Local Exact Controllability of Micropolar Fluids

This paper is devoted to prove the local exact controllability to the trajectories of micropolar fluids with distributed controls supported in small sets. First, we deduce new Carleman inequalities for associated linearized systems which leads to null controllability at any time T > 0. Then, we deduce a local result concerning the exact controllability to the trajectories of the whole nonlinear system. The arguments are presented separately in the two-dimensional and three-dimensional cases, since the required techniques are different.     

Null‐Controllability of a System of Linear Thermoelasticity

We consider a linear system of thermoelasticity in a compact, C ^infin, n-dimensional connected Riemannian manifold. This system consists of a wave equation coupled to a heat equation. When the boundary of the manifold is non‐empty, Dirichlet boundary conditions are considered. We study the controllability properties of this system when the control acts in the hyperbolic equation (and not in the parabolic one) and has its support restricted to an open subset of the manifold. We show that, if the control time and the support of the control satisfy the geometric control condition for the wave equation, this system of thermoelasticity is null-controllable. More precisely, any finite‐energy solution can be driven to zero at the control time. An analogous result is proved when the control acts on the parabolic equation. Finally, when the manifold has no boundary, the null‐controllability of the linear system of three‐dimensional thermoelastic ity is proved. (Accepted June 13, 1996)     

On the Nonexistence of Some Special Eigenfunctions for the Dirichlet Laplacian and the Lamé System

Let Ω be a bounded Lipschitz domain in ℝ ⁿ with n ≥ 3. We prove that the Dirichlet Laplacian does not admit any eigenfunction of the form u(x) =ϕ(x′)+ψ(x _n) with x′=(x1, ..., x _n−1). The result is sharp since there are 2-d polygonal domains in which this kind of eigenfunctions does exist. These special eigenfunctions for the Dirichlet Laplacian are related to the existence of uniaxial eigenvibrations for the Lamé system with Dirichlet boundary conditions. Thus, as a corollary of this result, we deduce that there is no bounded Lipschitz domain in 3-d for which the Lamé system with Dirichlet boundary conditions admits uniaxial eigenvibrations. This revised version was published online in August 2006 with corrections to the Cover Date.     

Conditions for pressure build-up due to buoyancy effects on forced convection in vertical eccentric annuli under thermal boundary condition of first kind

This paper presents a numerical investigation for the conditions at which the buoyancy effects (represented by the buoyancy parameter (Gr/Re)) result in pressure build-up due to mixed convection in vertical eccentric annuli under thermal boundary conditions of first kind. In this regard, the critical values of buoyancy parameter (Gr/Re)_crt at which the pressure gradient vanishes and starts to become positive leading to the pressure build-up are obtained numerically for radius ratio N=0.5 and eccentricity E=0.1–0.7. Results of practical applications such as the locations at which the negative pressure gradient becomes zero changing its sign to be positive, the locations of zero pressure defect and the fully developed length under different operating conditions are drawn and presented. For sufficiently large values of Gr/Re≫(Gr/Re)_crt, possibilities and locations of flow reversal incipient are determined. Information of technical relevance is presented.     

Analysis of regular and chaotic dynamics of the Euler-Bernoulli beams using finite difference and finite element methods

Chaotic vibrations of flexible non-linear Euler-Bernoulli beams subjected to harmonic load and with various boundary conditions(symmetric and non-symmetric)are studied in this work.Reliability of the obtained results is verified by the finite difference method(FDM)and the finite element method(FEM)with the Bubnov-Galerkin approximation for various boundary conditions and various dynamic regimes(regular and non-regular).The influence of boundary conditions on the Euler-Bernoulli beams dynamics is studied mainly,dynamic behavior vs.control parameters { ωp,q0 } is reported,and scenarios of the system transition into chaos are illustrated.     

Two notes on nonlocal field theory

In this paper, two fundamental problems completely unsolved in nonlocal field theory are studied. The first is the dependence of nonlocal residuals. By studying this problem, a theorem concerning the relationship between the residuals of nonlocal body force and nonlocal moment of momentum is given and proven. The other problem is how to give the stress boundary conditions in the linear theory of nonlocal elasticity. The stress boundary conditions obtained in this paper can not only answer why the nonlocal stress solution satisfying the boundary conditionst _ji ^(s) n _j ¦O ₂ =p _i (p _i is a constant) on the surface of crack does not exist but also give a model of the molecular cohesive stress on the crack tip.     

