The liner parabolic equation \hbox{$\frac{\pp y}{\pp t}-\frac12\,\D y+F\cdot\na y={\vec{1}}_{\calo_0}u$} $\frac{\mathit{\partial y}}{\mathit{\partial t}}\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}\hspace{0.17em}\mathit{\Delta y}\mathrm{+}\mathit{F}\mathrm{·}\mathrm{\nabla }\mathit{y}\mathrm{=}{{1}}_{{\mathrm{𝒪}}_{\mathrm{0}}}\mathit{u}$ with Neumann boundary condition on a convex open domain 𝒪 ⊂ ℝd with smooth boundary is exactly null controllable on each finite interval if 𝒪0 is an open subset of 𝒪 which contains a suitable neighbourhood of the recession cone of \hbox{$\ov\calo$} $\overline{\mathrm{𝒪}}$ . Here, F : ℝd → ℝd is a bounded, C1-continuous function, and F = ∇g, where g is convex and coercive.

The parabolic equations driven by linearly multiplicative Gaussian noise are stabilizable in probability by linear feedback controllers with support in a suitably chosen open subset of the domain. This procedure extends to Navier − Stokes equations with multiplicative noise. The exact controllability is also discussed.

One shows that the linearized Navier-Stokes equation in ${\mathcal{O}}{\subset} R^d,\;d \ge 2$ , around an unstable equilibrium solution is exponentially stabilizable in probability by an internal noise controller $V(t,\xi)=\displaystyle\sum\limits_{i=1}^{N} V_i(t)\psi_i(\xi) \dot\beta_i(t)$ , $\xi\in{\mathcal{O}}$ , where $\{\beta_i\}^N_{i=1}$ are independent Brownian motions in a probability space and $\{\psi_i\}^N_{i=1}$ is a system of functions on ${\mathcal{O}}$ with support in an arbitrary open subset ${\mathcal{O}}_0\subset {\mathcal{O}}$ . The stochastic control input $\{V_i\}^N_{i=1}$ is found in feedback form. One constructs also a tangential boundary noise controller which exponentially stabilizes in probability the equilibrium solution.

One proves that the steady-state solutions to Navier–Stokes equations with internal controllers are locally exponentially stabilizable by linear feedback controllers provided by a LQ control problem associated with the linearized equation.

The internal and boundary exact null controllability of nonlinear convective heat equations with homogeneous Dirichlet boundary conditions are studied. The methods we use combine Kakutani fixed point theorem, Carleman estimates for the backward adjoint linearized system, interpolation inequalities and some estimates in the theory of parabolic boundary value problems in Lk .