| Contents |
| We consider blowing-up solutions $y^{0}(t)$, $t\in
\lbrack 0,T_{y^{0}}),$ of some problems of ODEs, $\frac{%
dy}{dt}(t)=f(y(t)),y(0)=y_{0}$, where $f:$ $\mathbb{R}%
^{d}\rightarrow \mathbb{R}^{d}$ is a locally Lipschitz function and $d\geq
1$. The controllability question we analyze in this work is the following:
Given $\epsilon >0$, can we find a continuous deformation of the trajectory $%
y^{0}(t)$, built as solution of the control perturbed problem obtained by replacing $f(y(t))$ by $f(y(t))+u(t),$ for a suitable control $u\in
L_{loc}^{1}(0,+\infty :\mathbb{R}^{d})$ such that $y(t)=y^{0}(t)$ for any $%
t\in \lbrack 0,T_{y^{0}}-\epsilon ]$ and such that $y(t)$ is continued to the whole $[0,+\infty )?$
We shall see that the answer is positive, by improving some previous work by the authors.
The key ingredient is the following powerful nonlinear variation of constants formula, extending previous results [Alekseev (1961), as in Laksmikantham and Leela (1969), for nonlinear terms of class $C^{2}$] to more general ones: let $f$ be a globally Lipschitz function and let $\beta (t,y)$ a family of maximal monotone operators on the space $H=% \mathbb{R}^{d}$, with $\beta (t,.)\in
L_{loc}^{1}(0,+\infty :\mathbb{R}^{d}),$ such that the solutions of the perturbed problem, $P(f,\beta )$, $ \frac{dy}{dt}(t) \in f(y(t))+\beta
(t,y(t)),y(t_{0})=\xi, $ are well defined as absolutely continuous functions on $[0,T],$ for a given $T>0.$ Let $y=\phi (t,t_{0},\xi )$ be the unique solution of the ODE in $\mathbb{R}^{d}$, $y^{\prime }=f(y(t)),y(t_{0})=\xi,$ and let $\Phi (t,t_{0},\xi )=\partial _{\xi }\phi (t,t_{0},\xi ),$ where $\partial _{_{\xi }}$ denotes partial differentiation. We shall show then that $\phi $ is Lipschitz continuous (a generalization of the differentiability Peano's theorem)$,$ that $\Phi $ is absolutely continuous and that the solution $y(t)$ of the perturbed problem $P(f,\beta ),$ for any $t\in \lbrack 0,T],$ has the integral representation $y(t)=y^{0}(t)+\int_{t_{0}}^{t}\Phi (t,s,y(s))\beta (s,y(s))ds,$ where $y^{0}(t)=\phi (t,t_{0},y_{0})$ is the unperturbed solution. A suitable similar expresion can be obtained if $\beta (t,.)$ is multivalued (as for some variational inequalities). We will give applications to some nonlinear blowing-up parabolic problems. |
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