Multiobjective Model Predictive Control of a Parabolic AdvectionDiffusionReaction Equation
Abstract
:1. Introduction
2. The Multiobjective Optimal Control Problem
3. Multiobjective Model Predictive Control (Mompc)
Algorithm 1 Multiobjective model predictive control 
Given: Initial state ${y}_{\circ}\in H$. 

 (1)
 The statement from Theorem 3 is important, since it gives us performance bounds on the cost functions values of the MOMPC feedback control μ already after choosing the initial control ${\overline{u}}_{0}\in {\mathcal{U}}_{opt}^{0}\left({y}_{\circ}\right)$, but before performing the MOMPC Algorithm 1. Thus, one strategy is to compute the entire Pareto set ${\mathcal{U}}_{opt}^{0}\left({y}_{\circ}\right)$, which is computationally cheap due to the small time horizon $({t}_{0},{t}_{0}^{\ell})$, and then to choose the initial control ${\overline{u}}_{0}\in {\mathcal{U}}_{opt}^{0}\left({y}_{\circ}\right)$ according to the desired upper bounds on the cost functions.
 (2)
 Theorem 3 holds for arbitrary $T>0$. In particular, by taking the limit $T\to \infty $, the result can also be shown for the infinitehorizon case.
Algorithm 2 Multiobjective Gradient Descent Method 
Given: Current iterate n, initial control $u\in {U}^{n}$, tolerance $\epsilon >0$, Armijo parameter $0<\beta \ll 1$. 

 (1)
 Note that we cannot prove that the sequence ${\left({u}_{k}\right)}_{k\in \mathbb{N}}$ has an accumulation point in the infinitedimensional case. However, we will not encounter this problem in our numerical implementation, since the space ${U}^{n}$ will be discretized. Therefore, Algorithm 2 will in practice terminate in a finite number of steps.
 (2)
 By construction of the algorithm it holds ${\widehat{J}}_{i}^{n}({\overline{u}}_{n},{y}_{\mathsf{a}})\le {\widehat{J}}_{i}^{n}(u,{y}_{\mathsf{a}})$, $i=1,2$, for any initial control $u\in {U}^{n}$, so that (5) is fulfilled, if we choose $u=\tilde{u}$.
Algorithm 3 Multiobjective Model Predictive Control 
Given: Initial state ${y}_{\circ}\in H$. 

4. Numerical Tests
4.1. Example 1
4.2. Example 2
5. Conclusions and Outlook
Author Contributions
Funding
Conflicts of Interest
Abbreviations
MO  Multiobjective optimization 
MOP  Multiobjective optimization problem 
MOCP  Multiobjective optimal control problem 
MPC  Model predictive control 
MOMPC  Multiobjective model predictive control 
ODE  Ordinary differential equation 
PDE  Partial differential equation 
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Banholzer, S.; Fabrini, G.; Grüne, L.; Volkwein, S. Multiobjective Model Predictive Control of a Parabolic AdvectionDiffusionReaction Equation. Mathematics 2020, 8, 777. https://doi.org/10.3390/math8050777
Banholzer S, Fabrini G, Grüne L, Volkwein S. Multiobjective Model Predictive Control of a Parabolic AdvectionDiffusionReaction Equation. Mathematics. 2020; 8(5):777. https://doi.org/10.3390/math8050777
Chicago/Turabian StyleBanholzer, Stefan, Giulia Fabrini, Lars Grüne, and Stefan Volkwein. 2020. "Multiobjective Model Predictive Control of a Parabolic AdvectionDiffusionReaction Equation" Mathematics 8, no. 5: 777. https://doi.org/10.3390/math8050777