Let $f:\mathbb{R}^d\times \mathbb{R}^n\rightarrow \mathbb{R}^d$ be a $C^2$-function and $u_t$ be a "sufficiently well-behaved smooth map from $[0,T]$ to $\mathbb{R}^n$ such that the (controlled) ODE: $$ \partial_t X_t^{x,u} = f(X_t^{x,u},u_t); \qquad X_0^{x,u}=x, $$ has a (unique) solution for each initial condition $X_0^{x,u}=x$. Fix $x_1,\dots,x_n,y_1,\dots,y_n,z \in \mathbb{R}^d$, $T,\epsilon>0$.
Is there a solution to the deterministic control problem, defined by minimizing: $$ J[u]\triangleq \frac1{n}\sum_{i=1}^n \|X_{t}^{x_i,u}- y_i\|_2 + \int_0^{t} \|u_s\|ds , $$ $$ \tau\triangleq \inf \left\{ t\geq 0 : \frac1{k}\sum_{j=1}^k \|X_t^{z_j,u}\|_2 >1 \right\}? $$