Understanding Lagrange Multipliers in Dynamic Optimization

In static optimization problems, the Lagrange multiplier is a constant value. However, in dynamic optimization, the Lagrange multiplier, denoted as λ(t), becomes a function of time. To eliminate terms containing the derivative of state variables (e.g., velocity, ẋ(t)), we can employ integration by parts. This technique allows us to rewrite the last term on the right-hand side of the equation and simplify the problem.