Authors:
Prof. Dr. Dmitriy Leykekhman | University of Connecticut | United States
Prof. Dr. Boris Vexler | TU Munich | Germany
We consider a parabolic optimal control problem:
\begin{equation} \label{eq:intro-obj}
\min_{q,u}J(q,u):= \frac{1}{2} \int_0^T \|u(t) - \widehat{u}(t)\|_{L^2(\Omega)}^2 dt
+ \frac{\alpha}{2} \int_0^T |q(t)|^2 dt
\end{equation}
subject to the second order parabolic equation
\begin{subequations} \label{eq:intro-state}
\begin{align}
u_t(t,x)-\Delta u(t,x) &= q(t)\delta_{x_0}, & (t,x) &\in I\times\Omega,\; \\
u(t,x) &= 0, & (t,x) &\in I\times\partial\Omega, \\
u(0,x) &= 0, & x &\in \Omega
\end{align}
\end{subequations}
and subject to pointwise control constraints
\begin{equation}\label{eq:control_constraints}
q_a \le q(t) \le q_b \quad \text{a.\,e. in } I.
\end{equation}
Here $I=[0,T]$, $\Omega$ is a convex polygonal or polyhedral domain, $x_0\in \Omega$ fixed, and $\delta_{x_0}$ is the Dirac delta function. The parameter $\alpha$ is assumed to be positive and the desired state $\widehat u$ fulfills $\widehat u \in L^2(I;L^\infty(\Omega))$. The control bounds $q_a,q_b \in \mathbb{R}\cup\{\pm \infty\}$ fulfill $q_a <q_b$.
To approximate the problem numerically we use the standard continuous piecewise linear approximation in space and the first order discontinuous Galerkin method in time. Despite low regularity of the state equation, we establish almost optimal $h^2+k$ convergence rate in 2D and $h+\sqrt{k}$ in 3D for the control in $L^2$ norm. I will explain the key regularity estimates and sharp a priori fully discrete global and local error estimates in $L^2([0,T]; L^\infty(\Omega))$ norms for parabolic problems. These new error estimates are essential in our analysis and rather technical, especially in 3D, and require a new technique which I will explain. Using these sharp results we improve almost twice the previously obtained error estimates in \cite{key-1}. The 2D result were published in \cite{key-2}, but 3D results are new.
\begin{thebibliography}{1}
\bibitem[1]{key-1}W. Gong, M. Hinze, Z. Zhou. A priori error analysis for finite element approximation of
parabolic optimal control problems with pointwise control. SIAM J. Control Optim. 52 (2014), 97\textendash 119.
\bibitem[2]{key-2}D. Leykekhman, B. Vexler. SOptimal a priori error estimates of parabolic optimal control
problems with pointwise control. SIAM J. Numer. Anal. 51 (2013), 2797\textendash 2821.
\end{thebibliography}