## Local exact controllability of the Navier-Stokes equations.(English)Zbl 0873.76020

Summary: Let $$\Omega\Subset\mathbb{R}^n$$ and $$\widehat u(t,x)$$ be a given solution of the equation $$\partial_tu(t,x)+ A(u)= f(t,x)$$, $$t\in(0,T)$$, $$x\in\Omega$$. The equation is called local exact controllable from boundary if for any initial condition $$u_0(x)$$ belonging to the $$\varepsilon$$-neighbourhood of the point $$\widehat u(0,\cdot)$$ ($$\varepsilon=\varepsilon(\widehat u))$$ there exists such boundary control $$\alpha$$ that a solution $$u$$ of the equation supplied with $$u|_{(0,T)\times\partial\Omega}=\alpha$$, $$u|_{t=0}= u_0$$ satisfies the condition $$u(T,x)\equiv\widehat u(T,x)$$. The local exact boundary controllability of the two-dimensional and three-dimensional Navier-Stokes as well as Boussinesq equations is established in this paper. For two-dimensional Navier-Stokes equations the same property is established also in the case when the control is defined on a boundary’s arbitrary subset. For two-dimensional Euler equations (respectively for Navier-Stokes equations), global exact (respectively approximate) controllability has been shown (with slip boundary conditions for Navier-Stokes equations) by J.-M. Coron [C. R. Acad. Sci., Paris, Ser. I 317, No. 3, 271-276 (1993; Zbl 0781.76013)].

### MSC:

 76D05 Navier-Stokes equations for incompressible viscous fluids 93C20 Control/observation systems governed by partial differential equations 35Q30 Navier-Stokes equations

### Keywords:

boundary control; Boussinesq equations

Zbl 0781.76013