In this occasion we talk, speak in local neural control able of the complete

n-dimensional Adychevskaja-Busineski model. Index of my presentation. We have

two models. The first model called number one is the classical model of

Busineski but we appear one term, no local term. No local term is v0 plus v1

nor right-wide in L2 squared. This term no local appears in the first equation,

equation is velocity and they appear in left side and side right of temperature

equation. We also other model. This model different or model one called model three.

This no local term is nor L2. This model one is nor L2. This model three is nor Lp with p greater than 3 less or equal to 6.

This respect with term no local term is one, v0. We have the classical Busineski system.

The classical Busineski system we have Pioner, Pioner Borg, Gonzalo Guerrero, Fernandez Cara,

Jean-Pierre Puel and Immanuel Lobe. Other words in Busineski system with no local term not.

We have words of Guerrero and words of Carreño. We with respect term no local. We have words when

first equation only first equation when only with equation is model is called Ladicheskaya-Smagoritsky system.

We proof in collaboration with Fernandez Cara, Silvano Meneses and me. We proofed in the

case Ladicheskaya-Smagoritsky system. No controllability for local in this case. In this working we studying n equal 2 and n equal 3.

We consider classical space, classical space H and VP. For n equal 2 and n equal 3 we have well presented for the nonlinear system 1 definition.

We let initial states y0, theta 0 belong this space. Respectfully y0, theta 0 belong the water space such that d no less than delta.

Respectively y0, theta 0 less than delta for system nonlinear 3 and nonlinear system 1. The axis control B belong L2 W times 0T and B0 belong L2 W0T.

Such that the solution yP theta satisfying null controllability property. We have 3 main results. Theorem 2, the nonlinear system 1 is local null controllable in any capital C rather 0.

Theorem 3, the nonlinear system 3 is local null controllable in capital C rather 0. Remember the nonlinear system 2 is local nonlinear system for nonlinear system 1.

For nonlinear system 1 have a term no local in L2. Theorem 3 is local null controllable for nonlinear system 3 when nullic alter in LP.

We first proof theorem 2 and theorem 3. We first proof result of null controllable label of the linear system associate.

The linear system associate theorem 2 and theorem 3 is same. But we recreated proof different result the nonlinear system.

Other result is large time in null controllability. This result is proof for n equal to. For n equal to is proof of theorem 4.

For proof this linear system is recreated well presented result preliminaries. We have regularity of the linear heat equation.

The result of regularity of linear equation is the spice LLS. This result is proofed by professor Matias Heidl here present.

We use regularity for stock equation. For stock equation we recreated irregularity and other regularity in spice LP1, LP2.

This result is working in paper of Giga. We proof the linear system is necessary proof using Karleman inequality for the I-joint system.

Karleman estimate consider ways n zero and L classical function by definition of the ways necessary by Karleman inequality.

While define L alpha and C E alpha star maximum and alpha hot minimum. We choose a large enoke M in definition alpha such that 36 alpha hot graded are 33 alpha star.

This hypothesis is technical necessary for the proof nonlinear problem. Consider the I-joint system for the linear system.

The linear system remember is sum for nonlinear problem 1 and nonlinear problem 3. Nonlinear problem 1 is we term nonlocal L2 and nonlinear system 3 have nonlocal term in LP.

We have Karleman inequality. This Karleman inequality we proof in work Sergio Guerrero in 26. We now proof of the linear system.

We define it standard ways, standard ways, other standard ways and proposition says 1. Proposition says 1 is control the linear problem necessary for proof of theorem 2.

If n equal 2 and n equal 3 with this condition we can fin state control such that 7 is the linear problem, such that the linear problem is satisfying this inequality.

In particular we have Y in capital C equal 0 and T in capital C equal 0. Looking this condition this way, this way is plot in T equal to T.

But this integral finite then we have Y in T equal 0. These ways plot in capital T and integral finite. This consequence Y in T equal 0 and T in T equal 0.

This proposition says 1 is the control of the linear problem necessary by proof of theorem 2. We have a continuation proposition says 2 is the control regularity for,

the local controllability for the linear system 7 but we need to proof of theorem 3. Proposition says 2. With proposition say 1 and proposition say 2 we proof the classical Busineski system.

Proposition say 1 and proposition say 2 is sufficient proof local controllability for classical Busineski system. But this model Ladishevkaia-Busineski have no locality.

No locality is very difficult. We need more regularity. We have a proposition say 3. Proposition say 3 is more regularity for control the proposition say 1.

The assumption in proposition say 1 be satisfied we have this control and state have this regularity. Proposition say 3 is more regularity for proposition say 1 the linear system.

This regularity proposition say 1 and proposition say 3 is good for proof of theorem 2. The proof of theorem analogous theorem 6.2 needs more regularity.

And more regularity is this regularity looking for control and regularity for states. Good regularity for temperature and work with ways.

We have a technical result. We proof proposition 6.6 associate with the linear system.

We proof proposition 6.6 associate with the linear system. The proof of theorem 3. We take this property. This term is consequence.

This technical result in general maximum regularity is this two terms and this term appear the result technical of proposition say 1.

This proof the linear system proposition 6.1 and 6.3 need for the proof theorem 2. Proposition say 2, say 4 and say 6 needs the proof of theorem 3.

Well, proof of theorem 2. Proof of theorem 2. Local null controllability for the system 1 with null local term is normal driving in L2.

We define it as spice. Spice in. Why define it as spice in? Such that we given properties such that this term in here we can have regularity of the linear system.