  Prompt Engineering (PE) has emerged as a critical technique for guiding Large
Language Models (LLMs) in solving intricate tasks. Its importance is
highlighted by its potential to significantly enhance the efficiency and
effectiveness of human-machine interaction. As tasks grow increasingly complex,
recent advanced PE methods have extended beyond the limitations of single-round
interactions to embrace multi-round interactions, which allows for a deeper and
more nuanced engagement with LLMs. In this paper, we propose an optimal control
framework tailored for multi-round interactions with LLMs. This framework
provides a unified mathematical structure that not only systematizes the
existing PE methods but also sets the stage for rigorous analytical
improvements. Furthermore, we extend this framework to include PE via ensemble
methods and multi-agent collaboration, thereby enlarging the scope of
applicability. By adopting an optimal control perspective, we offer fresh
insights into existing PE methods and highlight theoretical challenges that
warrant future research. Besides, our work lays a foundation for the
development of more effective and interpretable PE methods.

Machine Learning;Optimization and Control