1 Introduction to Nonlinear Optimal Control

  • Bernard Bonnard
  • Jean-Baptiste Caillau
Part of the Lecture Notes in Control and Information Science book series (LNCIS, volume 328)


The maximum principle is presented in the weak and general forms. The standard proofs are detailed, and the connection with the shooting method for numerical resolution is made. A brief introduction to the micro-local analysis of extremals is also provided. Regarding second-order conditions, small timeoptimality is addressed by means of high order generalized variations. As for local optimality of extremals, the conjugate point theory is introduced both for regular problems and for minimum time singular single input affine control systems. The analysis is applied to the minimum time control of the Kepler equation, and the numerical simulations for the corresponding orbit transfer problems are given. In the case of state constrained optimal control problems, necessary conditions are stated for boundary arcs. The junction and reflection conditions are derived in the Riemannian case.


Maximum Principle Optimal Control Problem Reference Trajectory Conjugate Point Lebesgue Point 
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Authors and Affiliations

  • Bernard Bonnard
    • 1
  • Jean-Baptiste Caillau
    • 2
  1. 1.Université de Bourgogne, Bâtiment Sciences Mirande, BP 47870, F-21078 Dijon. 
  2. 2.ENSEEIHT-IRIT (UMR CNRS 5505), 2 Rue Camichel, F-31071 Toulouse. 

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