Fast controls and minimum time (Q2767806)

The system is NEWLINE\[NEWLINE y'(t) = Ay(t) + Bu(t) \tag{1}NEWLINE\]NEWLINE where \(A\) is the infinitesimal generator of a strongly continuous semigroup \(S(t)\) in a Banach space \(X;\) the controls \(u(\cdot)\) belong to \(L^p(0, T; U)\) \((U\) another Banach space, \(1 \leq p \leq \infty)\) and \(B : U \to X\) is a linear bounded operator. The system (1) is assumed null controllable, which implies that given \(T, \rho > 0\) there exists \(\alpha = \alpha(T, \rho)\) such that every \(y\) in the ball with center \(0\) and radius \(\alpha(T, \rho)\) can be driven to zero by a \(u(\cdot) \in L^p(0, T; U)\) with \(\|u(\cdot)\|_{L^p(0, T; U)} \leq \rho.\) NEWLINENEWLINENEWLINEThe authors show that the following problems are equivalent: \((a)\) estimates of \(\alpha(T, \rho)\) for \(T\) small, \((b)\) estimates of the minimum time function, \((c)\) estimates of the minimum \(L^p\) norm for \(T\) small.