Optimal Control and the Calculus of Variations

1: Introduction 1.1: The maxima and minima of functions 1.2: The calculus of variations 1.3: Optimal control Part 2: Optimization in 2.1: Functions of one variable 2.2: Critical points, end-points, and points of discontinuity 2.3: Functions of several variables 2.4: Minimization with constraints 2.5: A geometrical interpretation 2.6: Distinguishing maxima from minima Part 3: The calculus of variations 3.1: Problems in which the end-points are not fixed 3.2: Finding minimizing curves 3.3: Isoperimetric problems 3.4: Sufficiency conditions 3.5: Fields of extremals 3.6: Hilbert's invariant integral 3.7: Semi-fields and the Jacobi condition Part 4: Optimal Control I: Theory 4.1: Introduction 4.2: Control of a simple first-order system 4.3: Systems governed by ordinary differential equations 4.4: The optimal control problem 4.5: The Pontryagin maximum principle 4.6: Optimal control to target curves Part 5: Optimal Control II: Applications 5.1: Time-optimal control of linear systems 5.2: Optimal control to target curves 5.3: Singular controls 5.4: Fuel-optimal controls 5.5: Problems where the cost depends on X (t l) 5.6: Linear systems with quadratic cost 5.7: The steady-state Riccai equation 5.8: The calculus of variations revisited Part 6: Proof of the Maximum Principle of Pontryagin 6.1: Convex sets in 6.2: The linearized state equations 6.3: Behaviour of H on an optimal path 6.4: Sufficiency conditions for optimal control Appendix: Answers and hints for the exercises Bibliography Index