An inner estimate of the attainability set of a nonlinear controlled object (Q2703872)

The paper investigates the problem of controllability in time \(T>0\): Namely, does \(0\) belong to the reachable set in time \(T>0\) (fixed) NEWLINE\[NEWLINED(T)= \left\{x(T)\;/\;x(-)\text{ solves }\begin{cases} x'= f(x,u)\\ x(0)= 0\end{cases}\right\}NEWLINE\]NEWLINE (with \(0\in U\) and \(f(0,0)= 0\))?NEWLINENEWLINENEWLINEThe author gives some sufficient conditions for the solution to the above problem. He underlines an inner estimation \(K\) of the interior of \(D(T)\) (namely \(0\in \text{Int}(k)\subset D(T)\)). Applications of the proposed method are discussed for the classical example of the controlled pendulum.