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Null-controllability of nonlinear infinite delay systems with restrained controls. (English) Zbl 0595.93007
For nonlinear infinite delay systems $\dot x(t)=L(t,x_ t)+B(t)u(t)+f(t,x(\cdot),u(\cdot))+\int^{0}_{- \infty}A(\theta)x(t\quad +\theta)d\theta,$ $x(t)=\psi (t),\quad t\in]-\infty,0]$ sufficient conditions for null-controllability are obtained. Here L(t,$$\psi)$$ is continuous in t, linear in $$\psi$$ and $$L(t,\psi)=\sum^{1}_{k=0}A_ k\psi (-t_ k)$$, where B(t) is a continuous matrix function, A($$\theta)$$ is an $$n\times n$$ matrix whose elements are Lebesgue integrable on $$]-\infty,0]$$; $$f: I\times B\times L_ 2\to E^ n$$ is a continuous function and the control $$u(t)\in \{u| u\in E^ m$$, $$| u_ j| \leq 1$$, $$j=1,...,m\}$$ is a square-integrable function.