Introduction to Optimal Control

Mathematical Modeling · Axiom Academy

LESSON Introduction to Optimal Control - Mathematical Modeling Introduction to Optimal Control Optimizing Dynamic Systems Over Time Many real-world problems involve optimizing systems that evolve over time. Unlike static optimization, we must find the best trajectory or policy : Aerospace: Find the thrust profile that minimizes fuel while reaching orbit Economics: Determine optimal investment and consumption over a lifetime Robotics: Plan the motor controls that move a robot arm along the fastest path Medicine: Design drug dosing schedules that maximize efficacy while minimizing side effects Ecology: Determine sustainable harvesting policies for renewable resources Optimal control theory provides the mathematical framework to solve these dynamic optimization problems. An optimal control problem consists of three key ingredients: Components of an Optimal Control Problem 1. State Variables x(t) : Describe the system's configuration at time t 2. Control Variables u(t) : The decisions or inputs we can choose 3. Dynamics: How the state evolves based on current state and control 4. Objective Functional: What we want to minimize or maximize 5. Boundary Conditions: Initial state x(0) = x_0 and possibly terminal conditions Understanding the Objective Functional The objective J has two parts: Terminal cost : Penalizes or rewards the final state Running cost L(x, u, t) : Accumulated cost along the trajectory Mayer problem: Only terminal cost ( L = 0 )

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