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DOI: 10.23952/cot.2026.32
Received February 22, 2025; Accepted June 26, 2025; Published online March 26, 2026
Abstract. In this paper, we study a Dirichlet control problem for the Poisson equation, where the control is assumed to be piecewise constant function which is allowed to take 
different values. The space of admissible Dirichlet controls is non-convex and therefore standard derivatives in Banach spaces are not applicable. Furthermore piecewise constant functions do not belong to 
and standard extension techniques to consider the weak solution of the Dirichlet problem do not apply. Therefore we resort to the notion of very weak solutions of the state equation in 
spaces. We then study the differentiability of the shape-to-state operator of this problem and derive the first order necessary optimality conditions using the topological state derivative. In fact we prove the existence of the weak topological state derivative introduced in [P. Baumann, I. Mazari-Fouquer, K. Sturm, The topological state derivative: An optimal control perspective on topology optimisation, J. Geom. Anal. 33 (2023) 243] for a multimaterial shape functional which is then expressed via an adjoint variable. The topological derivative resembles formulas found for derivative in the more standard Dirichlet control problems. In the final part of the paper we show how to apply a multimaterial level-set algorithm with the finite element software NGSolve [J. Schöberl, C++11 Implementation of Finite Elements in NGSolve, ASC Report No. 30/2014, 2014] and present several numerical examples in dimension three.
How to Cite this Article:
K. turm, A multimaterial topology optimisation approach to Dirichlet control with piecewise constant functions, Commun. Optim. Theory 2026 (2026) 32.