# CDS 110b: Optimal Control

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.

Q: Could you please explain what the psi function is or what it means if psi(x(T))=0 versus what it means if psi(x(T))=x(T)?

The psi function represents a general form of terminal constraint for the state variables. It gives a way of indicating which states have a terminal cost attached to them. For example, by defining psi_i(x(T))=x_i(T)-x_i,f for i=1,2,...n, we can impose terminal costs on all states (a fully constrained case) by letting p=n (n being the # of states). When we optimize over time and want x(T)=x_f, then x(T)-x_f=0, and so psi(x(T))=0.

Luis Soto, 22 Jan 08

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(t_f) 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 $u$ obtained?

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

Shaunak Sen, 12 Jan 06

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

Yes, you are correct.

Jeremy Gillula, 07 Jan 05