Chapter 7: Path Constraints, Dynamic Programming, and Feedback

Constraints, value functions, feedback synthesis, and neighboring-optimal corrections

Practical control design must handle restrictions, changing initial conditions, and deviations from a nominal trajectory. The chapter first develops equality and inequality path constraints, including constrained arcs and their numerical enforcement. It then changes viewpoint from trajectory optimization to the value function through dynamic programming and HJB theory, recovers LQR feedback, places optimal control within a guidance–navigation–control architecture, and concludes with analytical simplification and neighboring-optimal corrections.

Chapter contents¶

1. Equality Path Constraints in Optimal Control

2. Inequality Path Constraints, Complementarity, and Constrained Arcs

3. State Inequality Path Constraints and Direct Collocation

4. The Principle of Optimality and the Hamilton–Jacobi–Bellman Equation

5. HJB Formulation of the Finite-Horizon Linear Quadratic Regulator

6. Riccati Equation, Optimal Feedback Law, and Numerical Implementation

7. Optimal Control within Guidance, Navigation, Estimation, and Control

8. Analytical Direct Shooting and Problem Simplification

9. Bang–Bang Control, Switching Functions, and Singular Arcs

10. Neighboring Optimal Control

11. Exercises