On the null-controllability in function space of nonlinear systems of neutral functional differential equations with limited controls

Consider the following functional equations of neutral type: $$begin{gathered} (i) (d/dt)D(t,x_t ) = L(t,x_t ), hfill (ii) (d/dt)D(t,x_t ) = L(t,x_t ) + B(t)u(t), hfill (iii) (d/dt)D(t,x_t ) = L(t,x_t ) + B(t)u(t) + f(t,x(t),u(t)), hfill end{gathered} $$ whereD, L are bounded linear operators fromC([?h, 0],E ⁿ) intoE ⁿ for eacht?(σ, ∞) =J, B is ann ×m continuous matrix function,u:J →C ^m is square integrable with values in the unitm-dimensional cubeC ^m, andf(t, 0, 0)=0. We prove that, if the system (i) is uniformly asymptotically stable and if the controlled system (ii) is controllable, then the system (iii) is null-controllable with constraints, provided that $$f = f_1 + f_2 $$ , where $$begin{gathered} |f_1 (t,phi ,0)| leqslant varepsilon parallel phi parallel , |f_2 (t,phi ,0)| leqslant pi (t)parallel phi parallel , t geqslant sigma , hfill Pi = int_0^infty {pi (t)dt< infty .} hfill end{gathered} $$