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