Seminar in Makroökonomik Detail

Vortragende Person/Vortragende Personen:
Michael Kartmann

On 12^th November 2024 at 15:15, Michael Kartmann from the University of Konstanz will give a talk.

Abstract: In this talk, we are concerned with the solution of infinite-horizon optimal control problems of the form

\begin{equation}
    \begin{aligned}
        \min_{u \in L^2((0,\infty),\mathbb{R}^\rho)} \mathcal{J}(u) = \int_{0}^{\infty} \tfrac{1}{2} \|y(t)-y_d(t)\|^2_{\mathbb{R}^{\rho}}+ \tfrac{\lambda}{2}\|u(t)\|_{\mathscr{U}_n}^2 + \mu \|u(t) \|_{L^1(\Omega)}\,\mathrm{d}t,
    \end{aligned}
\end{equation}
subject to $(u, y)$ solving the following linear switched input-output system
\begin{equation}
    \left\{\quad
    \begin{aligned}
        \mathcal{M}_{\sigma(t)}\tfrac{\mathrm{d}}{\mathrm{dt}} \theta(t) + \mathcal{A}_{\sigma(t)}\theta(t) &= \mathcal{B}_{\sigma(t)}u(t) &&t\geq 0\\
        y(t) &= \mathcal{C}_{\sigma(t)}\theta(t) && t\geq 0\\
       \theta(0) &= \theta_{\circ}. && \\
    \end{aligned}\right.
\end{equation}
and control constraints $u\in \mathscr{U}_{\mathsf{ad}}$. Here $\sigma:[0,\infty) \to \{1,\ldots,L\}$ is a switching signal, that switches through different system operators $\mathcal{M}_i,\mathcal{A}_i,\mathcal{B}_i,\mathcal{C}_i$ for $i=1,\ldots,L$. \\
To approximate the solution of $(1)$, we apply Model Predictive Control (MPC): the optimal control problem is solved over smaller, receding time intervals $(t_n,t_n+T)$ for some prediction horizon $T>0$ and the solutions are concatenated in the sampling interval $(t_n,t_{{n+1}})$ for $0<t_{{n+1}}<t_n+T$. First, we derive optimality conditions for these small-horizon problems and discuss their suboptimality w.r.t. $(1)$. The difficulty here is that the cost functional $\mathcal{J}$ is not differentiable in the classical sense, due to the presence of the $L^1$-regularization. Second, the repeated solution of small-horizon optimal control problems motivates model reduction: we consider (Petrov-)Galerkin reduced-order models for $(2)$ to speed up the MPC process. To quantify the error, we do a full a posteriori error analysis for the optimal control, optimal state, and optimal value function of the small-horizon problems, which allows us to control the evolving error through the MPC iterations.  These estimates are then used to construct two certified ROM-MPC algorithms for the solution of $(1)$, that are up to $10$ times faster than the MPC relying on the full-order model. This is joint work with Stefan Volkwein (U. Konstanz), Mattia Manucci, and Benjamin Unger (U. Stuttgart).