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

Mathematische Annalen

### Note on the internal stabilization of stochastic parabolic equations with linearly multiplicative gaussian noise

ESAIM: Control, Optimisation and Calculus of Variations

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.

### Exact null internal controllability for the heat equation on unbounded convex domains

ESAIM: Control, Optimisation and Calculus of Variations

The liner parabolic equation $\frac{y}{t}-\frac{1}{2}\phantom{\rule{0.166667em}{0ex}}𝔻y+F·y={\stackrel{\to }{1}}_{{}_{0}}u$ ∂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 &#x1d4aa;is 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.

### The internal stabilization by noise of the linearized Navier-Stokes equation

ESAIM: Control, Optimisation and Calculus of Variations

One shows that the linearized Navier-Stokes equation in $𝒪\subset {R}^{d},\phantom{\rule{0.277778em}{0ex}}d\ge 2$, around an unstable equilibrium solution is exponentially stabilizable in probability by an internal noise controller $V\left(t,\xi \right)=\sum _{i=1}^{N}{V}_{i}\left(t\right){\psi }_{i}\left(\xi \right){\stackrel{˙}{\beta }}_{i}\left(t\right)$, $\xi \in 𝒪$, 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 $𝒪$ with support in an arbitrary open subset ${𝒪}_{0}\subset 𝒪$. 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...

### Feedback stabilization of Navier–Stokes equations

ESAIM: Control, Optimisation and Calculus of Variations

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.

### Differentiable distribution semi-groups

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

### Dissipative sets and nonlinear perturbated equations in Banach spaces

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

### The internal stabilization by noise of the linearized Navier-Stokes equation

ESAIM: Control, Optimisation and Calculus of Variations

One shows that the linearized Navier-Stokes equation in $𝒪\subset {R}^{d},\phantom{\rule{0.277778em}{0ex}}d\ge 2$, around an unstable equilibrium solution is exponentially stabilizable in probability by an internal noise controller $V\left(t,\xi \right)=\sum _{i=1}^{N}{V}_{i}\left(t\right){\psi }_{i}\left(\xi \right){\stackrel{˙}{\beta }}_{i}\left(t\right)$, $\xi \in 𝒪$, 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 $𝒪$ with support in an arbitrary open subset ${𝒪}_{0}\subset 𝒪$. 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. ...

### Feedback stabilization of Navier–Stokes equations

ESAIM: Control, Optimisation and Calculus of Variations

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.

### Null controllability of nonlinear convective heat equations

ESAIM: Control, Optimisation and Calculus of Variations

### Existence for implicit differential equations in Banach spaces

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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.

### An operatorial approach to stochastic partial differential equations driven by linear multiplicative noise

Journal of the European Mathematical Society

### Null controllability of nonlinear convective heat equations

ESAIM: Control, Optimisation and Calculus of Variations

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 .

### Existence and uniqueness of solutions to wave equations with nonlinear degenerate damping and source terms

Control and Cybernetics

### Some results for the reflection problems in Hilbert spaces

Control and Cybernetics

### Kolmogorov equation associated to the stochastic reflection problem on a smooth convex set of a Hilbert space II

Annales de l'I.H.P. Probabilités et statistiques

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 Results on Stochastic Porous Media Equations

Bollettino dell'Unione Matematica Italiana

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.

### Essential m-dissipativity of Kolmogorov operators corresponding to periodic $2D$-Navier Stokes equations

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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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