Characterization of controllability for a class of nonlinear retarded functional differential equations (Q1910023)

The paper deals with the problem of functional controllability for the following nonlinear time-delay system \[ \dot x(t)= f(t, x_t)+ B(t)u(t),\quad t\in [t_0, t_1], \] \[ (x_t: [-h, 0]\to \mathbb{R}^n) \] which is linear with respect to the control \(u\). Such a problem has a solution only in the exceptional case when \[ \text{rank }B(t)= n\quad\text{a.e.}\quad [t_0- h, t_1] \] which is not a generalization of the well-known Kalman criterion of controllability \[ \text{rank}[B, AB,\dots, A^{n-1} B]= n \] for linear stationary system \(\dot x(t)= Ax(t)+ Bu(t)\) with no delay. Unfortunately, the author does not present any practical examples in which such a kind of controllability is essentially used. Observe that there are several kinds of functional controllability (\(F\)-controllability) (A. Manitius), \((s,t)\)-controllability (V. Marchenko)) generalizing Kalman's type of controllability to systems with delay.