# No local optima in quantum control?

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 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.

Cross-posted from MO: https://mathoverflow.net/q/221383/41654 after no answer.

• Just a comment passing by: isn't there any topological argument that shows that it is impossible to not have any local optima? I mean, Euler characteristic + index theory or Betti numbers + Morse inequalities? – Evgeny Oct 22 '15 at 8:41
• I don't know if that is the case, that would be very interesting and is the sort of thing I'm after here but I lack the expertise in such things. – Benjamin Oct 22 '15 at 14:08
• I should add, $J$ certainly has local optima. This is not the question. The question is about 'extra' local optima being introduced by singularities of the end point map. – Benjamin Nov 3 '15 at 16:15
• Sorry, I still had no time to look closer at this problem... – Evgeny Nov 4 '15 at 10:02