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A semismooth Newton method for a class of semilinear optimal control problems with box and volume constraints

Abstract

In this paper we consider optimal control problems subject to a semilinear elliptic state equation together with the control constraints 0≤u≤1 and ∫u=m. Optimality conditions for this problem are derived and reformulated as a nonlinear, nonsmooth equation which is solved using a semismooth Newton method. A regularization of the nonsmooth equation is necessary to obtain the superlinear convergence of the semismooth Newton method. We prove that the solutions of the regularized problems converge to a solution of the original problem and a path-following technique is used to ensure a constant decrease rate of the residual. We show that, in certain situations, the optimal controls take 0–1 values, which amounts to solving a topology optimization problem with volume constraint.

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Appendices

Appendix A: Collectively compact sets of operators

Let \(\mathcal {X}\) be a Banach space and \(\mathcal{K}\) be a subset of \(\mathcal {L}(\mathcal {X})\), where \(\mathcal{L}(\mathcal {X})\) is the set of bounded linear operators from \(\mathcal {X}\) into itself.

Definition 2

We say that \(\mathcal{K}\) is collectively compact if the set \(\{Kx,x\in \mathcal {X},\Vert x\Vert\leq1,\allowbreak K\in\mathcal{K}\}\) is relatively compact.

Obviously, if \(\mathcal{K}\) is collectively compact, every \(K\in\mathcal {K}\) is compact. The following result may be found in [3, Theorem 1.6].

Theorem 10

Let K, \((K_{n})_{n\in\mathbb{N}}\in\mathcal{L}(\mathcal {X})\). Assume K n K pointwise, {K n } is collectively compact and K is compact. Then (IK)−1 exists if and only if for some n 0 and all nn 0 the operators (IK n )−1 exist and are uniformly bounded, in which case (IK n )−1→(IK)−1 pointwise.

The following result can be easily deduced from Theorem 10; see [2].

Theorem 11

Let \(\mathcal{K}\) be a collectively compact set of bounded linear operators of \(\mathcal {X}\). Assume further that \(\mathcal{K}\) is pointwise sequentially compact, i.e., for every sequence (K n ) of \(\mathcal{K}\) there exists a subsequence \((K_{n_{p}})\) and \(K\in\mathcal{K}\) such that \(K_{n_{p}} x \to Kx\) for all \(x\in \mathcal {X}\). If IK is invertible for all \(K\in\mathcal{K}\), then

$$ \sup_{K\in\mathcal{K}} \bigl\Vert(I-K)^{-1}\bigr\Vert<\infty. $$
(60)

Appendix B: Operator convergence

Lemma 6

If y n y in L (D) then, for all ηL 2(D),

$$ B(y_n)^{-1} \eta= \bigl[A+ \psi '(y_n)\bigr]^{-1} \eta\to\bigl[A+ \psi '(y)\bigr]^{-1} \eta=B(y)^{-1} \eta\quad \mathit{in}\ L^\infty(D). $$
(61)

Proof

With y n y in L (D) and using \(\Vert\psi'' \Vert_{L^{\infty}}\leq M^{2}_{\psi}\) we obtain

$$ \psi '(y_n) \to \psi '(y) \quad \mbox{in}\ L^\infty(D). $$
(62)

We write

$$B(y_n)^{-1} = A^{-1}\bigl[I+\psi '(y_n)A^{-1} \bigr]^{-1}. $$

The family of operators {ψ′(y n )A −1:L 2(D)→L 2(D)} is collectively compact due to the compactness of A −1 and the uniform boundedness of \(\|\psi '(y_{n})\bigr\|_{L^{\infty}}\). We have for all φL 2(D)

$$\bigl\langle\bigl(I+\psi '(y) A^{-1}\bigr) \varphi,A^{-1}\varphi\bigr\rangle= \bigl\langle A^{-1}\varphi, \varphi\bigr\rangle+ \bigl\langle \psi '(y) A^{-1}\varphi ,A^{-1}\varphi\bigr\rangle\geq\bigl\langle A^{-1}\varphi, \varphi\bigr\rangle, $$

hence I+ψ′(y)A −1 is injective and subsequently invertible by the Fredholm alternative. In view of (62), we also have the pointwise convergence ψ′(y n )A −1ψ′(y)A −1, we may thus apply Theorem 10 to obtain

$$\bigl[I+\psi '(y_n)A^{-1} \bigr]^{-1} \to \bigl[I+\psi '(y)A^{-1} \bigr]^{-1}\quad \mbox{pointwise in }L^2(D), $$

which in turn implies (61) by composition with A −1. □

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Amstutz, S., Laurain, A. A semismooth Newton method for a class of semilinear optimal control problems with box and volume constraints. Comput Optim Appl 56, 369–403 (2013). https://doi.org/10.1007/s10589-013-9555-6

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Keywords

  • Optimal control
  • Topology optimization
  • Semilinear equation
  • Semismooth Newton method
  • Volume constraint