# Difference between revisions of "CDS 110b: Optimal Control"

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This lecture provides an overview of optimal control theory. Beginning with a review of optimization, we introduce the notion of Lagrange multipliers and provide a summary of the Pontryagin's maximum principle.

## Frequently Asked Questions

Q: In Problem 2.4d, are the boundary conditions for the differentially-flat trajectory correct?

Please ignore the boundary conditions given in part 2.4d for the differentially-flat trajectory and instead use x(0)=1 for the initial condition and x(1)=0 for the condition at final time t=1. Moreover, use c=100 instead of c=1. Note: the x(tf) of the optimal solution won't be exactly 0, but will be close enough for the intent of this problem.

Luis Soto, 21 Jan 08

Q: In problem 2.4(d) of the homework, to what positive value should the parameter b be set?

Use b = 1 for part d when solving for and comparing the two trajectories found symbolically in previous parts.

Julia Braman, 18 Jan 08

Q: In the example on Bang-Bang control discussed in the lecture, how is the control law for ${\displaystyle u}$ obtained?

Pontryagin's Maximum Principle says that ${\displaystyle u}$ has to be chosen to minimise the Hamiltonian ${\displaystyle H(x,u,\lambda )}$ for given values of ${\displaystyle x}$ and ${\displaystyle \lambda }$. In the example, ${\displaystyle H=1+({\lambda }^{T}A)x+({\lambda }^{T}B)u}$. At first glance, it seems that the more negative ${\displaystyle u}$ is the more ${\displaystyle H}$ will be minimised. And since the most negative value of ${\displaystyle u}$ allowed is ${\displaystyle -1}$, ${\displaystyle u=-1}$. However, the co-efficient of ${\displaystyle u}$ may be of either sign. Therefore, the sign of ${\displaystyle u}$ has to be chosen such that the sign of the term ${\displaystyle ({\lambda }^{T}B)u}$ is negative. That's how we come up with ${\displaystyle u=-sign({\lambda }^{T}B)}$.

Shaunak Sen, 12 Jan 06

Q: Notation question for you: In the Lecture notes from Wednesday, I'm assuming that ${\displaystyle T}$ is the final time and ${\displaystyle T}$ (superscript T) is a transpose operation. Am I correct in my assumption?

Yes, you are correct.

Jeremy Gillula, 07 Jan 05