The maximum principle of Pontryagin in control and in optimal control

Control systems......Page 4 Two general problems in optimal control......Page 5 Some examples of optimal control problems......Page 6 The necessary condition of Euler–Lagrange.......Page 8 The necessary condition of Legendre.......Page 9 The necessary condition of Weierstrass.......Page 10 Discussion of calculus of variations......Page 11 The Skinner–Rusk formulation of the calculus of variations......Page 13 From the calculus of variations to the Maximum Principle, almost......Page 14 Preliminary definitions......Page 16 The mysterious parts and the useful parts of the Maximum Principle......Page 17 The variational and adjoint equations......Page 19 Needle variations......Page 21 Multi-needle variations......Page 23 Free interval variations......Page 24 The fixed interval tangent cone......Page 26 The free interval tangent cone......Page 28 Approximation by the fixed interval tangent cone.......Page 29 Approximation by the free interval tangent cone.......Page 30 The connection between tangent cones and the Hamiltonian......Page 31 The fixed interval case.......Page 33 The free interval case.......Page 34 The properties of the adjoint response and the Hamiltonian......Page 37 The transversality conditions......Page 38 Regular and singular extremals......Page 42 Tangent cones and linearisation......Page 43 Control systems.......Page 44 Tangent cones.......Page 45 The necessary conditions of the Maximum Principle......Page 47 The rôle of the Riccati differential equation......Page 48 The infinite horizon problem......Page 49 Linear quadratic optimal control as a case study of abnormality......Page 50 The Maximum Principle for linear time-optimal control......Page 53 An example......Page 54 Integration.......Page 57 Ordinary differential equations with measurable time-dependence......Page 58 Combinations and hulls......Page 60 Topology of convex sets and cones......Page 63 Simplices and simplex cones......Page 64 Separation theorems for convex sets......Page 66 Linear functions on convex polytopes......Page 67 The Hairy Ball Theorem......Page 70 The desired results......Page 72