On the closure of reachable sets for control systems (Q1337227)

Given the following control system: \[ x^{(n)}_ 1+ a_{n- 1} x^{(n- 1)}_ 1+ \cdots+ a_ 1 x_ 1'+ a_ 0 x_ 1\in \{\phi_ 1, \phi_ 2\}\text{ a.e. on }[a,b],\quad x_ 2'= f(x_ 1), \] where \(\phi_ 1,\phi_ 2\in L^ 1([a,b])\), with \(\phi_ 1\leq \phi_ 2\), one proves that the reachable set is closed for a class of maps that is dense in \({\mathcal C}(\mathbb{R})\) for the topology of the uniform convergence.