\(\def\Real{\mathbb{R}}\)

- Pontryagin maximum principle
- Examples
- Lagrangians to Hamiltonians: Fenchel-Legendre transform.
**Legendre (sometimes called Legendre-Fenchel) transform**\(H\) of a convex function \(L:\Real\to\Real\)) is defined as

\[

H(p):=\sup_x px-L(x).

\]

Both functions, \(L\) and \(H\) can take value \(+\infty\).If \(L\) is strictly convex, the gradient mapping

\[

G:x\mapsto \frac{dL}{dx}

\]

is one-to-one, and it is clear that the argmax in the definition of \(H\) solves \(p=G(x)\). Denote the functional inverse to \(G\) as \(F: p=G(x) \Leftrightarrow x=F(p)\). In this case,

\[

H(p)=pF(p)-L(F(p)).

\]

Differentiating, we obtain

\[

\frac{dH}{dp}=F(p)+\left(p-\frac{dL}{dx}\right)\frac{dF}{dp}=F(p)

\]Hence the graph of the derivative of \(L\) with respect to \(x\), in the \((x,p)\) space is the same as the graph of the derivative of \(H\) with respect to \(p\). It follows immediately that the Legendre transform is involutive:

\[

L(x)=\sup_p px-H(p).

\]If \(L\) is convex, one can turn it into a strictly convex function by adding \(\epsilon |x|^2/2\). The graph of the resulting gradient,

\[

p=G(x)+\epsilon x

\]

converges to the graph of \(p=G(x)\) as \(\epsilon \to 0\). So one can define the graph \(p=G(x)\) for all convex functions.One can also define Legendre transform for nonconvex \(L\), using the same formula. The correspondence won’t involutive anymore (as Legendre-Fenchel transforms, as defined above, are always convex). Still, one can reproduce many results, in particular, that the derivative of \(H\) is discontinuous at \(p\) iff \(L-px\) has two competing global maxima, see figure below.

These considerations are relevant for the Homework 5, as in Problem a,

\[

V(x,0)=\inf_X \frac{(x-X)^2}{2T}+\cos(X)=\frac{x^2}{2T}-\frac{1}{T}

\sup \left(xX-\left(\frac{X^2}{2}+T\cos(X)\right)\right),

\]

i.e. is (somewhat modified) Legendre transform of the function \(X^2+T\cos X\).For more, see chapter on Legendre transform in Arnold’s

*Mechanic*s, p 61*ff*. - Transversality: boundary conditions as Lagrangian manifolds
- Minimal time linear problems. Bang-bang control.
- Sketch of the proof of PMP.

Does this result imply that for a non convex function have 2 competing global maxima/minima, the function will be not continuously differentiable?

Yes, if px-L has two competing maxima, H is not smooth at p.

In the last equation, is cos(T) supposed to be cos(x(T))=cos(X)? I don’t see how it is possible to extract the T out. Please let me know.

Thank, Tony, – typo corrected.