state mapping G : q → u due to the regularization effect of a sufficiently large η. We can then introduce the reduced functional f : Q → R and establish differentiability and Lipschitz properties for G, f, and their first-and second-order derivatives. The main results are the Lipschitz continuity of this operator G and its derivatives G and G in Section 3.2, which allow us to deduce analogous properties for f. First, let us recall a result on unique solvability of (EL γ ,η ) and regularity results for the solution, known from cf. [6,8,47]. While [8] had to deal with possible nonuniqueness of solutions for the Euler-Lagrange equations, the presence of a sufficiently large η ≥ 0 as in [6] makes the energy minimization problem for the fracture growth strictly convex and guarantees the existence of a unique solution of (EL γ ,η ) for any given q ∈ Q. Lemma 3.1: Let Assumption 2.1 hold, and let 0 ≤ ϕ − ∈ W ϕ . Then for every q ∈ Q, (EL γ ,η ) has a unique weak solution u ∈ W, such that ϕ ∈ L ∞ and 0 ≤ ϕ ≤ 1. Additionally, u ∈ H 1+s × H 1+s , and the following stability properties are satisfied In particular,c can be chosen independently of γ and ϕ.
Proof: We refer to Section 1 and 2 of [6] for the existence and regularity results With Lemma 3.1, it is now possible to introduce a control-to-state operator associated with the nonlinear PDE (EL γ ,η ). Here, u solves (EL γ ,η ) for right-hand side q ∈ Q. We obtain a usual reduced problem formulation where we implicitly use that W is embedded in For what follows, we will discuss the linearized equations w.r.t. existence, uniqueness and sufficient regularity. Unique solvability in V was stated in [11, Section 2.2, Proposition 2.1] and proven for similar equations in [8] without viscous approximation, but additional trivial kernel assumption ker A = {0}. There, due to the lack of the viscous approximation term η(d ϕ , v ϕ ), only a Fredholm property of A was proven, which in combination with the trivial kernel assumption ensured coercivity. Replacing this by Assumption 2.1, we extend Lemma 5.1 and 5.2 in [8] to obtain the existence of unique solutions first in V then in W in the setting η > 0, and eventually utilize the ideas from the nonlinear setting in [47] to guarantee regularity of solutions in the space (W u ∩ H 1+s ) × W ϕ . The latter improved regularity result is not needed for our analysis of SSC, but interesting for future research. We state and repeat important technical steps of the proofs for the convenience of the reader. To motivate this, note that for instance a direct adaptation of the proof of [8, Lemma 5.1] to obtain coercivity of the underlying bilinear form to the case η > 0 would lead to where the negative term cannot be absorbed directly into the L 2 -regularization term. Therefore, some further technical estimates as in, e.g. [5] have to be used to prove coercivity in V.

