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Numerical Methods for Nonlinear Optimal Control Problems

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Encyclopedia of Systems and Control

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Abstract

In this article we describe the three most common approaches for numerically solving nonlinear optimal control problems governed by ordinary differential equations. For computing approximations to optimal value functions and optimal feedback laws, we present the Hamilton-Jacobi-Bellman approach. For computing approximately optimal open-loop control functions and trajectories for a single initial value, we outline the indirect approach based on Pontryagin’s maximum principle and the approach via direct discretization.

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Bibliography

  • Bardi M, Capuzzo Dolcetta I (1997) Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Birkhäuser, Boston

    Book  MATH  Google Scholar 

  • Betts JT (2010) Practical methods for optimal control and estimation using nonlinear programming, 2nd edn. SIAM, Philadelphia

    Book  MATH  Google Scholar 

  • Binder T, Blank L, Bock HG, Bulirsch R, Dahmen W, Diehl M, Kronseder T, Marquardt W, Schlöder JP, von Stryk O (2001) Introduction to model based optimization of chemical processes on moving horizons. In: Grötschel M, Krumke SO, Rambau J (eds) Online optimization of large scale systems: state of the art. Springer, Heidelberg, pp 295–340

    Chapter  MATH  Google Scholar 

  • Bock HG, Plitt K (1984) A multiple shooting algorithm for direct solution of optimal control problems. In: Proceedings of the 9th IFAC world congress, Budapest. Pergamon, Oxford, pp 242–247

    Google Scholar 

  • Bryson AE, Ho YC (1975) Applied optimal control. Hemisphere Publishing Corp., Washington, DC. Revised printing

    Google Scholar 

  • Falcone M (1997) Numerical solution of dynamic programming equations. In: Appendix A in Bardi M, Capuzzo Dolcetta I (eds) Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Birkhäuser, Boston

    Google Scholar 

  • Falcone M, Ferretti R (2013) Semi-Lagrangian approximation schemes for linear and Hamilton-Jacobi equations. SIAM, Philadelphia

    Book  MATH  Google Scholar 

  • Garg D, Patterson M, Hager WW, Rao AV, Benson DA, Huntington GT (2010) A unified framework for the numerical solution of optimal control problems using pseudospectral methods. Automatica 46(11):1843–1851

    Article  MathSciNet  MATH  Google Scholar 

  • Gerdts M (2012) Optimal control of ODEs and DAEs. De Gruyter textbook. Walter de Gruyter & Co., Berlin

    Book  MATH  Google Scholar 

  • Grass D, Caulkins JP, Feichtinger G, Tragler G, Behrens DA (2008) Optimal control of nonlinear processes. Springer, Berlin

    Book  MATH  Google Scholar 

  • Grüne L, Pannek J (2017) Nonlinear model predictive control: theory and algorithms, 2nd edn. Springer, Cham

    Book  MATH  Google Scholar 

  • Malanowski K, Büskens C, Maurer H (1998) Convergence of approximations to nonlinear optimal control problems. In: Fiacco AV (ed) Mathematical programming with data perturbations. Lecture notes in pure and applied mathematics, vol 195. Dekker, New York, pp 253–284

    Google Scholar 

  • Malanowski K, Maurer H, Pickenhain S (2004) Second-order sufficient conditions for state-constrained optimal control problems. J Optim Theory Appl 123(3):595–617

    Article  MathSciNet  MATH  Google Scholar 

  • Maurer H (1981) First and second order sufficient optimality conditions in mathematical programming and optimal control. Math Program Stud 14: 163–177

    Article  MathSciNet  MATH  Google Scholar 

  • McEneaney WM (2006) Max-plus methods for nonlinear control and estimation. Systems & control: foundations & applications. Birkhäuser, Boston

    Google Scholar 

  • Pesch HJ (1994) A practical guide to the solution of real-life optimal control problems. Control Cybern 23(1–2):7–60

    MathSciNet  MATH  Google Scholar 

  • Vinter R (2000) Optimal control. Systems & control: foundations & applications. Birkhäuser, Boston

    Google Scholar 

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Author information

Authors and Affiliations

  1. Mathematical Institute, University of Bayreuth, Bayreuth, Germany

    Lars Grüne

Authors
  1. Lars Grüne
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Correspondence to Lars Grüne .

Editor information

Editors and Affiliations

  1. College of Engineering, Boston University, Boston, MA, USA

    John Baillieul

  2. Technological Leadership Institute, University of Minnesota, Minneapolis, MA, USA

    Tariq Samad

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Grüne, L. (2021). Numerical Methods for Nonlinear Optimal Control Problems. In: Baillieul, J., Samad, T. (eds) Encyclopedia of Systems and Control. Springer, Cham. https://doi.org/10.1007/978-3-030-44184-5_208

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