L4.2 - Discrete-time LQ-optimal control - finite horizon, fixed final state 

The two-point BVP displayed at 0:56 is a prescription for computing x_1 and lambda_{N-1} if x_0 and lambda_N are known. Typically x_0 is known but lambda_N is generally unknown. But ignore this fact at first (and work just symbolically). Now that we have "computed" x_1 and lambda_N-1, we can repeat the previous computation (multiplication by a matrix) and compute x_2 and lambda_N-2. And so on, untill we get x_N and lambda_0. The matrix that is labelled as M at about 1:24 gives us a relation (a linear one) between x_N and lambda_0 on on side and x_0 and lambda N on the other. Out of the four guys, two are given here: x_0 and x_N, and two are unknown: lambda_N and lambda_0. The known guys are not both on the same side but this is not a major problem. We can still solve for the two unknowns (and the procedure is shown in the video). In fact, computing the lambda_N is enough because once we get it, we can solve for full trajectories of x and lambda.

aa4cc Thank you so much for the explanation, it was clear for me! My last question would be: I've been following nicely the introduction to MPC and optimization videos, but since it started this part of the series about the "2 point BVP" and "indirect approach" for optimal control, I come into confusion of what is this method for... I mean, is it an extension for MPC control or is it purely optimization math to compute the optimal inputs "u"? Hope you can clarify me the applications of this indirect approach and this BVP.

@@jesusmanuellevinsonrondon5107 These are just alternative approaches. Different ways to formulate and solve a problem of optimal control. It is similar to what happens if you want to minimize a function of a scalar argument: either you can view it DIRECTly as minimization (and come up with some numerical techniques) or you can reformulate it as a problem of finding a root of a nonlinear equation (remember, we set the derivative - provided it exists - to zero) and by solving this reformulated problem you are solving the original problem INDIRECTly.

Thank you both for your comments, it really cleared up this part for me.

I think that there s sth wrong here ... if you replace k in that expression with either 0 or N-1, it won't give you that result