Lemma 3.2:
Let Assumption 2.1 hold and u ∈ W be given. Then, for every f : and it holds If a fortiori f ∈ W × → V * , then additionally d u ∈ W, and it holds Further, the constant on the right-hand side of (14) depends on positive integer powers of the norm u W . If additionally u ∈ H 1+s × H 1+s and f u ∈ H −1+s , then additionally d u ∈ H 1+s .
Proof: (1) Let us start with unique solvability of (13) in V. In comparison to [8, Lemma 5.1], the only difference is that due to Assumption 2.1, the bilinear form induced by A (u)·, · is coercive. Similar to [8], the crucial part is to look at the only possibly nonpositive terms in A (u)d u , d u and show that they can be absorbed into the viscous regularization term. Recalling the definition of the operator in (4), we estimate which follows from Hölder's and Young's inequality for a δ 1 > 0, since the standard Sobolev embedding guarantees d ϕ ∈ H 1 → L r , for an r such that 1 r + 1 p = 1 2 , i.e. r ∈ (2, ∞). Since r > 2, the Gagliardo-Nirenberg inequality and Young's inequality ensure with a δ 2 > 0 and exponents r r−2 and r 2 . Inserting this in (15), we find for a constant C(δ 1 , δ 2 ) > 0. The remaining terms in A (u)d u , d u are handled analogously to [8, Lemma 5.1]. With c korn > 0 being the constant from Korn's inequality of the second kind for zero boundary functions, we end up at We recall that ε > 0 is an a priori fixed number that stems from the model problem formulation and is in particular independent of the other constants. Thus, choosing δ 1 small enough such that κc korn − δ 1 c κ u 2 W > 0, then choosing δ 2 small enough such that ε − c κ, u 2 W δ 2 δ 1 > 0, we can then choose η large enough to bound the right-hand-side of (16) from below by c coer ( d u 2 1,2 + d ϕ 2 1,2 ) for a coercivity constant c coer > 0. Thus, by the Lax-Milgram lemma, there exists a unique solution d u := (d u , d ϕ ) ∈ V of (13) with (2) To show the improved regularity result in W, we follow [8,47] and test (13) first with (0, v ϕ ) ∈ V, to obtain and secondly with (v u , 0) ∈ V, to obtain Note that the term corresponding to the viscous regularization appears on the left-hand side of (18). The right-hand sides of both equalities can be treated as in [8,Lemma 5.2]. We only need to adapt the proof of [8,Lemma 5.2] to the presence of data f ∈ V * to eventually obtain where r ∈ (1, 2), 1 = 1 r + 1 r . A standard elliptic regularity result, cf. [48, Theorem 4.7 and Chapter 7.2.1], applied to (18) yields d ϕ ∈ H 1 ∩ L ∞ , and for c 1 (u, ϕ − , η) subsuming the constants of the right-hand side of (20), the following estimate holds Using this, we find Combining this with (19), using f u ∈ W −1,p and [49, Proposition 1.1], we find d u ∈ W 1,p and Using the additional regularity f ϕ ∈ L q and improved regularity results d ϕ ∈ L ∞ and d u ∈ W 1,p from (21) and (23), setting r = q = p 2 in (20), the righthand side of (18) is in fact in L q . Instead of (20), we obtain Due to the W 2,q regularity of , by the same argument as in [47, Corollary 2, Section 7], we find d ϕ ∈ W 2,q , and the following estimate holds where c 3 (u, ϕ − , η) subsumes the constant in (24). The dependence of the constant in (14) on u is a direct consequence of (20), (22), and (24). (3) Finally we prove d u ∈ H 1+s , exploiting the additional assumption (u, ϕ) ∈ H 1+s × H 1+s and again the same idea as for (EL γ ,η ) from the nonlinear setting of [47, Corollary 2, Section 7]. It is important to recognize that in (19), we have to ensure that g(ϕ) is a multiplier in the sense of [47] on H s . By [47, Lemma 1, Section 5], it suffices that g(ϕ) ∈ C θ for θ = 1 + s − 2 p , which itself follows from ϕ ∈ C θ . This holds true from the standard Sobolev embedding and our assumption on ϕ. The right-hand side of (19) is an element of H −1+s , since f u ∈ H −1+s by assumption and ϕC(u)d ϕ ∈ H s , since ϕ, d ϕ ∈ L ∞ and u ∈ H 1+s . Now from [47, Theorem 1, Section 2], we get that in fact d u ∈ H 1+s .

Differentiability results
Differentiability of G has been used in earlier publications and follows from standard techniques. Since this property is used for the Lipschitz results in Section 3.2, we present some steps of the proof.
with G(q) = u. For the second-order derivative, we find (27) where again G(q) = u, as well as G (q)d q j = d u j , j = 1, 2.
Proof: The claim follows by standard techniques utilizing the implicit function theorem (cf. [50,Theorem 4.B]) applied to F : (cf. the remarks from Section 2 regarding the definition of A and the allowed test functions). Note that Lemma 3.1 already ensures for every q ∈ Q the existence of a unique u ∈ W, such that F(q, u) = 0. The mapping F, and thus G, eventually, is continuously Fréchet differentiable from Q × W into W × , (29) The only interesting part of the proof is the differentiability of A from W into W × . A straightforward calculation verifies with A as in (4) and remainder To show that rem A is of order o( d u W ), we calculate exemplarily as in the proof of Lemma 3.2, cf. (22). The remaining terms in rem A (u, d u ) can be bounded analogously, overall it holds We skip the calculations, since the steps are very similar to the proof of the Lipschitz continuity results in Lemma 3.9, which we carry out in detail. Now, note that and for (f u , f ϕ ) = (B(d q ), 0) we see that F u (q, u)d u = d q is equivalent to (13). Thus by Lemma 3.2, F u (q, u) is invertible in W. Altogether, this proves that the control-to-state operator G is continuously Fréchet differentiable from Q into W in every q ∈ Q with derivative d u = G (q)d q given by (26). The second-order continuous Fréchet differentiability of G follows from the continuous differentiability of the operatorF(q, u) : = F (q, u) Proof: This follows from Lemma 3.2, by estimating the right-hand side of (27) in W × termwise, recalling the definitions of A and R from (6) and (7). Using Hölder's inequality with 1 q = 1 p + 1 p , recalling that p > 2 and q = p 2 , where in the final inequalities we used the norm estimates from Lemmas 3.1 and 3.2. The remaining terms are calculated analogously.
In order to prove first-and second-order optimality conditions, we collect some rather straightforward results on the reduced objective functional f.
where u = G(q) and d u = G (q)d q , and Proof: This immediately follows from Proposition 3.3 and the chain rule.

