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.

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 (I−K)⁻¹ exists if and only if for some n ₀ and all n≥n ₀ the operators (I−K _n )⁻¹ exist and are uniformly bounded, in which case (I−K _n )⁻¹→(I−K)⁻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 I−K 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 η∈L 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 φ∈L 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 ⁻¹→ψ′(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. □