Local null controllability of the control-affine nonlinear systems with time-varying disturbances

The problem of local null controllability for the control-affine nonlinear systems $\dot x(t)=f(x(t))+Bu(t)+w(t),$ $t\in[0,T]$ is considered in this paper. The principal requirements on the system are that the LTI pair $\left((\partial f/\partial x)(0), B\right)$ is controllable and the disturbance is limited by the constraint $|f(0)+w(t)|\leq M_d\left(1-\frac{t}{T}\right)^\eta,$ $M_d\geq0$ and $\eta>0.$ These properties together with one technical assumption yield a complete answer to the problem of deciding when the null controllable region have a nonempty interior. The criteria obtained involve purely algebraic manipulations of vector field $f,$ input matrix $B$ and bound on the disturbance $w(t).$ To prove the main result we have derived a new Gronwall-type inequality allowing the fine estimates of the closed-loop solutions. The theory is illustrated and the efficacy of proposed controller is demonstrated by the examples where the null controllable region is explicitly calculated. Finally we established the sufficient conditions to be the system under consideration (with $w(t)\equiv 0$) globally null controllable.