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One shows that the linearized Navier-Stokes equation in
, around an unstable equilibrium
solution is exponentially stabilizable in probability by an
internal noise controller , , where are
independent Brownian motions in a probability space and
is a system of functions on with
support in an arbitrary open subset . The
stochastic control input is found in feedback
form. One constructs also a tangential boundary noise controller
which exponentially stabilizes in probability the equilibrium
solution.
...
The Cauchy problem for a stochastic partial differential equation with a spatial correlated Gaussian noise is considered. The "drift" is continuous, one-sided linearily bounded and of at most polynomial growth while the "diffusion" is globally Lipschitz continuous. In the paper statements on existence and uniqueness of solutions, their pathwise spatial growth and on their ultimate boundedness as well as on asymptotical exponential stability in mean square in a certain Hilbert space of weighted functions...
We apply an approximation by means of the method of lines for hyperbolic stochastic functional partial differential equations driven by one-dimensional Brownian motion. We study the stability with respect to small -perturbations.
We describe several results obtained recently on stochastic nonlinear Schrödinger equations. We show that under suitable smoothness assumptions on the noise, the nonlinear Schrödinger perturbed by an additive or multiplicative noise is well posed under similar assumptions on the nonlinear term as in the deterministic theory. Then, we restrict our attention to the case of a focusing nonlinearity with critical or supercritical exponent. If the noise is additive, smooth in space and non degenerate,...
We study ergodic properties of stochastic dissipative systems with additive noise. We show that the system is uniformly exponentially ergodic provided the growth of nonlinearity at infinity is faster than linear. The abstract result is applied to the stochastic reaction diffusion equation in with .
We derive and analyze adaptive solvers for boundary value problems in which the
differential operator depends affinely on a sequence of parameters. These methods converge
uniformly in the parameters and provide an upper bound for the maximal error. Numerical
computations indicate that they are more efficient than similar methods that control the
error in a mean square sense.
We derive and analyze adaptive solvers for boundary value problems in which the
differential operator depends affinely on a sequence of parameters. These methods converge
uniformly in the parameters and provide an upper bound for the maximal error. Numerical
computations indicate that they are more efficient than similar methods that control the
error in a mean square sense.
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