The Linearized Crocco Equation

In this paper, we study the existence and uniqueness of a degenerate parabolic equation, with nonhomogeneous boundary conditions, coming from the linearization of the Crocco equation [12]. The Crocco equation is a nonlinear degenerate parabolic equation obtained from the Prandtl equations with the so-called Crocco transformation. The linearized Crocco equation plays a major role in stabilization problems of fluid flows described by the Prandtl equations [5]. To study the infinitesimal generator associated with the adjoint linearized Crocco equation – with homogeneous boundary conditions – we first study degenerate parabolic equations in which the x-variable plays the role of a time variable. This equation is doubly degenerate: the coefficient in front of ∂_x vanishes on a part of the boundary, and the coefficient of the elliptic operator vanishes in another part of the boundary. This makes very delicate the proof of uniqueness of solution. To overcome this difficulty, a uniqueness result is first obtained for an equation in which the elliptic operator is symmetric, and it is next extended to the original equation by combining an iterative process and a fixed point argument (see Th. 4.9). This kind of argument is also used to prove estimates, which cannot be obtained in a classical way.     

Nulle contrôlabilité régionale pour des équations de la chaleur dégénéréesRegional null controllability for degenerate heat equations

We are interested in a null controllability problem for a class of strongly degenerate heat equations.First for all T>0, we prove a regional null controllability result at time T at least in the region where the equation is not degenerate. The proof is based on an adequate observability inequality for the homogeneous adjoint problem. This inequality is obtained by application of Carleman estimates combined with the introduction of cut-off functions.Then we improve this result: for all T′>T, we obtain a result of persistent regional null controllability during the time interval [T,T′]. Finally we give similar results for the (non degenerate) heat equation in unbounded domain. To cite this article: P. Cannarsa et al., C. R. Mecanique 330 (2002) 397–401.     

Navier–Stokes Equations: Controllability by Means of Low Modes Forcing

We study controllability issues for 2D and 3D Navier–Stokes (NS) systems with periodic boundary conditions. The systems are controlled by a degenerate (applied to few low modes) forcing. Methods of differential geometric/Lie algebraic control theory are used to establish global controllability of finite-dimensional Galerkin approximations of 2D and 3D NS and Euler systems, global controllability in finite-dimensional projection of 2D NS system and L2-approximate controllability for 2D NS system. Beyond these main goals we obtain results on boundedness and continuous dependence of trajectories of 2D NS system on degenerate forcing, when the space of forcings is endowed with so called relaxation metric.     

Removing Collision Singularities from Action Minimizers for the N-Body Problem with Free Boundaries

For the planar and spatial N-body problems, it has been proved by Marchal and Chenciner that solutions for the minimizing problem with fixed ends are free from interior collisions. This important result has been extended by Ferrario & Terracini to Newtonian-type problems and equivariant problems. It has also been used to construct many symmetric solutions for the N-body problem. In this paper we are interested in action minimizing solutions in function spaces with free boundaries. The function spaces are imposed with boundary conditions, such that every mass point starts and ends on two transversal proper subspaces of ℝ^d, d≥2. We will prove that solutions for this minimizing problem with free boundaries are always free from collisions, including boundary collisions. This result can be used to construct certain classes of relative periodic solutions of the N-body problem.     

Navier–Stokes Equations with Shear-Thickening Viscosity. Regularity up to the Boundary

In this article we prove some sharp regularity results for the stationary and the evolution Navier–Stokes equations with shear dependent viscosity, see (1.1), under the no-slip boundary condition(1.4). We are interested in regularity results for the second order derivatives of the velocity and for the first order derivatives of the pressure up to the boundary, in dimension n ≥ 3. In reference [4] we consider the stationary problem in the half space \mathbbR₊ⁿ{\mathbb{R}}_+^n under slip and no-slip boundary conditions. Here, by working in a simpler context, we concentrate on the basic ideas of proofs. We consider a cubic domain and impose our boundary condition (1.4) only on two opposite faces. On the other faces we assume periodicity, as a device to avoid unessential technical difficulties. This choice is made so that we work in a bounded domain Ω and, at the same time, with a flat boundary. In the last section we provide the extension of the results from the stationary to the evolution problem.     

