I have a suggestion or else. So I'm going to talk about work, which is all of it is done

in collaboration with Enrique and some of it, some parts with Carlos Esteve and Daniel Piguillin.

So I'm going to split the talk in two parts. The first part is I'm going to present a method we

developed for proving the term by property for some specific nonlinear problem. So some of you may be

familiar with many talks myself or mainly have given on control and machine learning and typically

we show videos of how we interpret machine learning as an optimal control problem and we

have some theoretical results which explain these videos and these theoretical results are actually

based on this method, this proof which actually I don't think we've ever really presented it at a

talk so I'm going to try to do it now today in a more general setting of control theory not

necessarily focusing on machine learning and then I'm going to present another work which we did

with Enrique which is just it's not really very fundamental or technically developed

but it's a proposal which I think proposal on a classical topic, classical I'd say

established topic in control theory which I think is gonna is raising some questions

some interesting questions. So I'm going to start with the turnpike part. So what is just to motivate

the turnpike property let us consider a very standard optimal control problem in which we're

given say the heat equation in a bounded domain omega so okay omega is a bounded domain smooth I

don't go too much into these details and what matters here is that I have a tracking term so I

I want to minimize the discrepancy from a given steady target

and I also penalize the control so I have a control u the control u is actuating only inside

a small domain omega and then I observe basically this discrepancy from another domain omega naught

there is no necessary correlation between omega and omega naught and I solve this problem but typically

typically you you would be interested in solving something which is a bit simpler has much less

degrees of freedom which is the steady problem so we remove time and we consider the steady

Poisson equation no time and we just remove the time integrals in the in the functional so obviously

this problem has many many much less degrees of freedom it's a compressed version of the original

problem and this is typically what many practitioners solve in practice because if for instance you

imagine that instead of having the heat equation which is a relatively simple system to solve

numerically instead of having the heat equation you have something which is much more complicated

like Navier-Stokes or Euler or whatever then time time stepping and integrals in time are going to

be very costly so you're going to be interested in solving something like this the second problem

but can you actually do this does this make sense and actually the term by property is exactly what

tells you that this is a reasonable assumption basically to consider a steady problem and the

term by property tells you that if we're given a target yd and we're given these domains and we're

given an initial condition then there exists a couple of constants which are independent of the

time horizon for which we know that the optimal trajectory time-evolving trajectory and optimal

time-evolving control are going to be close to the correspondence steady counterparts

over most of the time so this is explained by this estimate and this estimate how do you read it well

basically you see that this function is going to be a very small in for all time which is far from

zero and capital T so you have two arcs and the way people motivate the the name term by property

is basically by looking back to economics where from where actually this this property comes from

originally and in economics you're interested in typically designing policies for for capital

you're given an initial capital level a final capital level and you want to find over time what

is the best strategy you can design so that you you maximize your profits over time and actually

turns out that people like paul samuelson solo and also even this goes back to for noyman ramsey etc

so there's a big history on this problem they found out that the optimal way to do this is basically

to to go to the steady path for most of the time and just match the starting point and the end point

at the initial and final time respect okay so if you're interested in learning more about the term

by property i'm not going to go too much into details on on the history and on on the established

theory but i encourage you to look at our paper with anika where we we talk about all of this in

in in detail there's many remarks many open problems etc okay just to motivate the results