## Exact controllability of the Navier-Stokes and Boussinesq equations.(English. Russian original)Zbl 0970.35116

Russ. Math. Surv. 54, No. 3, 565-618 (1999); translation from Usp. Mat. Nauk 54, No. 3, 93-146 (1999).
The paper deals with the exact, local exact and approximate controllabilities of the Navier-Stokes and Boussinesq equations, which describe the flow of a viscous incompressible fluid without and with heat processes taken into account, respectively. The main object of this paper is the controllability problem for the Boussinesq system on the cylinder $$Qw=(0,T)^Aw$$, where $$(0,T)$$ is the time interval and $$w$$-is an arbitrary subdomain of 2- or 3-dimensional torus. Results for the Navier-Stokes equation are obtained as simple corollaries of the corresponding results for the Boussinesq system. However, for simplicity, the authors widely discuss the results for the Navier-Stokes equation. A brief survey of existing results for the considered problem and the theory of Carleman estimates are given. The authors prove the exact controllability for arbitrary small $$T$$ and any open $$w$$ containing the support of distributed or boundary controls. Approximate controllability holds in the interval $$(0,T1)$$, where $$T1=T1(e)^A0$$ as $$e^A0$$. The Navier-Stokes and Boussinesq equations are irreversible in time and problems of controllability for these equations will not be soluble in general. The exact controllability theorems imply that if it is possible to stabilize unstable steady-state solutions and that there exist chaotic solutions of the considered system .

### MSC:

 35Q35 PDEs in connection with fluid mechanics 35B37 PDE in connection with control problems (MSC2000) 93B05 Controllability 93C20 Control/observation systems governed by partial differential equations 76D05 Navier-Stokes equations for incompressible viscous fluids
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