Given a manifold $M$ and a set of smooth functions of one real variable $\mathcal{A}$ and a 'control system' type first order differential equation:
$\frac{d x(t)}{dt} = F(x,u)$
one can consider the 'end-point map' $V_T:\mathcal{A}\rightarrow M$ which takes a control and sends it to the associated solution at time $T$.
Further given a smooth real valued function $J: M \rightarrow [0,1]$ with a single optimum taking the value $1$ ,is there a general principle for determining if a given $G(w)=J(V_T(w))$ has, assuming that the system is controllable, no local optima in the space of controls? It is clear to me that it can have.
One important example is that of quantum control. In this example $M=SU(n)$, $\mathcal{A}$ is some space of smooth functions (typically large or just taken as all smooth, bounded functions) the differential equation in equation is:
$\frac{d U_t}{dt}= (a + w(t)b)U_t$
where $a,b$ generate $\mathfrak{su}(n)$. In this case $J(U)=|Tr(U^{\dagger}G)|^2$ for some $G\in SU(n)$. Numerical evidence very strongly suggests that $J(V_T(w))$ never has local optima in these situations for $SU(4)$, but this seems very hard to prove. My attempts have all involved attempting to find the appropriate Hessian and understanding its index, but to no avail. My instinct is that for almost all $a,b$ there will no local optima.