Lipschitz continuity results
Next, we establish Lipschitz continuity of the operators G, G , and G , and subsequently also of the functionals f , f , and f . These technical results are the most crucial part on our way to establishing second-order sufficient optimality conditions. Lemma 3.6: Let q ∈ Q be given, then for all ρ > 0 there exists a constant c = c(q, ρ) > 0 such that for h ∈ Q, with h ≤ ρ, it holds where u h = G(q + h) and u = G(q).

Remark 3.7:
Note that the boundedness of Q ad yields a global Lipschitz continuity result for all q ∈ Q ad .
For any t ∈ [0, 1], set W d u th := G (q + th)d q and W (u th , ϕ th ) = u th := G(q + th). Utilizing Lemma 3.1 yields for a constantc(q, ρ) > 0 independent of t and h since th with a constant c(q, ρ) > 0. Collecting all estimates yields the assertion.

Lemma 3.8: Let q, d q ∈ Q be given, then for all
where d u h = G (q + h)d q and d u = G (q)d q .
Proof: Let q, h, d q ∈ Q be as stated. Due to the second-order Fréchet differentiability of G from Proposition 3.3, estimating as in the proof of Lemma 3.6 yields, for a t ∈ [0, 1], Again, for any t ∈ [0, 1], set W d u The assertion is obtained after collecting all estimates.
Proof: Letd u h andd u be as stated. By linearity of (27), it holds where Again, we want to utilize the norm estimate from Lemma 3.2, thus we estimate the right-hand side of (40) in W × . Let us start with the difference of the A terms, in particular, for all test functions v ∈ V, we obtain For the last term of the right-hand side, we exploit Lemma 3.6 for ϕ h − ϕ ∞ and u h − u 1,p , Lemma 3.1 for u h 1,p and ϕ ∞ , and Corollary 3.4 for d ϕ h to obtain The remaining terms on the right-hand side of (41) can be bounded in the same way ; overall, we obtain For the difference in R , exploiting local Lipschitz continuity of [(·) + ] 2 , and using Lemma 3.6 and Corollary 3.4, we estimate Next, we look at the difference in the A terms, i.e. at Recalling the definition of the operator A from (6), we see that in each of the six terms of A , each of the involved functions u, d u 1 , d u 2 occurs linearly. Thus, for all test functions v ∈ V, for the first term of the right-hand side of (42) it holds Let us estimate the last term of the right-hand side of (43), using Hölder's inequality with 1 q = Analogously, we bound the remaining terms, and subsequently, estimate (42) by Due to the local Lipschitz continuity of the Nemytskii operator [(·) + ], an easy calculation shows Combining all estimates, the right-hand side of (40) can be bounded in the W ×norm by c(q, ρ) d h , the claim now follows from Lemma 3.2.
Lipschitz continuity results for the reduced objective functional and its derivatives are now easily obtained.
Proof: For q, h, d q j , j = 1, 2 as stated, we will denote G(q + h) = u h , G(q) = u, ] =d u . All three results follow from the definition of f from (12) and the representation of f and f from Lemma 3.5, combined with the norm-and Lipschitz estimates for G, G , and G . For the reduced functional, a quick standard calculation yields where in the last inequality we used Lemma 3.6 to estimate u h − u 1,p , Lemma 3.1 for u h 1,p and u 1,p , and the boundedness of u d in L 2 .
For the first derivative, we estimate similarly Here, we additionally used Lemma 3.9.

