Gugat M, Qian M, Sokolowski J (2023)

**Publication Language:** English

**Publication Type:** Conference contribution

**Publication year:** 2023

**Publisher:** IEEE Xplore

**Pages Range:** 320-325

**Conference Proceedings Title:** 2023 27th International Conference on Methods and Models in Automation and Robotics

**ISBN:** 979-8-3503-1107-5

**DOI:** 10.1109/MMAR58394.2023.10242484

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.

Martin Gugat
Lehrstuhl für Dynamics, Control, Machine Learning and Numerics (Alexander von Humboldt-Professur)

East China Normal University (ECNU) / 华东师范大学
China (CN)
Polska Akademia Nauk (PAN) / Polish Academy of Sciences
Poland (PL)

**APA:**

Gugat, M., Qian, M., & Sokolowski, J. (2023). Topological derivative method for control of wave equation on networks. In IEEE Xplore (Eds.), *2023 27th International Conference on Methods and Models in Automation and Robotics* (pp. 320-325). Międzyzdroje, PL: IEEE Xplore.

**MLA:**

Gugat, Martin, Meizhi Qian, and Jan Sokolowski. "Topological derivative method for control of wave equation on networks." *Proceedings of the 27th International Conference on Methods and Models in Automation and Robotics (MMAR), Międzyzdroje* Ed. IEEE Xplore, IEEE Xplore, 2023. 320-325.

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