TY - INPR A1 - Gugat, Martin A1 - Qian, Meizhi A1 - Sokolowski, Jan T1 - Topological derivative method for control of wave equation on networks N2 - The dynamical, boundary optimal control problems on networks are considered. The domain of definition for the distributed parameter system is given by a graph G. The optimal cost function for control problem is further optimized with respect to the shape and topology of the graph Ω. The small cycle is introduced and the topological derivative of the cost with respect to the size of the cycle is determined. In this way, the singular perturbations of the graph can be analyzed in order to change the topology Ω. The topological derivative method in shape and topology optimization is a new tool which can be used to minimize the shape functionals under the Partial Differential Equations (PDEs) constraints. The topological derivative is used as well for solution of optimum design problems for graphs. In optimal control problems the topological derivative is used for optimum design of the domain of integration of the state equation. As an example, optimal control problems are considered on a cross with a small cycle. The state equation is the wave equation on the graph. The boundary control problem by Neumann conditions at a boundary vertex is solved for a tracking cost function. The shape functional is given by the optimal value of the control cost. The topological derivative of the shape functional is determined for the steady state model with the size of a cycle ε → 0. Numerical results for a model problem are presented. KW - distributed parameter system KW - optimal control KW - shape optimization KW - topological derivative KW - network modelling Y1 - 2023 ER - TY - JOUR A1 - Gugat, Martin A1 - Qian, Meizhi A1 - Sokolowski, Jan T1 - Network Design and Control: Shape and Topology Optimization for the Turnpike Property for the Wave Equation N2 - The optimal control problems for the wave equation are considered on networks. The turnpike property is shown for the state equation, the adjoint state equation as well as the optimal cost. The shape and topology optimization is performed for the network with the shape functional given by the optimality system of the control problem. The set of admissible shapes for the network is compact in finite dimensions, thus the use of turnpike property is straightforward. The topology optimization is analysed for an example of nucleation of a small cycle at the internal node of network. The topological derivative of the cost is introduced and evaluated in the framework of domain decomposition technique. Numerical examples are provided. KW - Turnpike Property KW - Wave Equation KW - Optimal Control KW - Spectral Method KW - Network Optimum Design Y1 - 2024 VL - J. Geom. Anal. IS - 34 ER -