Costate Variables in Optimal Control: Combining Objective Function and State Equations

In optimal control problems, a powerful method combines the objective function and the state equations using costate variables. These variables, denoted as λ(t), function similarly to Lagrange multipliers in static optimization. However, unlike constant Lagrange multipliers, costate variables are functions of time, capturing the dynamic evolution of the system. This approach, akin to using a Lagrangian in static optimization, allows for a comprehensive formulation that accounts for both the desired objective and the system's governing equations over time.