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 x˙(t)=f(x(t))+Bu(t)+w(t),t ∈ [0, T] is considered in this paper. The principal requirements on the system are that the LTI pair ((∂f/∂x)(0), B) is controllable and the disturbance is limited by the constraint |f(0)+w(t)|≤Md(1−tT)η,Md ≥ 0 and η > 0. These properties together with one technical assumption yield an answer to the problem of deciding when the null controllable region has a nonempty interior. The obtained criterion is built on the purely algebraic and/or differential manipulations with vector field f, input matrix B and a 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 example where the null controllable region is explicitly calculated. Finally, we established the sufficient conditions to be the system under consideration with w(t) ≡ 0 globally null controllable.