Academic Editor: Youssef EL FOUTAYENI
Received |
Accepted |
Published |
21 December 2019 |
05 January 2020 |
10 March 2020 |
Abstract: Let $X$ be a real Banach space with a norm $\Vert \cdot\Vert$. We consider the control system \begin{equation} %\label{eq1,Initialprb} \label{Initialprb} y^{\prime}(t) \in A y(t) +F(t,y(t)), \end{equation} where $A: D(A)\subset X\rightsquigarrow X$ is an $m$--dissipative set--valued operator, and $F:I\times X\rightsquigarrow X$ is a given set--valued map where $I=[a,b )\subset \mathbb{R}$ with $b\leq +\infty$. Approximate null--controllability of \eqref{Initialprb}, i.e., whether it is possible to approximately steer an initial state into the origin in a finite time has been addressed by several authors and from several viewpoints. Most of previous works considered the case when $F$ is generalized by a parametrization i.e., when $F:I\times X\rightsquigarrow X$ is defined by $$ F(t,y(t))=\{f_u(t,y(t)),\,u\in \mathcal{U}(t,y(t))\}, $$ where $f_u:I\times X\rightarrow X$ and $\mathcal{U}:I\times X\rightsquigarrow X$ is a set valued map of controls (depending on the time and state). Here, we first show that the problem of approximate null controllability of \eqref{Initialprb}, can be treated as a problem of near viability of the epigraph of the norm of the Banach space $X$ with respect to an appropriate dynamical system. We recall that a set is said to be near viable with respect to a system if, starting from the set, one is able to find solutions of the system remaining arbitrary close to the set in some arbitrarily small neighborhood of the set. We provide also, on the base of a new tangency condition, sufficient conditions for a cylindrical domain to be near viable with respect to the \eqref{Initialprb}. This result generalizes those given in \cite{BC16,BC1116} Finally we show that some points neighborhood of the origin can approximately be steered, with respect to \eqref{Initialprb}, to the origin in a finite time $T$. Some examples with numerical schemes concerning nonlinear control systems are investigated.
