Andrey Sarychev

Controllability of 2D Euler and Navier-Stokes Equations by Degenerate Forcing

AbstractWe study controllability issues for the 2D Euler and Navier-Stokes (NS) systems under periodic boundary conditions. These systems describe the motion of the homogeneous ideal or viscous incompressible fluid on a two-dimensional torus

On finite-dimensional projections of distributions for solutions of randomly forced 2D Navier-Stokes equations

\mathbb{T}^2

Lie- and chronologico-algebraic tools for studying stability of time-varying systems

. We assume the system to be controlled by a degenerate forcing applied to a fixed number of modes.In our previous work [3,5,4] we studied global controllability by means of degenerate forcing for Navier-Stokes (NS) systems with nonvanishing viscosity (ν > 0). Methods of differential geometric/Lie algebraic control theory have been used for that study. In [3] criteria for global controllability of finite-dimensional Galerkin approximations of 2D and 3D NS systems have been established. It is almost immediate to see that these criteria are also valid for the Galerkin approximations of the Euler systems. In [5,4] we established a much more intricate sufficient criteria for global controllability in a finite-dimensional observed component and for L2-approximate controllability for the 2D NS system. The justification of these criteria was based on a Lyapunov-Schmidt reduction to a finite-dimensional system. Possibility of such a reduction rested upon the dissipativity of the NS system, and hence the previous approach can not be adapted for the Euler system.In the present contribution we improve and extend the controllability results in several aspects : 1) we obtain a stronger sufficient condition for controllability of the 2D NS system in an observed component and for L2-approximate controllability; 2) we prove that these criteria are valid for the case of an ideal incompressible fluid (ν=0); 3) we study solid controllability in projection on any finite-dimensional subspace and establish a sufficient criterion for such controllability.

Non — Blow — Up Conditions for Riccati — type Matrix Differential and Difference Equations

Extremals Flows and Infinite Horizon Optimization.- Laplace Transforms and the American Call Option.- Time Change, Volatility, and Turbulence.- External Dynamical Equivalence of Analytic Control Systems.- On Option-Valuation in Illiquid Markets: Invariant Solutions to a Nonlinear Model.- Predicting the Time of the Ultimate Maximum for Brownian Motion with Drift.- A Stochastic Demand Model for Optimal Pricing of Non-Life Insurance Policies.- Optimality of Deterministic Policies for Certain Stochastic Control Problems with Multiple Criteria and Constraints.- Higher-Order Calculus of Variations on Time Scales.- Finding Invariants of Group Actions oon Function Spaces, a General Methodology from Non-Abelian Harmonic Analysis.- Nonholonomic Interpolation for Kinematic Problems, Entropy and Complexity.- Instalment Options: A Closed-Form Solution and the Limiting Case.- Existence and Lipschitzian Regularity for Relaxed Minimizers.- Pricing of Defaultable Securities under Stochastic Interest.- Spline Cubatures for Expectations of Diffusion Processes and Optimal Stopping in Higher Dimensions (with Computational Finance in View).- An Approximate Solution for Optimal Portfolio in Incomplete Markets.- Carleman Linearization of Linearly Observable Polynomial Systems.- Observability of Nonlinear Control Systems on Time Scales - Sufficient Conditions.- Sufficient Optimality Conditions for a Bang-bang Trajectory in a Bolza Problem.- Modelling Energy Markets with Extreme Spikes.- Generalized Bayesian Nonlinear Quickest Detection Problems: On Markov Family of Sufficient Statistics.- Necessary Optimality Condition for a Discrete Dead Oil Isotherm Optimal Control Problem.- Managing Operational Risk: Methodology and Prospects.

Solid Controllability in Fluid Dynamics

The paper is devoted to studying the image of probability measures on a Hilbert space under finite-dimensional analytic maps. We establish sufficient conditions under which the image of a measure has a density with respect to the Lebesgue measure and continuously depends on the map. The results obtained are applied to the 2D Navier–Stokes equations perturbed by various random forces of low dimension. AMS subject classifications: 35Q30, 60H15, 93C20

Geometric Control Theory and sub-Riemannian geometry

AbstractWe study the possible mechanisms of occurrence of the Lavrentiev phenomenon for the basic problem of the calculus of variations

Existence and Lipschitzian Regularity for Relaxed Minimizers

\mathcal{J}(x) = \int\limits_0^1 {\mathcal{L}(t,x(t),\dot x(t))dt \to \inf ,x(0) = x_0 ,x(1) = x1}