Math 589B Final Programming Project (Spring 2026)

Problem

We consider the controlled pendulum:

$$ \ddot{\theta} = \sin(\theta) - \alpha \dot{\theta} + u \cos(\theta) $$

State: $x = (\theta, \phi)$

Cost:

$$ \int_0^\infty \left[(1 - \cos \theta) + \frac{1}{2}\phi^2 + \frac{1}{2}u^2\right] dt $$

Optimal control: $u^* = -\lambda_2 \cos(\theta)$

Given: $\theta, \phi, \alpha$

Return: $\lambda_1, \lambda_2, J$

./solver theta phi alpha

Output: lambda1 lambda2 J

More details coming this week.