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

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

In optimal control theory, transversality conditions play a critical role in identifying optimal trajectories. These conditions differ depending on whether the terminal value of the state variable is fixed or free.

Fixed Terminal Value:

If both the initial value 'x(t0)' and terminal value 'x(t1)' are fixed, meaning 'dx(t0)=dx(t1)=0', then no specific conditions on the costate variables 'λ(t0)' and 'λ(t1)' are required.

Free Terminal Value:

However, when the terminal value is free, a common scenario in many applications, an additional condition becomes necessary for optimality: 'λ(t1)=0'. This crucial condition is known as the transversality condition for a fixed horizon problem.

The Hamiltonian Connection:

Interestingly, these necessary conditions are essentially the same as those derived from the Hamiltonian of the system. This highlights the Hamiltonian's significance as a powerful tool for generating the first-order necessary conditions in optimal control problems.

In summary:

• Fixed terminal value: No specific conditions on 'λ(t0)' and 'λ(t1)' are needed.* Free terminal value: The transversality condition 'λ(t1)=0' is necessary.* The Hamiltonian provides a unified framework for deriving these conditions.