In the last lecture, we have introduced an important idea, is the concept of tarpike

that I remind you, it emphasizes the fact that when the time horizon is very long, very

often the optimal strategy is essentially of a steady state nature, so that when we are moving

from one place to another, when we are controlling a system from an inertial state to a final

destination in a long time horizon, the best we can do is to go from the initial datum to the

optimal steady state, stay in the steady state and then jump off again. This is like if you go from

your university in Chuncheon to the Academy of Sciences in Beijing, probably what you will

do is to take the fast train that will take two or three hours and then the control will be

moving from your office to the train station in Chuncheon. So there is an inertial arc which is

active, this is what we are highlighting here, there is an inertial arc in which the control

system has to be very active to get into a short time, the steady control will be the

fast train where you will sit for three hours, the train will drive you close to the final

destination, not exactly to the final destination because you will arrive to Beijing train station

and then you will jump. This was the key observations we made and that in practical

applications what it means is that if you are working in long time intervals, rather than

computing the control in the whole long time interval which would be very expensive, because

you remember in order to control an optimal control, applying a gradient descent algorithm

we have to implement a numerical methodology that was iterative, in which we first solve the

adjoint equation, take the control that the adjoint equation gives, plug it into the state equation,

then solve the forward state equation, check the terminal condition, compute the residual,

meaning the distance of the solution to the target that you want to minimize or to make it zero,

and with this residual adjust the solution of the adjoint equation and then solve the backwards

adjoint equation to then plug it again into the forward equation and so on. Therefore you are

solving in each step of the gradient iteration the adjoint equation and the state equation fully

in the whole time horizon. When the time horizon is very long this will be computationally very

expensive and in particular it will lead to errors that are substantial because of the fact

that numerical algorithms converge, but you have after all an estimate of global time on the

convergence of the numerical scheme which grows when the time interval is very long

and that will oblige you in long time ornithons to take the time step to be extremely narrow,

but if the time horizon is long and in the time step in the numerical algorithm is very very narrow

then the cost of solving each of these equations, the adjoint backward and the state forward

will have a very substantial computational cost. In practice this could be even impossible for

say large dimensional control systems appearing in application. This is why

the philosophy that emerges out of the d star pi principle is that before

rather than controlling the system actively in a time-dependent manner you first compute

the steady state optimal control in which the time has been chopped off. You implement this

steady optimal control in most of the time intervals and then you simply solve two short time

controllability problems in the beginning and in the end just to make sure that you are linking

the initial state to the steady state and then eventually the steady state into the final target.

So this is the tarpike philosophy that emerges out of this analysis. Of course this only makes

sense when time is very long because as I said when time is very short you need to act in a very

singular manner with either very large or oscillatory controls or with direct delta

controls, impulsional controls, but there is no room to take this such tarpike path.

But oftentimes in applications as I said the time that we are given is long,

is long for the process under consideration and therefore this tarpike strategy is the one

that is the most convenient one. Now when are these controls obtained through the

tarpike principle optimal? The important remark we made was that these controls are typically

optimal when you consider controls that are both optimal in the sense that they are of a small norm

but also the whole trajectory is penalized. And as I said in principle you could say that the whole

trajectory is not penalized because you say okay I wanted to go from Chanchun to Beijing,