L∞-Norm minimal control of the wave equation: on the weakness of the bang-bang principle

Martin Gugat; Gunter Leugering

ESAIM: Control, Optimisation and Calculus of Variations (2008)

  • Volume: 14, Issue: 2, page 254-283
  • ISSN: 1292-8119

Abstract

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For optimal control problems with ordinary differential equations where the L ∞ -norm of the control is minimized, often bang-bang principles hold. For systems that are governed by a hyperbolic partial differential equation, the situation is different: even if a weak form of the bang-bang principle still holds for the wave equation, it implies no restriction on the form of the optimal control. To illustrate that for the Dirichlet boundary control of the wave equation in general not even feasible controls of bang-bang type exist, we examine the states that can be reached by bang-bang-off controls, that is controls that are allowed to attain only three values: Their maximum and minimum values and the value zero. We show that for certain control times, the difference between the initial and the terminal state can only attain a finite number of values. For the problems of optimal exact and approximate boundary control of the wave equation where the L ∞ -norm of the control is minimized, we introduce dual problems and present the weak form of a bang-bang principle, that states that the values of L ∞ -norm minimal controls are constrained by the sign of the dual solutions. Since these dual solutions are in general given as measures, this is no restriction on the form of the control function: the dual solution may have a finite support, and when the dual solution vanishes, the control is allowed to attain all values from the interval between the two extremal control values.


How to cite

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Gugat, Martin, and Leugering, Gunter. "L∞-Norm minimal control of the wave equation: on the weakness of the bang-bang principle." ESAIM: Control, Optimisation and Calculus of Variations 14.2 (2008): 254-283. <http://eudml.org/doc/250310>.

@article{Gugat2008,
abstract = {
For optimal control problems with ordinary differential equations where the $L^\infty$-norm of the control is minimized, often bang-bang principles hold. For systems that are governed by a hyperbolic partial differential equation, the situation is different: even if a weak form of the bang-bang principle still holds for the wave equation, it implies no restriction on the form of the optimal control. To illustrate that for the Dirichlet boundary control of the wave equation in general not even feasible controls of bang-bang type exist, we examine the states that can be reached by bang-bang-off controls, that is controls that are allowed to attain only three values: Their maximum and minimum values and the value zero. We show that for certain control times, the difference between the initial and the terminal state can only attain a finite number of values. For the problems of optimal exact and approximate boundary control of the wave equation where the $L^\infty$-norm of the control is minimized, we introduce dual problems and present the weak form of a bang-bang principle, that states that the values of $L^\infty$-norm minimal controls are constrained by the sign of the dual solutions. Since these dual solutions are in general given as measures, this is no restriction on the form of the control function: the dual solution may have a finite support, and when the dual solution vanishes, the control is allowed to attain all values from the interval between the two extremal control values.
},
author = {Gugat, Martin, Leugering, Gunter},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Optimal control of pdes; optimal boundary control; wave equation; bang-bang; bang-bang-off; dual problem; dual solutions; $L^\infty$; measures; optimal control of PDEs; wave equation; },
language = {eng},
month = {3},
number = {2},
pages = {254-283},
publisher = {EDP Sciences},
title = {L∞-Norm minimal control of the wave equation: on the weakness of the bang-bang principle},
url = {http://eudml.org/doc/250310},
volume = {14},
year = {2008},
}

TY - JOUR
AU - Gugat, Martin
AU - Leugering, Gunter
TI - L∞-Norm minimal control of the wave equation: on the weakness of the bang-bang principle
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2008/3//
PB - EDP Sciences
VL - 14
IS - 2
SP - 254
EP - 283
AB - 
For optimal control problems with ordinary differential equations where the $L^\infty$-norm of the control is minimized, often bang-bang principles hold. For systems that are governed by a hyperbolic partial differential equation, the situation is different: even if a weak form of the bang-bang principle still holds for the wave equation, it implies no restriction on the form of the optimal control. To illustrate that for the Dirichlet boundary control of the wave equation in general not even feasible controls of bang-bang type exist, we examine the states that can be reached by bang-bang-off controls, that is controls that are allowed to attain only three values: Their maximum and minimum values and the value zero. We show that for certain control times, the difference between the initial and the terminal state can only attain a finite number of values. For the problems of optimal exact and approximate boundary control of the wave equation where the $L^\infty$-norm of the control is minimized, we introduce dual problems and present the weak form of a bang-bang principle, that states that the values of $L^\infty$-norm minimal controls are constrained by the sign of the dual solutions. Since these dual solutions are in general given as measures, this is no restriction on the form of the control function: the dual solution may have a finite support, and when the dual solution vanishes, the control is allowed to attain all values from the interval between the two extremal control values.

LA - eng
KW - Optimal control of pdes; optimal boundary control; wave equation; bang-bang; bang-bang-off; dual problem; dual solutions; $L^\infty$; measures; optimal control of PDEs; wave equation;
UR - http://eudml.org/doc/250310
ER -

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Citations in EuDML Documents

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  1. Larissa V. Fardigola, Controllability problems for the 1-D wave equation on a half-axis with the Dirichlet boundary control
  2. Larissa V. Fardigola, Controllability problems for the 1-D wave equation on a half-axis with the Dirichlet boundary control
  3. Karl Kunisch, Daniel Wachsmuth, On time optimal control of the wave equation, its regularization and optimality system
  4. Larissa V. Fardigola, Controllability problems for the 1-D wave equation on a half-axis with the Dirichlet boundary control
  5. Christian Clason, Kazufumi Ito, Karl Kunisch, A minimum effort optimal control problem for elliptic PDEs
  6. Lino J. Alvarez-Vázquez, Francisco J. Fernández, Aurea Martínez, Analysis of a time optimal control problem related to the management of a bioreactor
  7. Lino J. Alvarez-Vázquez, Francisco J. Fernández, Aurea Martínez, Analysis of a time optimal control problem related to the management of a bioreactor
  8. Christian Clason, Kazufumi Ito, Karl Kunisch, A minimum effort optimal control problem for elliptic PDEs

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