A general approach to the solution of the bounded control synthesis problem in a controllability problem

Consider a system of differential equations
$$ \dot x=f(x,u),\qquad u\in M, $$
where $x$ an $n$-dimensional vector, $u$ is an $r$-dimensional control, and $M$ is a subset of an $r$-dimensional space. A general approach to the solution of the following synthesis problem is proposed: construct a control $u=u(x)\in M$ such that the corresponding trajectory of the system $\dot x=f(x,u(x))$ starting at an arbitrary point $x_0$ terminates at the final time $T(x_0)$ at the point $x_1$.
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