- Pontryagin maximum principle
- Lagrangians to Hamiltonians: Fenchel-Legendre transform.
Legendre (sometimes called Legendre-Fenchel) transform \(H\) of a convex function \(L:\Real\to\Real\)) is defined as
Both functions, \(L\) and \(H\) can take value \(+\infty\).
If \(L\) is strictly convex, the gradient mapping
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,
Differentiating, we obtain
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:
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,
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,
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 Mechanics, p 61ff.
- Transversality: boundary conditions as Lagrangian manifolds
- Minimal time linear problems. Bang-bang control.
- Sketch of the proof of PMP.