Bonjour les doyens, j’ai un sous sur comment procéder pour résoudre un problème de formulation mixte issu d’un problème de contrôle optimal d’une EPD linéaire. Cette formulation mixte dépend de deux variables que je cherche à déterminer numériquement par la méthode des éléments finis.
Est-ce que vous pouvez m’aider ?
Hello Daniel,
You need to formulate your equations so that someone can help you…
You can use latex style between dollars to write equations.
Eventually you can discuss of a simplified problem. Please write in English so that everybody can understand.
Thank you very much for your advice.
This work is devoted to the problem of null controllability for the following parabolic equation:
\begin{eqnarray}
\begin{cases} \label{W32}
\varphi_t - \Delta \varphi + B(x, t) \cdot \nabla \varphi + a(x, t) \varphi = u \chi_O & \text{in } Q, \\
\varphi = 0 & \text{on } \Sigma, \\
\varphi(x,0) = \varphi_0(x) & \text{in } \Omega
\end{cases}
\end{eqnarray}
We transform this problem into an optimization problem and obtain:
\begin{equation} \label{W37}
\min_{y \in W_\epsilon} \hat{J}^\ast_\epsilon(y) := \frac{1}{2} \iint_{q} \rho_0^{-2} |y(x, t)|^2 \, dx \, dt + \frac{\epsilon}{2} \|y(\cdot, T)\|^2_{L^2(\Omega)} + (\varphi_0, y(\cdot, 0))_{L^2(\Omega)}.
\end{equation}
where
\[
W_\epsilon = \{ y \in \Phi_\epsilon : L^\ast y = 0 \text{ in } L^2(Q) \}
\]
with
\begin{eqnarray} \label{W36}
\begin{cases}
L^\ast y = -y_t - \Delta y + B(x, t) \cdot \nabla y + a(x, t) y 0 \quad \text{in} \; Q, \\
y = 0 \quad \text{on} \; \Sigma, \\
y(\cdot, T) = y_T \quad \text{in} \; \Omega,
\end{cases}
\end{eqnarray}
We consider the following mixed formulation: find \( (y_\epsilon, \lambda_\epsilon) \in \Phi_\epsilon \times L^2(Q) \) that solves
\begin{equation} \label{W38}
\begin{cases}
a_\epsilon(y_\epsilon, y) + b(y, \lambda_\epsilon) = l(y), \forall y \in \Phi_\epsilon, \\
b(y_\epsilon, \lambda) = 0, \forall \lambda \in L^2(Q),
\end{cases}
\end{equation}
where
\[
a_\epsilon : \Phi_\epsilon \times \Phi_\epsilon \rightarrow \mathbb{R}, \quad a_\epsilon(y, y') := \iint_{q} \rho_0^{-2} y y' \, dx \, dt + \epsilon (y(\cdot, T), y'(\cdot, T))_{L^2(\Omega)}
\]
\[
b : \Phi_\epsilon \times L^2(Q) \rightarrow \mathbb{R}, \quad b(y, \lambda) := - \iint_{Q} L^\ast y \, \lambda \, dx \, dt
\]
\[
l : \Phi_\epsilon \rightarrow \mathbb{R}, \quad l(y) := - (\varphi_0, y(\cdot, 0))_{L^2(\Omega)}.
\]
The continuation of the work shows that \(u_\epsilon := \rho_0^{-2} y_\epsilon \chi_\omega\),
\[
\iint_{q} u_\epsilon \, \bar{y} \, dx \, dt + (\epsilon y_\epsilon(\cdot, T), \bar{y}(\cdot, T)) - \iint_{Q} L^\ast \bar{y}(x, t) \lambda_\epsilon(x, t) \, dx \, dt = l(\bar{y}), \quad \forall \bar{y} \in \Phi_\epsilon.
\]
The objective is to numerically determine the functions $u$ et $\lambda$
Just give the pdf image of your problem. It’s easy to read.
I believe it is not possible to attach a PDF document
With E. Trelat and G. lance we write a book on this subject and
all exemples are here
t the pdf of the book is here: https://www.ljll.fr/hecht/ftp/tmp/PDE-constrained%20optimization%20within%20FreeFEM-07-07-24.pdf
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