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.

This paper proposes a integrated framework for optimal control of opinion dynamics in social networks, addressing three progressively challenging scenarios: Model-based stochastic control, where agent interactions follow known probability distributions, enabling analytical optimal policies; Model-free Reinforcement Learning (RL), where interaction randomness has unknown distributions but system dynamics are preserved; Data-driven RL for unknown systems, where time-varying network dynamics (with stochasticity constraints) are fully unknown, requiring purely observational learning. By designing an RL control framework grounded in convex quadratic optimization, we bridge model-based control and data-driven learning, offering new insights for social network manipulation and multi-agent coordination. Numerical simulations demonstrate the framework's effectiveness.