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Quadratic order conditions for bang-singular extremals
Error bounds for Euler approximation of linear-quadratic control problems with bang-bang solutions
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 |
References:
[1] |
W. Alt, On the approximation of infinite optimization problems with an application to optimal control problems,, Appl. Math. Optim., 12 (1984), 15.
doi: 10.1007/BF01449031. |
[2] |
W. Alt, Local stability of solutions to differentiable optimization problems in Banach spaces,, J. Optim. Theory Appl., 70 (1991), 443.
doi: 10.1007/BF00941297. |
[3] |
W. Alt, Discretization and mesh-independence of Newton's method for generalized equations,, in, (1997), 1.
|
[4] |
W. Alt, R. Baier, M. Gerdts and F. Lempio, Approximations of linear control problems with bang-bang solutions,, Optimization, (2011).
doi: 10.1080/02331934.2011.568619. |
[5] |
W. Alt and N. Bräutigam, Finite-difference discretizations of quadratic control problems governed by ordinary elliptic differential equations,, Comp. Optim. Appl., 43 (2009), 133.
doi: 10.1007/s10589-007-9129-6. |
[6] |
W. Alt and U. Mackenroth, Convergence of finite element approximations to state constrained convex parabolic boundary control problems,, SIAM J. Control Optim., 27 (1989), 718.
doi: 10.1137/0327038. |
[7] |
W. Alt and M. Seydenschwanz, Regularization and discretization of linear-quadratic control problems,, Control Cybernet., (). Google Scholar |
[8] |
R. Baier, I. A. Chahma and F. Lempio, Stability and convergence of Euler's method for state-constrained differential inclusions,, SIAM J. Optim., 18 (2007), 1004.
doi: 10.1137/060661867. |
[9] |
W. J. Beyn and J. Rieger, Numerical fixed grid methods for differential inclusions,, Computing, 81 (2007), 91.
doi: 10.1007/s00607-007-0240-4. |
[10] |
I. A. Chahma, Set-valued discrete approximation of state-constrained differential inclusions,, Bayreuth. Math. Schr., 67 (2003), 3.
|
[11] |
K. Deckelnick and M. Hinze, A note on the approximation of elliptic control problems with bang-bang controls,, Comp. Optim. Appl., (2010), 10589.
doi: 10.1007/s10589-010-9365-z. |
[12] |
V. Dhamo and F. Tröltzsch, Some aspects of reachability for parabolic boundary control problems with control constraints,, Comp. Optim. Appl., (2010), 10589.
doi: 10.1007/s10589-009-9310-1. |
[13] |
A. L. Dontchev and E. M. Farkhi, Error estimates for discretized differential inclusions,, Computing, 41 (1989), 349.
doi: 10.1007/BF02241223. |
[14] |
A. L. Dontchev and W. W. Hager, Lipschitzian stability in nonlinear control and optimization,, SIAM J. Control Optim., 31 (1993), 569.
doi: 10.1137/0331026. |
[15] |
A. L. Dontchev and W. W. Hager, The Euler approximation in state constrained optimal control,, Math. Comp., 70 (2001), 173.
doi: 10.1090/S0025-5718-00-01184-4. |
[16] |
A. L. Dontchev, W. W. Hager and K. Malanowski, Error bounds for Euler approximation of a state and control constrained optimal control problem,, Numer. Funct. Anal. Optim., 21 (2000), 653.
doi: 10.1080/01630560008816979. |
[17] |
I. Ekeland and R. Temam, "Convex Analysis and Variational Problems,", North Holland, (1976).
|
[18] |
U. Felgenhauer, On stability of bang-bang type controls,, SIAM J. Control Optim., 41 (2003), 1843.
doi: 10.1137/S0363012901399271. |
[19] |
U. Felgenhauer, The shooting approach in analyzing bang-bang extremals with simultaneous control switches,, Control Cybernet., 37 (2008), 307.
|
[20] |
U. Felgenhauer, Directional sensitivity differentials for parametric bang-bang control problems,, in, (2010), 264. Google Scholar |
[21] |
U. Felgenhauer, L. Poggiolini and G. Stefani, Optimality and stability result for bang-bang optimal controls with simple and double switch behaviour,, Control Cybernet., 38 (2009), 1305.
