Overview
Optimal Control focuses on optimize decisions in dynamic physical and engineered systems. In the map of OR, it connects State dynamics, Control, HJB to decisions that must be modeled, solved, explained, and revised as evidence changes.
Optimal control studies continuous-time dynamics, control policies, Hamilton-Jacobi-Bellman equations, and model predictive control. The practical use case is clearest in Robotics, Energy systems, Aerospace, Process control, where the method helps turn constraints and tradeoffs into a decision artifact someone can inspect.
Core ideas
State dynamics
State dynamics is a core checkpoint for Optimal Control: define it concretely, attach units or rules where possible, and test whether stakeholders interpret it the same way.
Control
Control is a core checkpoint for Optimal Control: define it concretely, attach units or rules where possible, and test whether stakeholders interpret it the same way.
HJB
HJB is a core checkpoint for Optimal Control: define it concretely, attach units or rules where possible, and test whether stakeholders interpret it the same way.
MPC
MPC is a core checkpoint for Optimal Control: define it concretely, attach units or rules where possible, and test whether stakeholders interpret it the same way.
Pontryagin principle
Pontryagin principle is a core checkpoint for Optimal Control: define it concretely, attach units or rules where possible, and test whether stakeholders interpret it the same way.
How to use it
- 1Start with Robotics: write the decision, time horizon, actors, and objective in operational language.
- 2Translate the problem into State dynamics, Control, and HJB; define units and data sources for each one.
- 3Build a small instance of Optimal Control that can be solved or simulated by hand inspection before using full production data.
- 4Compare the recommendation against a baseline policy, not just against mathematical optimality.
- 5Document assumptions, sensitivity results, and the conditions under which the recommendation should be revisited.
Applications
- Robotics: compare feasible policies, quantify the operating tradeoffs, and make the assumptions behind the recommendation visible.
- Energy systems: compare feasible policies, quantify the operating tradeoffs, and make the assumptions behind the recommendation visible.
- Aerospace: compare feasible policies, quantify the operating tradeoffs, and make the assumptions behind the recommendation visible.
- Process control: compare feasible policies, quantify the operating tradeoffs, and make the assumptions behind the recommendation visible.
Common pitfalls
- Applying Optimal Control because the label sounds appropriate while leaving the actual decision boundary vague.
- Treating State dynamics as a technical detail instead of a modeling choice that affects the recommendation.
- Reporting one answer without showing sensitivity to demand, capacity, costs, or behavioral assumptions.
- Ignoring implementation details such as data quality, explainability, ownership, and how users will override bad recommendations.
Resources
- Dynamic Programming and Optimal Control — Bertsekas
Topic-specific source curated for Optimal Control.
- MIT OCW 6.231 — Dynamic Programming and Stochastic Control
Dynamic programming and stochastic control lectures for sequential decision models.
- Stanford EE364A — Convex Optimization
Convex analysis, duality, least squares, quadratic programs, and conic optimization.
- Algorithms for Decision Making
Open book covering planning, MDPs, reinforcement learning, and decision algorithms.