American Institute of Mathematical Sciences

Advanced
2012, 2(3): 547-570. doi: 10.3934/naco.2012.2.547

Error bounds for Euler approximation of linear-quadratic control problems with bang-bang solutions

Walter Alt 1, , Robert Baier 2, , Matthias Gerdts 3, and  Frank Lempio 2,

1. 

Institut für Angewandte Mathematik, Friedrich-Schiller-Universität Jena, 07740 Jena, Germany
2. 

Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany, Germany
3. 

Institut für Mathematik und Rechneranwendung, Fakultät für Luft- und Raumfahrttechnik, Universität der Bundeswehr, 85577 Neubiberg/München, Germany

Received  July 2011 Revised  May 2012 Published  August 2012

We analyze the Euler discretization to a class of linear-quadratic optimal control problems. First we show convergence of order $h$ for the optimal values of the objective function, where $h$ is the mesh size. Under the additional assumption that the optimal control has bang-bang structure we show that the discrete and the continuous controls coincide except on a set of measure $O(\sqrt{h})$. Under a slightly stronger assumption on the smoothness of the coefficients of the system equation we obtain an error estimate of order $O(h)$.
Keywords: discretization., bang-bang control, Linear-quadratic optimal control.
Mathematics Subject Classification: Primary: 49J15; Secondary: 49M25, 49N10, 49J3.
Citation: Walter Alt, Robert Baier, Matthias Gerdts, Frank Lempio. Error bounds for Euler approximation of linear-quadratic control problems with bang-bang solutions. Numerical Algebra, Control & Optimization, 2012, 2 (3) : 547-570. doi: 10.3934/naco.2012.2.547
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Control Cybernet., 38 (2009), 1305-1325.  Google Scholar

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Robert E. Krieger Publ. Co., 1980.  Google Scholar

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Comp. Optim. Appl., 30 (2005), 45-61. doi: 10.1007/s10589-005-4559-5.  Google Scholar

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in "Mathematical Programming with Data Perturbations V" (ed. A. V. Fiacco), Lecture Notes in Pure and Applied Mathematics 195, Marcel Dekker, (1997), 253-284. Google Scholar

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Optimal Control Appl. Methods, 26 (2005), 129-156. doi: 10.1002/oca.756.  Google Scholar

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SIAM J. Control Optim., 42 (2004), 2239-2263. doi: 10.1137/S0363012902402578.  Google Scholar

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ESAIM, Math. Model. Numer. Anal., 44 (2010), 167-188. doi: 10.1051/m2an/2009045.  Google Scholar

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SIAM J. Contr. Optim., 43 (2004), 970-985. doi: 10.1137/S0363012903431608.  Google Scholar

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SIAM J. Control Optim., 35 (1997), 1470-1486. doi: 10.1137/S0363012995288987.  Google Scholar

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show all references

References:
[1]

Appl. Math. Optim., 12 (1984), 15-27. doi: 10.1007/BF01449031.  Google Scholar

[2]

J. Optim. Theory Appl., 70 (1991), 443-466. doi: 10.1007/BF00941297.  Google Scholar

[3]

in "Mathematical Programming with Data Perturbations V" (ed. A. V. Fiacco), Lecture Notes in Pure and Applied Mathematics 195, Marcel Dekker, (1997), 1-30.  Google Scholar

[4]

Optimization, DOI: 10.1080/02331934.2011.568619, (2011). doi: 10.1080/02331934.2011.568619.  Google Scholar

[5]

Comp. Optim. Appl., 43 (2009), 133-150. doi: 10.1007/s10589-007-9129-6.  Google Scholar

[6]

SIAM J. Control Optim., 27 (1989), 718-736. doi: 10.1137/0327038.  Google Scholar

[7]

W. Alt and M. Seydenschwanz, Regularization and discretization of linear-quadratic control problems,, Control Cybernet., ().   Google Scholar

[8]

SIAM J. Optim., 18 (2007), 1004-1026. doi: 10.1137/060661867.  Google Scholar

[9]

Computing, 81 (2007), 91-106. doi: 10.1007/s00607-007-0240-4.  Google Scholar

[10]

Bayreuth. Math. Schr., 67 (2003), 3-161.  Google Scholar

[11]

Comp. Optim. Appl., DOI: 10.1007/s10589-010-9365-z, (2010). doi: 10.1007/s10589-010-9365-z.  Google Scholar

[12]

Comp. Optim. Appl., DOI: 10.1007/s10589-009-9310-1, (2010). doi: 10.1007/s10589-009-9310-1.  Google Scholar

[13]

Computing, 41 (1989), 349-358. doi: 10.1007/BF02241223.  Google Scholar

[14]

SIAM J. Control Optim., 31 (1993), 569-603. doi: 10.1137/0331026.  Google Scholar

[15]

Math. Comp., 70 (2001), 173-203. doi: 10.1090/S0025-5718-00-01184-4.  Google Scholar

[16]

Numer. Funct. Anal. Optim., 21 (2000), 653-682. doi: 10.1080/01630560008816979.  Google Scholar

[17]

North Holland, Amsterdam-Oxford, 1976.  Google Scholar

[18]

SIAM J. Control Optim., 41 (2003), 1843-1867. doi: 10.1137/S0363012901399271.  Google Scholar

[19]

Control Cybernet., 37 (2008), 307-327.  Google Scholar

[20]

in "Lecture Notes Comp. Sci., Vol. 5910" (eds. I. Lirkov et al.), Springer-Verlag, (2010), 264-271. Google Scholar

[21]

Control Cybernet., 38 (2009), 1305-1325.  Google Scholar

[22]

Robert E. Krieger Publ. Co., 1980.  Google Scholar

[23]

Comp. Optim. Appl., 30 (2005), 45-61. doi: 10.1007/s10589-005-4559-5.  Google Scholar

[24]

in "Mathematical Programming with Data Perturbations V" (ed. A. V. Fiacco), Lecture Notes in Pure and Applied Mathematics 195, Marcel Dekker, (1997), 253-284. Google Scholar

[25]

Optimal Control Appl. Methods, 26 (2005), 129-156. doi: 10.1002/oca.756.  Google Scholar

[26]

SIAM J. Control Optim., 42 (2004), 2239-2263. doi: 10.1137/S0363012902402578.  Google Scholar

[27]

ESAIM, Math. Model. Numer. Anal., 44 (2010), 167-188. doi: 10.1051/m2an/2009045.  Google Scholar

[28]

SIAM J. Contr. Optim., 43 (2004), 970-985. doi: 10.1137/S0363012903431608.  Google Scholar

[29]

Wiley-Interscience, 1988.  Google Scholar

[30]

Springer-Verlag, 1994. Google Scholar

[31]

SIAM J. Control Optim., 35 (1997), 1470-1486. doi: 10.1137/S0363012995288987.  Google Scholar

[32]

Control Cybernet., 34 (2005), 967-982.  Google Scholar

[33]

SIAM J. Control Optim., 28 (1990), 1148-1161. doi: 10.1137/0328062.  Google Scholar

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