The adjoint equation
To conclude our preliminary results, we collect results for the adjoint equation for later use. Recalling that C is the rank-4 elasticity tensor Ce(v) : e(z) = Ce(z) : e(v) and hence symmetric for all v, z ∈ V u (c.f. [ and Further, the first derivative f , see (32), can equivalently be expressed by where z = (z u , z ϕ ) ∈ W is the unique weak solution to This equivalence of (32) and (49) in combination with (50) follows from usual calculations for adjoint operators, cf. to the first-order optimality conditions obtained in [8] for the case η = 0. Likewise, by standard calculations, we obtain an expression of the second derivative f from (33) with the help of the adjoint state, resulting in where G(q) = u ∈ W, G (q)d q j = d u j ∈ W, j = 1, 2, and z ∈ W is the solution to (50).
For completeness, we will quickly establish an auxiliary result for the adjoint equations here. Corollary 3.11: Let q ∈ Q and f h , f ∈ W × be given, then for every ρ > 0, there exists a constant c = c(q, ρ) > 0, such that for all h ∈ Q, with h ≤ ρ, it holds where z h ∈ W is the solution to (47) for W u h = G(q + h) and f adj = f h , and z ∈ W is the solution to (47) for W u = G(q) and f adj = f ∈ W × , respectively.
Proof: This is a straightforward extension of the necessary optimality conditions from [6,8] for settings without control constraints to the control-constrained setting. Since Q ad is convex, a local minimizerq satisfies the variational inequality Using (49) for d q = q −q results in the stated optimality system. Now we can prove our main results, the second-order optimality conditions, by applying an abstract result from [44]. We start by defining the cone of critical directions: The following second-order necessary result for (NLP γ ,η ) then holds.

Theorem 4.4:
Letq ∈ Q ad be a locally optimal control to (NLP γ ,η ). Then, it holds ∈ Q and q ∈ Q ad ∩ B Q (q), where B Q (q) is a Q = L 2 ( ) -neighborhood ofq. Next, we point out that the tracking-type objective functional results in a Legendre form Q : d q → f (q)(d q ) 2 from L 2 ( ) → R. Here, in particular, the regularity of the control-to-state operator and the compact embedding L 2 → W −1,p is used. In combination with the norm term stemming from the Tikhonov term, it immediately follows that f indeed suffices to the definition of Legendre forms from [44, p. 3]. Choosing U 2 = U ∞ = Q and K = Q ad , all conditions of Assumption (A2) in [44] are satisfied. Furthermore, for the choice of the cone of critical directions C(q), we refer to Remark 2.4 of the same paper. Hence all prerequisites of [44,Theorem 2.3] are satisfied, now applying the theorem yields the assertion.
The result on second-order sufficient optimality conditions for (NLP γ ,η ), which does not involve a two-norm discrepancy, follows directly from [44,Theorem 2.5]. We will assume the following second-order sufficient condition. Assumption 4.5 (SSC): Letq ∈ Q ad , together with the associated state and adjoint stateū,z ∈ W, fulfil the first-order necessary conditions given in Lemma 4.3. We assume (56) Theorem 4.6: Letq ∈ Q ad , with associated stateū and adjoint statez, satisfy Assumption 4.5. Then, there exist constants ≥ 0 and σ ≥ 0 such that the quadratic growth condition f (q) ≥ f (q) + σ q −q 2 holds for every q ∈ Q ad with q −q ≤ . In particular, this means thatq is a locally optimal control in the sense of L 2 .

Remark 4.7:
Comparing conditions (55) and (56), we see that the gap between the necessary and sufficient second-order conditions is minimal. This stems from the structure of our main problem (NLP γ ,η ). In fact, since the reduced objective functional f satisfies Assumption (A2) of [44], it holds that the positivity assumption (56) and the coercivity condition  , u), where u = G(q). Letq, with associatedū andz, satisfy Assumption 4.5, then the quadratic growth condition in Theorem 4.6 can be written equivalently as J(q, u) ≥ J(q,ū) + σ q −q 2 , for every q ∈ Q ad with q −q ≤ , and and σ as in Theorem 2.1.

Disclosure statement
No potential conflict of interest was reported by the author(s).