Finite-dimensional controllability for systems of Navier-Stokes type

We consider the problem of partial controllability for an evolution equation with a quadratic nonlinearity, in which one should provide, at a given time, a given projection of the solution onto some finite-dimensional subspace by using the action of external forces that belong to one and the same subspace at each time. On the basis of estimates for the solution of a subdifferential Cauchy problem for a system of Navier-Stokes type, we prove the controllability and the existence of a control with minimum norm. We consider applications to the Navier-Stokes equations of a viscous incompressible fluid and a heat convection model.     

Boundary controllability for conservative PDEs

Boundary observability and controllability problems for evolution equations governed by PDEs have been greatly studied in the past years. However, the problems were studied on a case-by-case basis, only for some particular types of boundary controls, and, moreover, several unnatural restrictions concerning lower-order terms were used.Our goal here is to give a general approach for boundary controllability problems, which is valid for all evolution PDEs of hyperbolic or ultrahyperbolic type, all boundary controls for which the corresponding homogeneous problem is well-posed, and all well-posedness spaces for the homogeneous problem. The first example of such equations is the class of hyperbolic equations, but valid examples are also equations such as the Schroedinger equation and various models for the plate equation.This work is essentially based on some apriori estimates of Carleman's type obtained by the author in a previous paper [29].This research was partially supported by the National Science Foundation under Grant NSF-DMS-8903747.     

Partial Boundary Regularity for the Navier-Stokes Equations

We prove two conditions of local Holder continuity for suitable weak solutions to the Navier-Stokes equations near a smooth curved part of the boundary of the domain. One of these conditions has the form of the Caffarelli-Kohn-Nirenberg condition for local boundedness of suitable weak solutions at interior points of the space-time cylinder. The corresponding results near a planar part of the boundary have been established earlier by Seregin. Bibliography: 21 titles. To Nina Nikolaevna Uraltseva on the occasion of her 70th birthday __________ Published in Zapiski Nauchnykh Seminarov POMI, Vol. 310, 2004, pp. 158–190.     

On the global null controllability of a Navier-Stokes system with Navier slip boundary conditions

In this paper, we deal with a two-dimensional Navier-Stokes system in a rectangle with Navier slip boundary conditions on the horizontal sides. We establish the global null controllability of the system by controlling the normal component and the vorticity of the velocity on the vertical sides. The linearized control system around zero is controllable but one does not know how to deduce global controllability results for the nonlinear system. Our proof uses the return method together with a local exact controllability result by Fursikov and Imanuvilov.     

Boundary controllability for the quasi-linear wave equations coupled in parallel

We study the boundary exact controllability for a system of two quasi-linear wave equations coupled in parallel with springs and viscous terms. We prove the locally exact controllability around superposition equilibria under some checkable geometrical conditions. We then establish the globally exact controllability in such a way that the state of the coupled quasi-linear system moves from a superposition equilibrium in one location to a superposition equilibrium in another location. Our results show that exact controllability is geometrical characters of a Riemannian metric, given by the coefficients and superposition equilibria of the system.     

Positive orthant scalar controllability of bilinear systems

For the bilinear control system $\dot x = \left( {A + uB} \right)x$ ,x ∈ ? ⁿ ,u ∈ ?, whereA is ann ×n essentially nonnegative matrix, andB is a diagonal matrix, the following controllability problem is investigated: can any two points with positive coordinates be joined by a trajectory of the system? Forn>2, the answer is negative in the generic case: hypersurfaces in ? ⁿ are constructed that are intersected by all the trajectories of the system in one direction.     

Boundary controllability of a viscoelastic beam

It is shown thai, a viscoelastic beam with long memory is exactly controllable by L₂ (0 ,T)-boundary controls in any time T > 0. Furthermore, a weak bang-hang principle is derived In the elastic case     

Boundary controllability of parabolic coupled equations

This paper is concerned with the boundary controllability of non-scalar linear parabolic systems. More precisely, two coupled one-dimensional parabolic equations are considered. We show that, in this framework, boundary controllability is not equivalent and is more complex than distributed controllability. In our main result, we provide necessary and sufficient conditions for the null controllability.     

Boundary value problem for multivalued differential equations and controllability

Control process of the type x = f(t, x, u), u?U(t, x), can be deparametrized by writing them in terms of multivalued differential equations of the form x?F(t, x) = {f(t, x, u): u?U(t, x)}. So, under suitable hypotheses, the controllability problem turns out to be equivalent to a two-point boundary value problem for a multivalued differential equation. In this paper an existence theorem is sought for the latter boundary value problem. The result is achieved by using the fixed point argument as a crucial tool.     

