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Consider the equation

$\frac{d Y_t}{dt} = (A + w(t)B) Y_t$

evolving on a compact semi-simple Lie group $G$ where $\frak{g}$ is the Lie algebra of $G$ and $A,B \in \frak{g}$ and:

$J:G \rightarrow [0,1]$

which has a single global maximum and minimum, and any number of saddle points. Also assume that $A$ and $B$ generate the whole algebra $\frak{g}$.

Is it true that $F[w]= J(V_T[w])$ (where $V_T$ is the end point map of that above differential equation, where $T$ is a final time large enough that the system is accessible) has no local optima?

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  • $\begingroup$ The expression $AY_t$ means the right translation of $A$ by $Y_t$? $\endgroup$
    – Ben McKay
    Commented Apr 6, 2016 at 17:11
  • $\begingroup$ Surely $F$ has optima at all paths $w$ that reach the max or min of $J$ at time $T$. Since you can access the whole Lie group, why would that not happen? I must be missing something. $\endgroup$
    – Ben McKay
    Commented Apr 6, 2016 at 17:17
  • $\begingroup$ You ask if $F[w]$ only has no local optima, but you say that you know it has global, hence local, optima. But then it does not have only no local optima. I assume that "only no" means "no". What else might "only no" mean? $\endgroup$
    – Ben McKay
    Commented Apr 6, 2016 at 18:47
  • $\begingroup$ I should have said: "Yes to the first question. Yes, those optima exist. However, I'm asking about local optima. I.e. w which are local optima of $F$, but for which $V_T[w]$ is NOT a global optima of $J$" $\endgroup$
    – Benjamin
    Commented Apr 6, 2016 at 19:34

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