Transversality Condition in Optimal Control: Fixed vs. Free Terminal Values

In optimal control problems, the transversality condition plays a crucial role in determining the optimal trajectory. It outlines the necessary conditions on the costate variables, often denoted by λ(t), at the initial and terminal time points.

When both the initial value x(t0) and terminal value x(t1) are fixed, meaning dx(t0) = dx(t1) = 0, no specific conditions are required for λ(t0) and λ(t1). This is because the fixed endpoints already constrain the problem sufficiently.

However, if the terminal value is free, which is a common scenario, an additional condition becomes necessary for optimality: λ(t1) = 0. This condition is known as the transversality condition for a fixed horizon problem. It essentially ensures that the costate variable, which represents the sensitivity of the objective function to changes in the state variable, becomes zero at the terminal time, indicating that no further improvement in the objective function is possible by adjusting the terminal state.

Understanding the transversality condition is essential for correctly formulating and solving optimal control problems, especially when dealing with free terminal values.