On Conditions for Controllability and Local Regularity of a System of Differential Equations
Ahmad Hadra Zuhri, Yudi Soeharyadi, Jalina Widjaja

Faculty of Mathematics and Natural Sciences, Bandung Institute of Technology
Jalan Ganesha 10, Bandung 40132, Indonesia

Abstract

We consider a system of differential equation on a Banach space \(X\) given by:
\begin{equation*}
\frac{d}{dt}x(t)=Ax(t)+u(t)f(t,x(t)),
x(0)=x_0,
\end{equation*}
where \(A\) is an infinitesimal generator of a \(C_0\)-semigroup, \(f:R_0^+ \times X \rightarrow X\) is a locally Lipschitz function, and \(u \in L^p ([0,T],R)\) is a control defined on \([0,T]\) with \(1<p<\infty\). Using compactness principle and the generalization of Gronwalls Lemma, the system is shown to be controllable when \(f\) is bounded by a quadratic function. Another result of this study is to examine the local existence and the uniqueness of the solution of the given equation for locally bounded function \(f\) through weighted \(\omega\)-norm. Examples related to the results of this study will be given.

Keywords: Differential Equation, Controllability, Local Existence

Topic: MATHEMATICS AND STATISTICS