B Is Implicit Function Theorem Useful in Optimal Control Theory?

[mech-eng]

Would you please explain what an implicit function in general is? Why $y^2+x^2=c$ is assumed as implicit even though it can be expressed in terms of $y$?

$y^2=c-x^2$ and then $y=\sqrt |x|$

Thank you.

 

[QuantumQuest]

In general, any function we get by taking a relation $f(x,y) = g(x,y)$ and solving for $y$ is called an implicit function for the relation at hand. But keep in mind that a relation may have more than one implicit function. The example you give i.e. $x^2 + y^2 = c$ has more than one implicit function. If you solve for $y^2$ as you did and you want to get $y$, you need to take the square root of the right hand side and this leads to one positive square root (one implicit function) and one negative (a second implicit function) for any value you give to $c$.

 

[FactChecker]

You have given an example where it is simple to solve for y (although you need to be careful about 2 possible solutions). But you can also leave it implicitly defined. There are other examples that are much harder to solve for y. If you do not or can not solve for y, then you have an implicit function.

 

[member 587159]

Very relevant is https://en.wikipedia.org/wiki/Implicit_function_theorem which tells us when a certain expression defines a function implicitely such that it is continuously differentiable and uniquely.

 

[nuuskur]

Let's work in $\Bbb R^2$, say. Suppose you had an equation $F(x,y)=0$. Is there such a subset $X\subseteq \Bbb R$ such that for a fixed $x\in X$ there is a unique $y$ such that the equation is satisfied? If so, we say $F$ determines implicitly a mapping $f:X\to\Bbb R$ (satisfying $F(x,f(x))=0, x\in X$). We don't want to pick some funny weird subsets $X$. We want it to be open so we could talk about differentiability (which is a very strong assumption - the course I took on optimal control theory Heavily relies on smoothness of the object function and the implicit function theorem becomes a powerful weapon - of course, it restricts the choice of $F$, as well).