# American Institute of Mathematical Sciences

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

 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 $O(h)$.
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
##### 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., (). [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. [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. [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). [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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##### 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., (). [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. [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. [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). [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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