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Given a function (functional actually) $f(x,g(x))$, can a notion of simplicity be attached with respect to the function $g(x)$? (all functions and args are real).

Specifically, intuitively one could say that the function $f(x,0)$ is simpler than the general function $f(x,g(x))$ - notice that $g(x)$ is null- because it eliminates terms. However, how could one quantify this using some measure of $f$? For example, energy arguments could be used. Is there a way to do this?

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  • $\begingroup$ You haven't given much information in your question. Why would you want this kind of definition, could you give a context, or a possible application? $\endgroup$
    – Ryan Budney
    Jan 2, 2012 at 5:14
  • $\begingroup$ Well, it is a problem in control theory and particularly in tracking control. In a nutshell, given a tracking controller $f(x,g(x))$ for a nonlinear system, which tracks arbitrary reference paths $g(x)$, if the reference is a straight line, can i simplify the controller? $\endgroup$
    – Jorge
    Jan 2, 2012 at 12:12

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