Morley FEM for a Distributed Optimal Control Problem Governed by the von Kármán Equations

Lemma A.1 (Maximum Norm Estimates).

Let Θ ¯ (resp. Θ ¯ M ) solve (2.5b) (resp. (5.2b)). Then, for a sufficiently small choice of the discretization parameter h, it holds that | | | Θ ¯ - Θ ¯ M | | | 𝐋 ∞ ⁢ ( Ω ) ≲ h γ .

Proof.

Introduce intermediate terms, and use a triangle inequality (twice), Lemmas 3.1, 3.5 and Theorem 4.2 to obtain

Lemma A.2.

Let Θ ¯ solve (2.5b) with control u ¯ , and let P h : C ⁢ ( Ω ¯ ) → U h be defined as in Definition 5.2. Then, under the assumption that | Ω h 2 | ≲ h , it holds that

Proof.

A use of the numerical integration property (see for example [30, Lemma 5.5], [25, Lemma 4]) of P h and the Cauchy–Schwarz inequality lead to

Since Θ ¯ ∈ C 0 , 1 ⁢ ( Ω ¯ ) , the definition of P h , the Cauchy–Schwarz inequality and the assumption | Ω h 2 | ≲ h lead to

This concludes the proof. ∎

Lemma A.3.

Let u ¯ = ( u ¯ , 0 ) (resp. u ¯ h = ( u ¯ h , 0 ) ) be the optimal control in the optimality system (2.5) (resp. (5.1)) at the continuous (resp. discrete) level. Then the following estimates hold true.

1. For u 1 , u 2 ∈ U ad and sufficiently close to u ¯ , and for v ∈ U h , ( j h ′′ ⁢ ( u 1 ) - j h ′′ ⁢ ( u 2 ) ) ⁢ ( v 2 ) ≲ ∥ u 1 - u 2 ∥ ⁢ ∥ v ∥ 2 .

2. For v ∈ U h , ( j h ′′ ⁢ ( u ¯ ) - j ′′ ⁢ ( u ¯ ) ) ⁢ ( v 2 ) ≲ h γ ⁢ ∥ v ∥ 2 , γ ∈ ( 1 2 , 1 ] being the index of elliptic regularity.

3. Let P h : C ⁢ ( Ω ¯ ) → U h be defined as in Definition 5.2 . Then ( j h ′ ⁢ ( P h ⁢ u ¯ ) - j h ′ ⁢ ( u ¯ h ) , P h ⁢ u ¯ - u ¯ h ) ≳ ∥ P h ⁢ u ¯ - u ¯ h ∥ 2 .

4. For u ∈ U ad sufficiently close to u ¯ , v 1 ∈ Ω h 1 , and v 2 ∈ Ω h 2 , j h ′′ ⁢ ( u ) ⁢ ( v 1 , v 2 ) ≲ ∥ v 1 ∥ L 1 ⁢ ( Ω h 1 ) ⁢ ∥ v 1 ∥ L 1 ⁢ ( Ω h 2 ) .

Proof.

(i) For i = 1 , 2 ,

where 𝐳 v , u i , M satisfies the linearized von Kármán equation in the direction of v given by

for all Φ M ∈ 𝐕 M . From (A.3) with i = 1 and i = 2 ,

Consider the first term on the right-hand side of (A.5). A use of the Cauchy–Schwarz inequality and the a priori bound for the solution of (A.4) yields

Note that

Consider the auxiliary problem that seeks 𝝃 𝐩 , M ∈ 𝐕 M such that, for all Φ M ∈ 𝐕 M ,

Choose Φ M = 𝐳 v , u 1 , M - 𝐳 v , u 2 , M in (A.8), and use (A.4) to obtain

A use of Lemma 3.6 (a), (3.16) and the a priori bounds for the solutions of (A.4) and (A.8), and then a substitution in (A.7) leads to | | | 𝐳 v , u 1 , M - 𝐳 v , u 2 , M | | | ≤ ∥ u 1 - u 2 ∥ ⁢ ∥ v ∥ . Therefore, a substitution of this estimate in (A.6) yields

To estimate the second term on the right-hand side of (A.5), use the definition in (A.2), and consider the first term in the expansion. For 𝐳 v , u i , M = ( z v , u i , M , 1 , z v , u i , M , 2 ) and Θ u i , M = ( θ u i , M , 1 , θ u i , M , 2 ) , a simple manipulation, the generalized Hölder inequality, Lemma 3.5 and the a priori bound for the solutions of (A.4) lead to

Change the auxiliary problem (A.8) (resp. (A.4)) with duality pairing ( ⋅ , ⋅ ) 𝐕 M ′ , 𝐕 M on the right-hand side, choose Φ M = 𝐳 v , u 1 , M - 𝐳 v , u 2 , M (resp. Φ M = Θ u 1 , M - Θ u 2 , M ), and modify the arguments to obtain

A substitution of (A.11) in (A.10) leads to

The remaining terms that contribute to the right-hand side of (A.5) can be estimated in a similar way. A combination of (A.5), (A.9), (A.12) and (A.2) yields the desired result in (i).

