Error estimates for a multiobjective optimal control of a pointwise tracking problem

Abstract

We analyze a pointwise tracking multiobjective optimal control problem subject to the Poisson problem and bilateral control constraints. To approximate Pareto optimal points and the Pareto front numerically, we consider two different finite element-based scalarization techniques, namely the weighted-sum method and the reference point method, where in both methods many scalar-constrained optimization problems have to be solved. We prove a priori error estimates for both scalarizations. The underlying subproblems of either method are solved with a Barzilai-Borwein gradient method. Numerical experiments illustrate the accuracy of the proposed method.

Figures (3)

• Figure 1: Pareto front and approximation error $\|\mathcal{J}(\bar{y}_{\alpha}^{h},\bar{u}_{\alpha}^{h}) - \mathcal{J}(\bar{y}_{\alpha},\bar{u}_{\alpha})\|_{2}$ with $50$ different values of $\alpha$ considering mesh refinement under the WSM for the cases $\lambda_1=\lambda_2=1$ (1.A)--(1.B), $\lambda_1=0.1$ and $\lambda_2=1$ (1.C)--(1.D), $\lambda_1=1$ and $\lambda_2=0.1$ (1.E)--(1.F), and $\lambda_1=\lambda_2=0.1$ (1.G)--(1.H).
• Figure 2: Approximate optimal control $\bar{u}_{\alpha}^{h}$ when $\alpha=(0.2,0.8)$ (2.A), $\alpha=(0.4,0.6)$ (2.B), $\alpha=(0.6,0.4)$ (2.C), and $\alpha=(0.8,0.2)$ (2.D). Here, $h=2^{-6}$ and $\lambda_1=\lambda_2=0.1$
• Figure 3: Approximate optimal control $\bar{u}_{\zeta}^{h}$ for $\zeta^{9}$ (3.A), $\zeta^{7}$ (3.B), $\zeta^{4}$ (3.C), and $\zeta^{2}$ (3.D). Here, $h=2^{-6}$ and $\lambda_1=\lambda_2=0.1$

Theorems & Definitions (25)

• Proposition 2.1: Differentiability of $j_1$ and $j_2$
• proof
• Lemma 2.2: Auxiliary result
• proof
• Proposition 2.3: Existence of a solution (WSM)
• proof
• Theorem 2.4: First-order optimality condition (WSM)
• Proposition 2.5: Existence of a solution (RPM)
• proof
• Theorem 2.6: First-order optimality condition (RPM)