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 -

References

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  1. A. Barvinok, A course in convexity. AMS, Providence, Rhode Island (2002).  Zbl1014.52001
  2. J.K. Bennighof and R.L. Boucher, Exact minimum-time control of a distributed system using a traveling wave formulation. J. Optim. Theory Appl73 (1992) 149–167.  Zbl0794.49004
  3. F.H. Clarke, Optimization and Nonsmooth Analysis. John Wiley, New York (1983).  Zbl0582.49001
  4. A. Dovretzki, On Liapunov's convexity theorem. Proc. Natl. Acad. Sci91 (1994) 2145.  
  5. V. Drobot, An infinte-dimensional version of Liapunov's convexity theorem. Michigan Math. J17 (1970) 405–408.  Zbl0192.48801
  6. C. Fabre, J.-P. Puel and E. Zuazua, Contrôlabilité approchée de l'équation de la chaleur linéaire avec des contrôles de norme l minimale. (Approximate controllability for the linear heat equation with controls of minimal l norm). C. R. Acad. Sci., Paris, Sér. I316 (1993) 679–684.  
  7. C. Fabre, J.-P. Puel and E. Zuazua, Approximate controllability of the semilinear heat equation. Proc. R. Soc. Edinb., Sect. A125 (1995) 31–61.  Zbl0818.93032
  8. M. Gugat, Time-parametric control: Uniform convergence of the optimal value functions of discretized problems. Contr. Cybern28 (1999) 7–33.  Zbl0951.49032
  9. M. Gugat and G. Leugering, Regularization of l -optimal control problems for distributed parameter systems. Comput. Optim. Appl22 (2002) 151–192.  Zbl1015.49020
  10. M. Gugat, G. Leugering and G. Sklyar, lp-optimal boundary control for the wave equation. SIAM J. Control Optim44 (2005) 49–74.  Zbl1083.49017
  11. H. Hermes and J. Lasalle. Functional analysis and time optimal control. Academic Press (1969).  Zbl0203.47504
  12. T. Kato, Linear evolution equations of hyperbolic type. Univ. Tokyo Sec. I17 (1970) 241–258.  Zbl0222.47011
  13. T. Kato, Perturbation theory for linear operators, Corr. printing of the 2nd edn. Springer (1980).  Zbl0435.47001
  14. W. Krabs, On moment theory and controllability of one-dimensional vibrating systems and heating processes, Lecture Notes in Control and Information Science173. Springer-Verlag, Heidelberg (1992).  Zbl0955.93501
  15. W. Krabs, Optimal Control of Undamped Linear Vibrations. Heldermann Verlag, Lemgo, Germany (1995).  Zbl0839.73002
  16. C.M. Lee and F.D.K. Roberts, A comparison of algorithms for rational l approximation. Math. Comp27 (1973) 111–121.  Zbl0257.65018
  17. E.B. Lee and L. Markus, Foundations of Optimal Control Theory. Wiley, New York (1968).  Zbl0159.13201
  18. J.-L. Lions, Exact controllability, stabilization and perturbations of distributed systems. SIAM Rev30 (1988) 1–68.  Zbl0644.49028
  19. A. Lyapunov, Sur les fonctions-vecteurs complètement additives. Bull. Acad. Sci. URSS, Sér. Math4 (1940) 465–478.  Zbl0024.38504
  20. J. Macki and A. Strauss, Introduction to Optimal Control Theory. Springer-Verlag, New York (1982).  Zbl0493.49001
  21. V.J. Mizel and T.I. Seidman, An abstract bang-bang principle and time-optimal boundary control of the heat equation. SIAM J. Control Optim35 (1997) 1204–1216.  Zbl0891.49014
  22. N. Papageorgiu, Measurable multifunctions and their applications to convex integral functionals. Internat. J. Math. Math. Sciences12 (1989) 175–192.  
  23. G.K. Pedersen, Analysis Now. Springer-Verlag, New York (1989).  Zbl0668.46002
  24. R.T. Rockafellar, Convex Analysis. Princeton University Press (1970).  Zbl0193.18401
  25. K. Yosida, Functional Analysis. Springer, Berlin (1965).  Zbl0126.11504
  26. E. Zuazua, Optimal and approximate control of finite-difference approximation schemes for the 1d wave equation. Rend. Mat. Appl24 (2004) 201–237.  Zbl1085.49041
  27. E. Zuazua, Propagation, observation. and control of waves approximated by finite difference methods. SIAM Rev47 (2005) 197–243.  Zbl1077.65095
  28. E. Zuazua, Controllability of partial differential equations: Some results and open problems, in Handbook of Differential Equations: Evolutionary Differential Equations, C. Dafermos and E. Feireisl Eds., Elsevier Science (2006).  

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. Christian Clason, Kazufumi Ito, Karl Kunisch, A minimum effort optimal control problem for elliptic PDEs
  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. Lino J. Alvarez-Vázquez, Francisco J. Fernández, Aurea Martínez, Analysis of a time optimal control problem related to the management of a bioreactor

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