Optimal Impulse Control and Impulse Game for Continuous-Time Deterministic Systems: A Review

Optimal impulse control and impulse games provide the cutting-edge frameworks for modeling systems where control actions occur at discrete time points, and optimizing objectives under discontinuous interventions. This review synthesizes the theoretical advancements, computational approaches, emerging challenges, and possible research directions in the field. Firstly, we briefly review the fundamental theory of optimal control theory for continuous-time systems, including Pontryagin's maximum principle (PMP) and dynamic programming principle (DPP). Secondly, we present the foundational results in optimal impulse control, including necessary conditions and sufficient conditions. Thirdly, we systematize impulse game methodologies, from Nash equilibrium existence theory to the connection between Nash equilibrium and systems stability. Fourthly, we summarize the numerical algorithms including the intelligent computation approaches. Finally, we examine the new trends and challenges in theory and applications as well as computational considerations.