This question is a repost from mo.
Let $0 \leq r(t,s) \leq 1$, $t, \ s \in [0, T]$ be a smooth enough function, such that
- $r(t,t)$ increases in $t$
- $r(t, s) = r(s, t)$ decreases as $t$ and $s$ move away from each other (that is, as $|t - s|$ grows larger)
and let $\mathbf{w} \in \mathbb{R}^2_+$. Define $$ Q(t,s) = \mathbf{w}^\top \begin{pmatrix} r(t, t) & -r(t, s) \\ -r(t,s) & r(s, s) \end{pmatrix}^{-1} \mathbf{w}. $$
What other assumptions should I impose on $r$ (and possibly $\mathbf{w}$), so that $Q(t,s)$ have exactly one minimum in $0 < s < t < T$? Is it possibly to have a one-dimensional minimal manifold for some $r$?
Is there some standard trick to deal with problems like this? I tried to reduce the problem to some convex optimization, but have not succeeded.
I found a nice intuitive interpretation of this problem. It is quite obvious that $t$ and $s$ are repelled both from $0$ and each other, so this seems to be in a sense a question about equilibrium of two points.
Can something be said about a similar optimization problem $$ Q(t,s,\mu) = \mathbf{w}^\top \begin{pmatrix} r(t, t) & -r(t, s) & r(t, \mu) \\ -r(t,s) & r(s, s) & -r(s, \mu) \\ r(t, \mu) & -r(s, \mu) & r(\mu, \mu) \end{pmatrix}^{-1} \mathbf{w} \quad ? $$