|
[22] |
M. R. Hestenes, "Calculus of Variations and Optimal Control Theory,", Robert E. Krieger Publ. Co., (1980).
|
[23] |
M. Hinze, A variational discretization concept in control constrained optimization: The linear-quadratic case,, Comp. Optim. Appl., 30 (2005), 45.
doi: 10.1007/s10589-005-4559-5. |
[24] |
K. Malanowski, C. Büskens and H. Maurer, Convergence of approximations to nonlinear optimal control problems,, in, (1997), 253. Google Scholar |
[25] |
H. Maurer, C. Büskens, J.-H. R. Kim and C. Y. Kaya, Optimization methods for the verification of second-order sufficient conditions for bang-bang controls,, Optimal Control Appl. Methods, 26 (2005), 129.
doi: 10.1002/oca.756. |
[26] |
H. Maurer and N. P. Osmolovskii, Second order sufficient conditions for time optimal bang-bang control,, SIAM J. Control Optim., 42 (2004), 2239.
doi: 10.1137/S0363012902402578. |
[27] |
P. Merino, F. Tröltzsch and B. Vexler, Error estimates for the finite element approximation of a semilinear elliptic control problem with state constraints and finite dimensional control space,, ESAIM, 44 (2010), 167.
doi: 10.1051/m2an/2009045. |
[28] |
C. Meyer and A. Rösch, Superconvergence properties of optimal control problems,, SIAM J. Contr. Optim., 43 (2004), 970.
doi: 10.1137/S0363012903431608. |
[29] |
B. Sendov and V. A. Popov, "The Averaged Moduli of Smoothness,", Wiley-Interscience, (1988).
|
[30] |
J. Stoer and R. Bulirsch, "Introduction to Numerical Analysis,", Springer-Verlag, (1994). Google Scholar |
[31] |
V. M. Veliov, On the time-discretization of control systems,, SIAM J. Control Optim., 35 (1997), 1470.
doi: 10.1137/S0363012995288987. |
[32] |
V. M. Veliov, Error analysis of discrete approximations to bang-bang optimal control problems: the linear case,, Control Cybernet., 34 (2005), 967.
|
[33] |
P. R. Wolenski, The exponential formula for the reachable set of a Lipschitz differential inclusion,, SIAM J. Control Optim., 28 (1990), 1148.
doi: 10.1137/0328062. |
show all references
References:
[1] |
W. Alt, On the approximation of infinite optimization problems with an application to optimal control problems,, Appl. Math. Optim., 12 (1984), 15.
doi: 10.1007/BF01449031. |
[2] |
W. Alt, Local stability of solutions to differentiable optimization problems in Banach spaces,, J. Optim. Theory Appl., 70 (1991), 443.
doi: 10.1007/BF00941297. |
[3] |
W. Alt, Discretization and mesh-independence of Newton's method for generalized equations,, in, (1997), 1.
|
[4] |
W. Alt, R. Baier, M. Gerdts and F. Lempio, Approximations of linear control problems with bang-bang solutions,, Optimization, (2011).
doi: 10.1080/02331934.2011.568619. |
[5] |
W. Alt and N. Bräutigam, Finite-difference discretizations of quadratic control problems governed by ordinary elliptic differential equations,, Comp. Optim. Appl., 43 (2009), 133.
doi: 10.1007/s10589-007-9129-6. |
[6] |
W. Alt and U. Mackenroth, Convergence of finite element approximations to state constrained convex parabolic boundary control problems,, SIAM J. Control Optim., 27 (1989), 718.
doi: 10.1137/0327038. |
[7] |
W. Alt and M. Seydenschwanz, Regularization and discretization of linear-quadratic control problems,, Control Cybernet., (). Google Scholar |
[8] |
R. Baier, I. A. Chahma and F. Lempio, Stability and convergence of Euler's method for state-constrained differential inclusions,, SIAM J. Optim., 18 (2007), 1004.
doi: 10.1137/060661867. |
[9] |
W. J. Beyn and J. Rieger, Numerical fixed grid methods for differential inclusions,, Computing, 81 (2007), 91.
doi: 10.1007/s00607-007-0240-4. |
[10] |
I. A. Chahma, Set-valued discrete approximation of state-constrained differential inclusions,, Bayreuth. Math. Schr., 67 (2003), 3.
|
[11] |
K. Deckelnick and M. Hinze, A note on the approximation of elliptic control problems with bang-bang controls,, Comp. Optim. Appl., (2010), 10589.
