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Spatial decay of perturbations in hyperbolic equations with optimal boundary control

Benedikt Oppeneiger, Manuel Schaller, Karl Worthmann

TL;DR

The paper addresses robustness of optimal control for a chain of hyperbolic transport equations under boundary and point controls by establishing a criterion for domain-uniform stabilization. It develops a state-space/mild-solution framework and proves that the system is $0$-domain-uniform exponentially stabilizable if and only if the distance between neighboring control access points is uniformly bounded by some $L_0>0$, independent of domain length. The results are extended from Dirichlet to Neumann boundary control via a Dirichlet-transformed input, preserving the same spacing criterion and its dual detectability implications. Numerical examples corroborate that equidistant placement of control points yields spatially localized decay of perturbations in optimal-control setups, while sparse control slows or impedes decay. The findings inform the design of boundary-control schemes for long chains of transport systems, ensuring robust, spatially localized responses in optimized trajectories.

Abstract

Recently, domain-uniform stabilizability and detectability has been the central assumption %in order robustness results on the to ensure robustness in the sense of exponential decay of spatially localized perturbations in optimally controlled evolution equations. In the present paper we analyze a chain of transport equations with boundary and point controls with regard to this property. Both for Dirichlet and Neumann boundary and coupling conditions, we show a necessary and sufficient criterion on control domains which allow for the domain-uniform stabilization of this equation. We illustrate the results by means of a numerical example.

Spatial decay of perturbations in hyperbolic equations with optimal boundary control

TL;DR

The paper addresses robustness of optimal control for a chain of hyperbolic transport equations under boundary and point controls by establishing a criterion for domain-uniform stabilization. It develops a state-space/mild-solution framework and proves that the system is -domain-uniform exponentially stabilizable if and only if the distance between neighboring control access points is uniformly bounded by some , independent of domain length. The results are extended from Dirichlet to Neumann boundary control via a Dirichlet-transformed input, preserving the same spacing criterion and its dual detectability implications. Numerical examples corroborate that equidistant placement of control points yields spatially localized decay of perturbations in optimal-control setups, while sparse control slows or impedes decay. The findings inform the design of boundary-control schemes for long chains of transport systems, ensuring robust, spatially localized responses in optimized trajectories.

Abstract

Recently, domain-uniform stabilizability and detectability has been the central assumption %in order robustness results on the to ensure robustness in the sense of exponential decay of spatially localized perturbations in optimally controlled evolution equations. In the present paper we analyze a chain of transport equations with boundary and point controls with regard to this property. Both for Dirichlet and Neumann boundary and coupling conditions, we show a necessary and sufficient criterion on control domains which allow for the domain-uniform stabilization of this equation. We illustrate the results by means of a numerical example.

Paper Structure

This paper contains 6 sections, 2 theorems, 68 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Dirichlet boundary control
  4. Neumann boundary control
  5. Numerical example
  6. Conclusions

Key Result

Theorem 2

Assume $x_0^L \in H^1(0,L)$. Then the following statements are equivalent:

Figures (4)

  • Figure 1: Sketch of a chain of coupled transport equations
  • Figure 2: Optimal state trajectory of transport equation with control domain (1) respectively (2) and domain size $L=2$ respectively $L=10$ ($\alpha = 0.156$, $T=5$)
  • Figure 3: Optimal Dirichlet boundary control input for transport equation with equidistant control access points ($L=10$, $\alpha = 0.156$, $T=5$)
  • Figure 4: Relation between $L^2(0,T;L_\mu^2(0,L))$-norm of optimal state (solid) and costate (dotted) and domain size for $T = 5$, $\alpha = 0.156$, $\mu = 0.5$.

Theorems & Definitions (6)

  • Definition 1: Domain-uniform stabilizability
  • Theorem 2
  • proof
  • Remark 3
  • Theorem 4
  • proof