Network design and control: shape and topology optimization for the turnpike property for the wave equation (Q6572139)

In this paper, the optimal shape and topology design problem, with dynamics governed by the wave equation on networks, is considered. The authors incorporate the Turnpike Property into the optimal control of a PDE alongside shape analysis.\N\NThis approach provides a framework where the complex dynamics and optimization of these systems can be reduced to studying and controlling around steady-state (turnpike) solutions, significantly simplifying the problem by focusing on long-term behaviors and stable configurations.\N\NUnder some regularity assumptions, the authors establish that the difference between dynamic and static problems exhibits the exponential Turnpike Property for the state equation, adjoint equation, and cost.\N\NThe shape and topological derivatives of a given cost for the network are defined and evaluated within the framework of the domain decomposition technique. Two examples of singular network perturbations by the nucleation of a small cycle are presented.\N\NNumerical examples, solved using MATLAB software, are provided. The results of the analysis are presented in plots.\N\NIt should be emphasized that the study of the Turnpike Property in the context of shape optimization is a novel approach.