Chapter 3: Linear–Quadratic, Indirect, and Singular Optimal Control

From Riccati feedback to shooting methods, bang–bang control, and singular arcs

With the necessary conditions in hand, we can study what they reveal and where they become difficult to use. The chapter begins with finite- and infinite-horizon linear–quadratic control, moves to indirect boundary-value methods and Hamiltonian structure, and then examines bang–bang, singular, and path-constrained behavior. The sequence connects an analytically tractable benchmark to the structural and numerical complications of general optimal-control problems.

Chapter contents¶

1. Linear–Quadratic Optimal Control and the Riccati Differential Equation

2. Infinite-Horizon Linear Quadratic Regulation

3. Indirect Numerical Methods, Multiple Shooting, and Pontryagin’s Minimum Principle

4. Hamiltonian Conservation, Minimum-Time Control, and Bang–Bang Solutions

5. Bolza, Mayer, and Lagrange Forms, Minimum-Time Conventions, and Singular Arcs

6. Singular Optimal Control Theory

7. The Goddard Rocket Problem and Singular Thrust

8. Coupled State–Costate Dynamics, Hamiltonian Matrices, and Shooting Instability

9. Path Constraints, Active Sets, and Augmented Hamiltonians

10. Exercises