This paper considers an optimal control problem in opinion dynamics, where the time interval is assumed to be divided into two segments. In one segment, the agent is under the player’s control, while in the other, the player does not exert control. The goal of the player is to bring the agent’s opinion close to a specific target value, while the player aims to minimize the payment function. By solving the player’s closed-loop optimal control using the HJB equation method and applying the Pontryagin Maximum Principle to derive the player’s open-loop optimal control, the simulation results indicate that the player incurs higher costs when implementing closed-loop control.

This paper investigates the optimal control problem of opinion dynamics within complex networks. By introducing a state transformation, the original problem is reformulated within a discounted Linear Quadratic Regulator (LQR) framework, establishing a connection between opinion control and classical control theory. Within this unified framework, the optimal control law can be obtained by solving the discrete-time algebraic Riccati equation, thereby circumventing the complexity of dealing with linear terms inherent in traditional dynamic programming approaches. Numerical experiments validate the effectiveness of the algorithm in a benchmark case, a 20-node complete network, and complex topologies. They also reveal the influence mechanisms of network heterogeneity on convergence speed and control energy consumption, providing a theoretical basis for public opinion guidance strategies under different network structures.