doi: 10.1007/s10589-010-9365-z. |
[12] |
V. Dhamo and F. Tröltzsch, Some aspects of reachability for parabolic boundary control problems with control constraints,, Comp. Optim. Appl., (2010), 10589.
doi: 10.1007/s10589-009-9310-1. |
[13] |
A. L. Dontchev and E. M. Farkhi, Error estimates for discretized differential inclusions,, Computing, 41 (1989), 349.
doi: 10.1007/BF02241223. |
[14] |
A. L. Dontchev and W. W. Hager, Lipschitzian stability in nonlinear control and optimization,, SIAM J. Control Optim., 31 (1993), 569.
doi: 10.1137/0331026. |
[15] |
A. L. Dontchev and W. W. Hager, The Euler approximation in state constrained optimal control,, Math. Comp., 70 (2001), 173.
doi: 10.1090/S0025-5718-00-01184-4. |
[16] |
A. L. Dontchev, W. W. Hager and K. Malanowski, Error bounds for Euler approximation of a state and control constrained optimal control problem,, Numer. Funct. Anal. Optim., 21 (2000), 653.
doi: 10.1080/01630560008816979. |
[17] |
I. Ekeland and R. Temam, "Convex Analysis and Variational Problems,", North Holland, (1976).
|
[18] |
U. Felgenhauer, On stability of bang-bang type controls,, SIAM J. Control Optim., 41 (2003), 1843.
doi: 10.1137/S0363012901399271. |
[19] |
U. Felgenhauer, The shooting approach in analyzing bang-bang extremals with simultaneous control switches,, Control Cybernet., 37 (2008), 307.
|
[20] |
U. Felgenhauer, Directional sensitivity differentials for parametric bang-bang control problems,, in, (2010), 264. Google Scholar |
[21] |
U. Felgenhauer, L. Poggiolini and G. Stefani, Optimality and stability result for bang-bang optimal controls with simple and double switch behaviour,, Control Cybernet., 38 (2009), 1305.
|
[22] |
M. R. Hestenes, "Calculus of Variations and Optimal Control Theory,", Robert E. Krieger Publ. Co., (1980).
|
[23] |
M. Hinze, A variational discretization concept in control constrained optimization: The linear-quadratic case,, Comp. Optim. Appl., 30 (2005), 45.
doi: 10.1007/s10589-005-4559-5. |
[24] |
K. Malanowski, C. Büskens and H. Maurer, Convergence of approximations to nonlinear optimal control problems,, in, (1997), 253. Google Scholar |
[25] |
H. Maurer, C. Büskens, J.-H. R. Kim and C. Y. Kaya, Optimization methods for the verification of second-order sufficient conditions for bang-bang controls,, Optimal Control Appl. Methods, 26 (2005), 129.
doi: 10.1002/oca.756. |
[26] |
H. Maurer and N. P. Osmolovskii, Second order sufficient conditions for time optimal bang-bang control,, SIAM J. Control Optim., 42 (2004), 2239.
doi: 10.1137/S0363012902402578. |
[27] |
P. Merino, F. Tröltzsch and B. Vexler, Error estimates for the finite element approximation of a semilinear elliptic control problem with state constraints and finite dimensional control space,, ESAIM, 44 (2010), 167.
doi: 10.1051/m2an/2009045. |
[28] |
C. Meyer and A. Rösch, Superconvergence properties of optimal control problems,, SIAM J. Contr. Optim., 43 (2004), 970.
doi: 10.1137/S0363012903431608. |
[29] |
B. Sendov and V. A. Popov, "The Averaged Moduli of Smoothness,", Wiley-Interscience, (1988).
|
[30] |
J. Stoer and R. Bulirsch, "Introduction to Numerical Analysis,", Springer-Verlag, (1994). Google Scholar |
[31] |
V. M. Veliov, On the time-discretization of control systems,, SIAM J. Control Optim., 35 (1997), 1470.
doi: 10.1137/S0363012995288987. |
[32] |
V. M. Veliov, Error analysis of discrete approximations to bang-bang optimal control problems: the linear case,, Control Cybernet., 34 (2005), 967.
|
[33] |
P. R. Wolenski, The exponential formula for the reachable set of a Lipschitz differential inclusion,, SIAM J. Control Optim., 28 (1990), 1148.
doi: 10.1137/0328062. |
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