This paper proposes a task-specific trajectory optimization framework for human-robot collaboration, enabling adaptive motion planning based on human interaction dynamics. Unlike conventional approaches that rely on predefined desired trajectories, the proposed framework optimizes the collaborative motion dynamically using the inverse differential Riccati equation, ensuring adaptability to task variations and human input. The generated trajectory serves as the reference for a neuro-adaptive PID controller, which leverages a neural network to adjust control gains in real time, addressing system uncertainties while maintaining low computational complexity. The combination of trajectory planning and the adaptive control law ensures stability and accurate joint-space tracking without requiring extensive parameter tuning. Numerical simulations validate the proposed approach.