Azərbaycan Dövlət Neft və Sənaye Universiteti Optimal Control

Variational Approach to the Fixed-Time, Free-Endpoint Problem

We now want to see how far a variational approach – i.e. an approach based on analyzing the first (and second) variation of the cost functional – can take us in studying the optimal control problem formulated in the previous lecture.

Our goal is to derive necessary conditions for optimality.

Thus, in the optimal control context it is more natural to directly perturb the control instead.

And then define perturbed state trajectories in terms of perturbed controls.

Motivated by Lagrange's idea for treating such constraints in calculus of variations, expressed by an augmented cost, let us rewrite our cost as indicated below.

Clearly, the extra term inside the integral does not change the value of the cost.

This is just a reformulation of the property already discussed by us in the context of calculus of variations.

which you can recognize as the system of Hamilton's canonical equations.

Let’s summarize the results obtained so far and see how to apply them in practice.

The integration of these equations leads to two constants whose values can be found from the known boundary conditions of the problem.

Thank you for attention