-
Abstract
We consider explosive solutions $y^{0}(t)$, $t\in \lbrack 0,T_{y^{0}}),$ of
some ordinary differential equations
\begin{equation*}
P(T_{y^{0}}):
\begin{array}{lc}
\frac{dy}{dt}(t)=f(y(t)),y(0)=y_{0}, &
\end{array}
\end{equation*}
where $f:$ $\mathbb{R}^{d}\rightarrow \mathbb{R}^{d}$ is a locally
Lipschitz superlinear function and $d\geq 1$. In this work we analyze the
following question of controlability: given $\epsilon >0$, a continuous
deformation $y(t)$ de $y^{0}(t)$, built as a solution of the perturbed
control problem obtained by replacing $f(y(t))$ by $f(y(t))+u(t),$ for a
suitable control $u$, such that $y(t)=y^{0}(t)$ for any $t\in \lbrack
0,T_{y^{0}}-\epsilon ]$ and such that $y(t)$ also blows up in $t=T_{y_{0}}$
but in such a way that $y(t)$ could be extended beyond $T_{y_{0}}$ as a
function $y\in L_{loc}^{1}(0,+\infty :\mathbb{R}^{d})$?
Mathematics Subject Classification: Primary: 35R10, 35R35; Secondary: 35K20.
\begin{equation} \\ \end{equation}
-
References
|
[1]
|
R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. 
|
|
[2]
|
V. M. Alekseev, An estimate for the perturbations of the solutions of ordinary differential equations (Russian), Vestnik Moskov Univ. Ser. I Mat. Meh., 2 (1961), 28-36.
|
|
[3]
|
H. Brezis, Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Mathematical Studies, Amsterdam, 1973.
|
|
[4]
|
A. Casal, J. I. Díaz, and J. M. Vegas, Blow-up in some ordinary and partial differential equations with time-delay. Dynam. Systems Appl. 18(1) (2009), 29-46.
|
|
[5]
|
J. I. Díaz and A. V. Fursikov, A simple proof of the approximate controllability from the interior for nonlinear evolution problems. Applied Mathematics Letters 7(5) (1994), 85-87.
|
|
[6]
|
V. Laksmikantham and S. Leela, Differential and Integral Inequalities, Theory and Applications, Vols. I and II, Academic Press, New York, 1969.
|
|
[7]
|
S. Mirica, On differentiability with respect to initial data in the theory of differential equations. Rev. Roumaine des Math. Pures Appl., 48(2) (2003),153-171.
|
-
Access History
{{journal.titleEn}} 
Open Access Under a Creative Commons 
DownLoad: