Yuexin Cao: Forward and Inverse Problems in Optimal Control

Abstract: In this thesis, we investigate two related problems in systems theory: the control and optimization in networked systems, and the reconstruction of unknown cost functions in optimal control. Both problems arise broadly in the real-world applications, such as engineering, biological, and social systems. The first part focuses on optimal control placement in large-scale networks. We establish graph-theoretic necessary and sufficient conditions for controllability in systems derived from Turing’s model by analyzing the eigenstructure of the system matrix. We then study energy-efficient control placement by exploiting network symmetries, enabling efficient computation of the controllability Gramian spectrum and providing closed-form algorithms for maximizing the Gramian trace under limited controls. The second part addresses inverse problems in optimal control. For continuous-time LQR with known dynamics, we derive conditions for uniquely recovering state, control, and terminal cost matrices, and characterize the solution space in ill-posed cases. In the model-free setting, we propose an adaptive IRL algorithm based on primal–dual interior-point iterations using off-policy data, achieving efficient online learning with provable convergence.