Journal of Optimization Theory and Applications

, Volume 42, Issue 3, pp 397–420

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

• J. U. Onwuatu
Contributed Papers

## Abstract

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],En) intoEn for eachtɛ(σ, ∞) =J, B is ann ×m continuous matrix function,u:JCm is square integrable with values in the unitm-dimensional cubeCm, 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
$$\begin{gathered} |f_1 (t,\phi ,0)| \leqslant \varepsilon \parallel \phi \parallel , |f_2 (t,\phi ,0)| \leqslant \pi (t)\parallel \phi \parallel , t \geqslant \sigma , \hfill \\ \Pi = \int_0^\infty {\pi (t)dt< \infty .} \hfill \\ \end{gathered}$$

## Key Words

Null-controllability with constraints uniform asymptotic stability controllability complete linear systems reachable sets

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