Pontryagin Principle

Question

Is there any reference from which I can clearly understand The Pontryagin Maximum Principle (with some clear examples). I need it to apply to dynamical systems (differential equations) and P.D.E (eventually).

Answer 1

I suggest you to read the book from Lenhart/Workman Optimal control applied to biological models. It's a very good book for a first sight into optimal control.

But, basically, the idea is that you'll suppose the existence $u^{\epsilon}(t) = u^*(t)+\epsilon$, where $u^*(t)$ is an optimal control. Then, you'll consider the next function $$J(u^\epsilon)=\int_a^b f\left(t,x\left(t\right),u^\epsilon\left(t\right)\right)dt$$. Then you'll differenciate and equal to $0$ the $J$ function, in the point $\epsilon\rightarrow 0$. With the equation $$\frac{\partial J}{\partial\epsilon}\Bigg|_{\epsilon=0}=0,$$

after some algebraic manipulations and math tricks, you'll get the Pontryagin Principle.I am just giving to you the spirit of the principle and you should go for a book.

NOTE: Pontryagin only takes local optimals. Easily it can fails.