(ii) A subtraction of the continuous and discrete second-order cost functionals leads to

The proof then follows from arguments similar to (A.9) and (A.10) and by using Theorem 4.2 for Θ ¯ and Θ u ¯ , M (resp. 𝐳 v , u ¯ and 𝐳 v , u ¯ , M ).

(iii) The proof follows from [25, Lemma 6] together with (i), (ii), Lemma A.1 and [30, Lemma 4.4].

(iv) Since 𝒯 h 1 ∩ 𝒯 h 2 = ∅ , v 1 ∈ Ω h 1 (resp. v 2 ∈ Ω h 2 ), we have ∫ Ω v 1 ⁢ v 2 ⁢ d ⁢ x = 0 . Hence a use of the definition of j h ′′ ⁢ ( u ) ⁢ ( v 1 , v 2 ) yields

where 𝐳 v i , M = ( z v i , M , 1 , z v i , M , 2 ) solves (A.4) with 𝐯 = 𝐯 i for i = 1 , 2 . The bound ∥ 𝐳 v i , M ∥ NC ≲ ∥ 𝐯 i ∥ L 1 ⁢ ( Ω h i ) for i = 1 , 2 would now lead to the desired estimate. ∎

Sketch of the proof of Lemma 5.6.

The proof of Lemma 5.6 follows from [25, Lemmas 9 and 10] together with Theorem 4.2, Proposition 5.10, Lemmas A.2 and A.3. We will sketch the steps of the proof.

A use of the pointwise version of the continuous optimality condition with x = S T and u = u ¯ h ⁢ ( S T ) leads to

Apply the above variational inequality to T ∈ 𝒯 h 2 with x T instead of S T to obtain

An integration of the above two inequalities over T, a summation over all T ∈ 𝒯 h and the definition of P h yield

Since P h ⁢ u ¯ ∈ U h , ad , choose 𝐮 h = P h ⁢ 𝐮 ¯ in the discrete optimality condition (5.2) to obtain

An addition of (A.13) and (A.14), and introduction of intermediate terms yield

Lemma A.2, the Cauchy–Schwarz inequality, Theorem 4.2 (c) and Proposition 5.10 lead to an estimate for the first term in (A.15) as ( ( P h ⁢ Θ ¯ - Θ ¯ ) + ( Θ ¯ - Θ u ¯ , M ) + ( Θ u ¯ , M - Θ P h ⁢ u ¯ , M ) , 𝐮 ¯ h - P h ⁢ 𝐮 ¯ ) ≲ h 3 / 2 ⁢ | | | P h ⁢ 𝐮 ¯ - 𝐮 ¯ h | | | . This and (A.15) result in

The term on the left-hand side of (A.16) is estimated using the discrete version of (A.1) and Lemma A.3 (iii) to obtain ( Θ P h ⁢ u ¯ , M + α ⁢ P h ⁢ 𝐮 ¯ , P h ⁢ 𝐮 ¯ - 𝐮 ¯ h ) - ( Θ ¯ M + α ⁢ 𝐮 ¯ h , P h ⁢ 𝐮 ¯ - 𝐮 ¯ h ) ≳ | | | P h ⁢ 𝐮 ¯ - 𝐮 ¯ h | | | 2 . A substitution of this estimate in (A.16) leads to | | | P h ⁢ 𝐮 ¯ - 𝐮 ¯ h | | | ≲ h 3 / 2 helps to deduce

For Ω h 1 , follow the same steps up to (A.16) with L 2 ⁢ ( Ω ) replaced with L 2 ⁢ ( Ω h 1 ) to obtain

The left-hand side of (A.18) is estimated as

For the details of this proof (that uses the second-order sufficient optimality conditions [30, Theorem 2.17] and Lemma A.3), see [25, Lemma 10]. The last inequality substituted in (A.18) and (A.17) concludes the proof. ∎