Part XIII: Optimal Control & Congestion Pricing
From Pontryagin's Maximum Principle for traffic signal optimization through LQR via the Riccati equation to Pigouvian congestion tolls and model predictive control — the mathematical machinery for actively steering urban traffic systems toward optimal performance.
Part Overview
Pontryagin Maximum Principle for traffic signal optimization, LQR via Riccati, Pigouvian congestion tolls, and model predictive control. Derives bang-bang signal timing and Green Wave as a singular arc, then bridges to real-time pricing with \(\tau^* = f \cdot dt/df\) and receding-horizon MPC for adaptive network control.
Key Topics
- • PMP for store-and-forward signal networks
- • Bang-bang signal timing
- • Green Wave as singular arc
- • LQR and algebraic Riccati equation
- • Pigouvian toll \(\tau^* = f \cdot dt/df\)
- • Model Predictive Control
3 chapters | From optimal control to congestion pricing | Steering traffic with mathematics
Chapters
Chapter 1: Pontryagin Maximum Principle
Applies PMP to the store-and-forward traffic model, deriving bang-bang signal timing from the Hamiltonian switching function and identifying Green Wave coordination as a singular arc solution.
Chapter 2: LQR & Riccati Equation
Linear-quadratic regulator for traffic state feedback via the algebraic Riccati equation. Derives the steady-state gain matrix \(K = R^{-1}B^T P\) and its stability guarantees for linearized traffic networks.
Chapter 3: Congestion Pricing & MPC
Pigouvian congestion tolls \(\tau^* = f \cdot dt/df\) that internalize the externality of marginal delay, combined with model predictive control for receding-horizon adaptive pricing and signal optimization.