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.

Keywords: optimal control of pdes, optimal boundary control, wave equation, bang-bang, bang-bang-off, dual problem, dual solutions, $L^\infty $, measures

@article{COCV_2008__14_2_254_0, author = {Leugering, Gunter and Gugat, Martin}, title = {$L^\infty $-norm minimal control of the wave equation : on the weakness of the bang-bang principle}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {254--283}, publisher = {EDP-Sciences}, volume = {14}, number = {2}, year = {2008}, doi = {10.1051/cocv:2007044}, mrnumber = {2394510}, zbl = {1133.49006}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv:2007044/} }

TY - JOUR AU - Leugering, Gunter AU - Gugat, Martin TI - $L^\infty $-norm minimal control of the wave equation : on the weakness of the bang-bang principle JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2008 SP - 254 EP - 283 VL - 14 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2007044/ DO - 10.1051/cocv:2007044 LA - en ID - COCV_2008__14_2_254_0 ER -

%0 Journal Article %A Leugering, Gunter %A Gugat, Martin %T $L^\infty $-norm minimal control of the wave equation : on the weakness of the bang-bang principle %J ESAIM: Control, Optimisation and Calculus of Variations %D 2008 %P 254-283 %V 14 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2007044/ %R 10.1051/cocv:2007044 %G en %F COCV_2008__14_2_254_0

Leugering, Gunter; Gugat, Martin. $L^\infty $-norm minimal control of the wave equation : on the weakness of the bang-bang principle. ESAIM: Control, Optimisation and Calculus of Variations, Volume 14 (2008) no. 2, pp. 254-283. doi : 10.1051/cocv:2007044. http://www.numdam.org/articles/10.1051/cocv:2007044/

[1] A course in convexity. AMS, Providence, Rhode Island (2002). | Zbl

,[2] Exact minimum-time control of a distributed system using a traveling wave formulation. J. Optim. Theory Appl 73 (1992) 149-167. | MR | Zbl

and ,[3] Optimization and Nonsmooth Analysis. John Wiley, New York (1983). | MR | Zbl

,[4] On Liapunov's convexity theorem. Proc. Natl. Acad. Sci 91 (1994) 2145. | MR | Zbl

,[5] An infinte-dimensional version of Liapunov's convexity theorem. Michigan Math. J 17 (1970) 405-408. | MR | Zbl

,[6] Contrôlabilité approchée de l’équation de la chaleur linéaire avec des contrôles de norme ${l}^{\infty}$ minimale. (Approximate controllability for the linear heat equation with controls of minimal ${l}^{\infty}$ norm). C. R. Acad. Sci., Paris, Sér. I 316 (1993) 679-684. | MR | Zbl

, and ,[7] Approximate controllability of the semilinear heat equation. Proc. R. Soc. Edinb., Sect. A 125 (1995) 31-61. | MR | Zbl

, and ,[8] Time-parametric control: Uniform convergence of the optimal value functions of discretized problems. Contr. Cybern 28 (1999) 7-33. | MR | Zbl

,[9] Regularization of ${l}^{\infty}$-optimal control problems for distributed parameter systems. Comput. Optim. Appl 22 (2002) 151-192. | MR | Zbl

and ,[10] ${l}^{p}$-optimal boundary control for the wave equation. SIAM J. Control Optim 44 (2005) 49-74. | MR | Zbl

, and ,[11] Functional analysis and time optimal control. Academic Press (1969). | MR | Zbl

and .[12] Linear evolution equations of hyperbolic type. Univ. Tokyo Sec. I 17 (1970) 241-258. | MR | Zbl

,[13] Perturbation theory for linear operators, Corr. printing of the 2nd edn. Springer (1980). | MR | Zbl

,[14] On moment theory and controllability of one-dimensional vibrating systems and heating processes, Lecture Notes in Control and Information Science 173. Springer-Verlag, Heidelberg (1992). | MR | Zbl

,[15] Optimal Control of Undamped Linear Vibrations. Heldermann Verlag, Lemgo, Germany (1995). | MR | Zbl

,[16] A comparison of algorithms for rational ${l}_{\infty}$ approximation. Math. Comp 27 (1973) 111-121. | MR | Zbl

and ,[17] Foundations of Optimal Control Theory. Wiley, New York (1968). | MR | Zbl

and ,[18] Exact controllability, stabilization and perturbations of distributed systems. SIAM Rev 30 (1988) 1-68. | MR | Zbl

,[19] Sur les fonctions-vecteurs complètement additives. Bull. Acad. Sci. URSS, Sér. Math 4 (1940) 465-478. | JFM | MR | Zbl

,[20] Introduction to Optimal Control Theory. Springer-Verlag, New York (1982). | MR | Zbl

and ,[21] An abstract bang-bang principle and time-optimal boundary control of the heat equation. SIAM J. Control Optim 35 (1997) 1204-1216. | MR | Zbl

and ,[22] Measurable multifunctions and their applications to convex integral functionals. Internat. J. Math. Math. Sciences 12 (1989) 175-192. | MR | Zbl

,[23] Analysis Now. Springer-Verlag, New York (1989). | MR | Zbl

,[24] Convex Analysis. Princeton University Press (1970). | MR | Zbl

,[25] Functional Analysis. Springer, Berlin (1965).

,[26] Optimal and approximate control of finite-difference approximation schemes for the 1d wave equation. Rend. Mat. Appl 24 (2004) 201-237. | MR | Zbl

,[27] Propagation, observation. and control of waves approximated by finite difference methods. SIAM Rev 47 (2005) 197-243. | MR | Zbl

,[28] 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).

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