| Contents |
| The main aim of our presentation is to study exact non-local controllability of solutions for semilinear inclusion in a reflexive Banach space $\mathbb{E}$. More precisely, we shall establish sufficient conditions for controllability of non-linear evolution system with generalized non-local condition of the form:
\begin{equation} \label{eq:ink1}
\begin{cases}
\dot{y}(t) \in A\big(t, y(t)\big) y(t) + F\big(t, y(t)\big) + Bu(t), \quad \mbox{for} \enspace t \in \mathcal{J} := [0, T], \enspace T > 0, \\
y(0) = M(y), \\
\end{cases}
\end{equation}
where $F \colon \mathcal{J} \times \mathbb{E} \to \mathbb{E}$ is a bounded, closed, multivalued map with convex values, $A(t, y)$ is a linear continuous operator on $\mathbb{E}$, for each $(t, y) \in \mathcal{J} \times \mathbb{E}$, control $u(\cdot)$ is a given function from $L^{2}(\mathcal{J}, \mathbb{U})$, a space of admissible control functions with $\mathbb{U}$ as reflexive Banach space, and $B$ is a bounded linear operator from $\mathbb{U}$ to $\mathbb{E}$.
The assumptions regarding the operator $M \colon \mathcal{C}(\mathcal{J}, \mathbb{E}) \to \mathbb{E}$ will be stated explicitly in the presentation.
Sufficient conditions are formulated and proved using a fixed point theorem. Finally, we present an example to illustrate application of the proposed method. |
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