Boundary controllability of integrodifferential systems in Banach spaces

Sufficient conditions for boundary controllability of integrodifferential systems in Banach spaces are established. The results are obtained by using the strongly continuous semigroup theory and the Banach contraction principle. Examples are provided to illustrate the theory.     

Boundary controllability of non-linear stochastic differential inclusions

The article is concerned with the boundary controllability of non-linear stochastic differential inclusions in a Banach space. Sufficient conditions for the boundary controllability are obtained by using a fixed point theorem for condensing maps due to Leray–Schauder non-linear alternative.     

Boundary controllability for the semilinear Schrödinger equations on Riemannian manifolds

We study the boundary exact controllability for the semilinear Schrödinger equation defined on an open, bounded, connected set Ω of a complete, n-dimensional, Riemannian manifold M with metric g. We prove the locally exact controllability around the equilibria under some checkable geometrical conditions. Our results show that exact controllability is geometrical characters of a Riemannian metric, given by the coefficients and equilibria of the semilinear Schrödinger equation. We then establish the globally exact controllability in such a way that the state of the semilinear Schrödinger equation moves from an equilibrium in one location to an equilibrium in another location.     

Boundary controllability for the time-fractional nonlinear Korteweg-de Vries (KdV) equation

In this paper, we study the time-fractional nonlinear Korteweg-de Vries (KdV) equation. By using the theory of semigroups, we prove the well-posedness of the time-fractional nonlinear KdV equation. Moreover, we present the boundary controllability result for the problem.     

Admissibility and controllability of diagonal Volterra equations with scalar inputs

This article studies Volterra evolution equations from the point of view of control theory, in the case that the generator of the underlying semigroup has a Riesz basis of eigenvectors. Conditions for admissibility of the system's control operator are given in terms of the Carleson embedding properties of certain discrete measures. Moreover, exact and null controllability are expressed in terms of a new interpolation question for analytic functions, providing a generalization of results known to hold for the standard Cauchy problem. The results are illustrated by examples involving heat conduction with memory.     

On the Stokes and Navier-Stokes System for Domains with Noncompact Boundary in Lq-spaces

We consider the L^q-theory of weak solutions of the Stokes and Navier-Stokes equations in two classes of unbounded domains with noncompact boundary, namely in perturbed half spaces which are obtained by a perturbation of the half space IRⁿ, and in aperture domains consisting of two disjoint half spaces separated by a wall but connected by a hole (aperture) through this wall. The proofs rest on the cut-off procedure and a new multiplier approach to the half space problem. In an aperture domain we additionally prescribe either the flux through the wall or the pressure drop at infinity to single out a unique solution. The nonlinear problem is solved for sufficiently small data and requires q =n/2, n ≥ 3, to estimate the nonlinearity.     

Exact controllability to trajectories for entropy solutions to scalar conservation laws in several space dimensions

We describe a new method that allows us to obtain a result of exact controllability to trajectories of multidimensional conservation laws in the context of entropy solutions and under a mere non-degeneracy assumption on the flux and a natural geometric condition.     

Boundary layers for compressible Navier-Stokes equations with outflow boundary condition

In this paper, we study the asymptotic relation between the solutions to the initial boundary value problem of the one-dimensional compressible full Navier-Stokes equations with outflow boundary condition and the associated Euler equations. We assume all the three characteristics to the corresponding Euler equations are all negative up to some small time, then we prove the existence and the stability of the boundary layers as long as the strength of the boundary layers is suitably small.     

An Exact Navier-Stokes Solution for Three-Dimensional,Spanwise-Homogeneous Boundary Layers

The plane stagnation flow onto (Hiemenz boundary layer, HBL) and the asymptotic suction boundary layer flow over a flat wall (ASBL) are two boundary layer flows for which the incompressible Navier-Stokes equations are amenable to exact similarity solutions. The Hiemenz solution has been extended to swept Hiemenz flows by superposition of a third, spanwise-homogeneous sweep velocity. This solution becomes singular as the chordwise, tangential base flow component vanishes. In this limit, the homogeneous ASBL solution is valid, which however cannot describe the swept Hiemenz flow, because it does not contain any chordwise velocity. This work presents a generalized three-dimensional similarity solution which describes three-dimensional spanwise homogeneously impinging boundary layers at arbitrary wall-normal suction velocities, using a rescaled similarity coordinate. The HBL and the ASBL are shown to be two limits of this solution. Further extensions consist of oblique impingement or different boundary suction directions, such as slip or stretching walls. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)