Слайд 2Variational Approach to the Fixed-Time, Free-Endpoint Problem
![Variational Approach to the Fixed-Time, Free-Endpoint Problem](/_ipx/f_webp&q_80&fit_contain&s_1440x1080/imagesDir/jpg/861774/slide-1.jpg)
Слайд 3We now want to see how far a variational approach – i.e.
![We now want to see how far a variational approach – i.e.](/_ipx/f_webp&q_80&fit_contain&s_1440x1080/imagesDir/jpg/861774/slide-2.jpg)
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.
Слайд 6
Our goal is to derive necessary conditions for optimality.
![Our goal is to derive necessary conditions for optimality.](/_ipx/f_webp&q_80&fit_contain&s_1440x1080/imagesDir/jpg/861774/slide-5.jpg)
Слайд 8
Thus, in the optimal control context it is more natural to directly
![Thus, in the optimal control context it is more natural to directly perturb the control instead.](/_ipx/f_webp&q_80&fit_contain&s_1440x1080/imagesDir/jpg/861774/slide-7.jpg)
perturb the control instead.
Слайд 9And then define perturbed state trajectories in terms of perturbed controls.
![And then define perturbed state trajectories in terms of perturbed controls.](/_ipx/f_webp&q_80&fit_contain&s_1440x1080/imagesDir/jpg/861774/slide-8.jpg)
Слайд 16
Motivated by Lagrange's idea for treating such constraints in calculus of variations,
![Motivated by Lagrange's idea for treating such constraints in calculus of variations,](/_ipx/f_webp&q_80&fit_contain&s_1440x1080/imagesDir/jpg/861774/slide-15.jpg)
expressed by an augmented cost, let us rewrite our cost as indicated below.
Слайд 17Clearly, the extra term inside the integral does not change the value
![Clearly, the extra term inside the integral does not change the value of the cost.](/_ipx/f_webp&q_80&fit_contain&s_1440x1080/imagesDir/jpg/861774/slide-16.jpg)
Слайд 31
This is just a reformulation of the property already discussed by us
![This is just a reformulation of the property already discussed by us](/_ipx/f_webp&q_80&fit_contain&s_1440x1080/imagesDir/jpg/861774/slide-30.jpg)
in the context of calculus of variations.
Слайд 32
which you can recognize as the system of Hamilton's canonical equations.
![which you can recognize as the system of Hamilton's canonical equations.](/_ipx/f_webp&q_80&fit_contain&s_1440x1080/imagesDir/jpg/861774/slide-31.jpg)
Слайд 35
Let’s summarize the results obtained so far and see how to apply
![Let’s summarize the results obtained so far and see how to apply them in practice.](/_ipx/f_webp&q_80&fit_contain&s_1440x1080/imagesDir/jpg/861774/slide-34.jpg)
them in practice.
Слайд 43
The integration of these equations leads to two constants whose values can
![The integration of these equations leads to two constants whose values can](/_ipx/f_webp&q_80&fit_contain&s_1440x1080/imagesDir/jpg/861774/slide-42.jpg)
be found from the known boundary conditions of the problem.