### Necessary Conditions for Multiple Integral Problem in the Calculus of Variations.

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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.

The liner parabolic equation $\frac{y}{t}-\frac{1}{2}\phantom{\rule{0.166667em}{0ex}}\mathbb{D}y+F\xb7y={\overrightarrow{1}}_{{}_{0}}u$ ∂y ∂t − 1 2 Δy + F · ∇ y = 1 x1d4aa; 0 u with Neumann boundary condition on a convex open domain x1d4aa; ⊂ ℝ with smooth boundary is exactly null controllable on each finite interval if 𝒪is an open subset of x1d4aa; which contains a suitable neighbourhood of the recession cone of $$ x1d4aa; . Here, : ℝ → ℝ is a bounded, -continuous function, and = ∇, where is convex and coercive.

One shows that the linearized Navier-Stokes equation in $\mathcal{O}\subset {R}^{d},\phantom{\rule{0.277778em}{0ex}}d\ge 2$, around an unstable equilibrium solution is exponentially stabilizable in probability by an internal noise controller $V(t,\xi )=\sum _{i=1}^{N}{V}_{i}\left(t\right){\psi}_{i}\left(\xi \right){\dot{\beta}}_{i}\left(t\right)$, $\xi \in \mathcal{O}$, where ${\left\{{\beta}_{i}\right\}}_{i=1}^{N}$ are independent Brownian motions in a probability space and ${\left\{{\psi}_{i}\right\}}_{i=1}^{N}$ 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 ${\left\{{V}_{i}\right\}}_{i=1}^{N}$ is found in feedback form. One constructs also a tangential boundary noise controller which exponentially stabilizes in probability the equilibrium...

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.

One shows that the linearized Navier-Stokes equation in $\mathcal{O}\subset {R}^{d},\phantom{\rule{0.277778em}{0ex}}d\ge 2$, around an unstable equilibrium solution is exponentially stabilizable in probability by an internal noise controller $V(t,\xi )=\sum _{i=1}^{N}{V}_{i}\left(t\right){\psi}_{i}\left(\xi \right){\dot{\beta}}_{i}\left(t\right)$, $\xi \in \mathcal{O}$, where ${\left\{{\beta}_{i}\right\}}_{i=1}^{N}$ are independent Brownian motions in a probability space and ${\left\{{\psi}_{i}\right\}}_{i=1}^{N}$ 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 ${\left\{{V}_{i}\right\}}_{i=1}^{N}$ 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 control problem associated with the linearized equation.

We prove two existence results on abstract differential equations of the type $d\left(Bu\right)/dt+A\left(u\right)=f$ and we give some applications of them to partial differential equations.

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 .

This work is concerned with the existence and regularity of solutions to the Neumann problem associated with a Ornstein–Uhlenbeck operator on a bounded and smooth convex set of a Hilbert space . This problem is related to the reflection problem associated with a stochastic differential equation in .

Some recent results about nonnegative solutions of stochastic porous media equations in bounded open subsets of ${\mathbb{R}}^{3}$ are considered. The existence of an invariant measure is proved.

We prove the essential m-dissipativity of the Kolmogorov operator associated with the stochastic Navier-Stokes flow with periodic boundary conditions in a space ${L}^{2}\left(H,\nu \right)$ where $\nu $ is an invariant measure

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