Use the implicit function theorem to prove smoothness

Quick description

Suppose you have a function and you would like to see that is . One way to show this is to find a function such that and The implicit function theorem will imply that is . Incomplete This article is incomplete. This article is missing large parts of what is required of a Tricki article. This is only one way I know the implicit function theorem is used to show regularity. This may not be the best way to state this idea.

Calculus

Example 1

Deterministic and Stochastic Optimal Control, Wendell H. Fleming, Raymond W. Rishel, page 8. Take a cost function and consider the calculus of variation problem of minimizing (1)

over piecewise functions with the given fixed end points and . A piecewise function is called an extremal of (1) if it satisfies (2)

for where is a constant and where denotes the variable of for which is substituted in (1) (the third variable).

Now let us assume that and consider an extremal of (1) that is also , i.e., is continuous. We will show using the implicit function theorem that must be , i.e., if the extremal is then it has to be as smooth as the cost function .

Remark Before we proceed further a quick remark about the assumption . This assumption precludes from sudden changes and forces it to be , even if this were not assumed. In order not to detract from the argument based on the implicit function theorem we assumed that is . The assumption will also come into play in the invocation of the implicit function theorem.

Now let us continue with our argument that must be as smooth as . The argument proceeds by induction. We already have the base case that is . Now let us suppose that it is for a . Then is as well. By (2) (3)

for some constant . Define  is and and is , it follows that is . We can rewrite (3) as  and this is strictly positive by assumption. These imply that is at least as smooth as , thus is . It follows that is .

are there other examples of the argument in example one?

The above example has a more sophisticated method than the one I tried to describe in the quick description section. In the generalization of the method of example 1, there would be a function and the goal would be to improve our understanding of its smoothness. To that end, we use itself to define another function such that , and is as smooth as . Now the implicit function theorem says is as smooth as and hence . The argument is continued as many steps as possible. Does anyone know of another application of this argument, or an argument similar to this?

Notation

The notation " " isn't very clear. My first thought was: "partial derivative with respect to ". After this, I figured you meant the operator on given by the decomposition of (the Jacobian of ), since this is the condition required by the theorem. Still, the notation gives rise the possibility .

What does the notation mean in equation 2?

Instead of using , why not use ?

Notation

Thanks for the comment. I like the notation and frequently use it because it consisely states everything related to the operation (we are taking a derivative with respect to a variable.) The one you suggest ( ) is also good I think. As for . This is common notation in many books, including Fleming and Rishel. I don't prefer it because I think it is confusing to use the same symbol to mean two entirely different things on the same page. In this case, if we use the notation, will mean the derivative of the function with respect to time and also the name of a free variable. is a is a free variable representing a function satisfying the Euler-Lagrange equation. It probably is not a good choice of notation because as you point out upon reading it one thinks it must be related to the in its context. I changed it to . This I think causes an abuse of notation, but hopefully not a confusing one.