Energy decay for the neutrino equation in the exterior of a torus

Local energy decay is established for the solutions of the neutrino equation in the exterior G of a torus for a class of boundary conditions, described as follows: To each energy conserving boundary condition at a point x on G there corresponds a vector in the tangent plane to G at x. The result has been proved when the torus and boundary conditions are axially symmetric and when the paths generated by this vector field are closed. What is novel about this problem is the fact that the boundary conditions are nowhere coercive.This research was sponsored in part by the National Science Foundation under Grants GP 8857 and GP-17526, and by the United State Air Force under contract F 44620-68-C-0054.     

Carleman Estimate for Elliptic Operators with Coefficients with Jumps at an Interface in Arbitrary Dimension and Application to the Null Controllability of Linear Parabolic Equations

In a bounded domain of R ⁿ⁺¹, n ≧ 2, we consider a second-order elliptic operator, ${A=-{\partial_{x_0}^2} - \nabla_x \cdot (c(x) \nabla_x)}In a bounded domain of R ⁿ⁺¹, n ≧ 2, we consider a second-order elliptic operator, A=-?_x0² - ?_x ·(c(x) ?_x){A=-{\partial_{x_0}^2} - \nabla_x \cdot (c(x) \nabla_x)}, where the (scalar) coefficient c(x) is piecewise smooth yet discontinuous across a smooth interface S. We prove a local Carleman estimate for A in the neighborhood of any point of the interface. The “observation” region can be chosen independently of the sign of the jump of the coefficient c at the considered point. The derivation of this estimate relies on the separation of the problem into three microlocal regions and the Calderón projector technique. Following the method of Lebeau and Robbiano (Comm Partial Differ Equ 20:335–356, 1995) we then prove the null controllability for the linear parabolic initial problem with Dirichlet boundary conditions associated with the operator ?_t - ?_x ·(c(x) ?_x){{\partial_t - \nabla_x \cdot (c(x) \nabla_x)}} .     

Stabilization using fractional-order PI and PID controllers

This paper presents a solution to the problem of stabilizing a given fractional dynamic system using fractional-order PIλ and PI^λD^μ controllers. It is based on plotting the global stability region in the (k _p, k _i)-plane for the PI^λ controller and in the (k _p , k _i , k _d)-space for the PIλDμ controller. Analytical expressions are derived for the purpose of describing the stability domain boundaries which are described by real root boundary, infinite root boundary and complex root boundary. Thus, the complete set of stabilizing parameters of the fractional-order controller is obtained. The algorithm has a simple and reliable result which is illustrated by several examples, and hence is practically useful in the analysis and design of fractional-order control systems.     

On singularly perturbed quasilinear systems

In this paper, the objective is to give sufficient conditions for the existence of solution of the nonlinear two-point boundary value problem(1.1). And we employ these results to consider the boundary layer phenomena of the quasilinear weakly coupled singularly perturbed system (DP)_q.     

Effect of inelastic collisions on the first-order slip coefficients for a diatomic gas with rotational degrees of freedom

When solving problems of inhomogeneous gas dynamics in the slip regime, it is necessary to know the boundary conditions for the velocity, temperature, heat fluxes, etc., that is, the boundary conditions for the gas macroparameters. In particular, such problems arise in developing the theory of thermophoresis of moderately large aerosol particles [1].The problem of monatomic and molecular (di- and polyatomic) gas slip along a boundary surface is considered in many publications (see, for example, [2–8]). The first-order effects include the isothermal and thermal gas slips characterized by the coefficients C_m and K_TS, respectively.In contrast to a monatomic gas, the molecules of diatomic and polyatomic gases have internal degrees of freedom, which considerably complicates the kinetic equation [9]. The lack of reliable models for the intermolecular interaction potential predetermines the need to construct model kinetic equations [10].In this study, for a diatomic gas whose molecules have rotational degrees of freedom, we propose a model kinetic equation obtained by developing the approach described in [6]. With the use of this model equation, the problem of diatomic gas slip along a plane surface is solved. As a result, for diatomic gases the coefficients C_m and K_TS, which depend on the thermophysical gas parameters and the intensity of inelastic collisions, are obtained.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, 2004, pp. 176–182. Original Russian Text Copyright © 2004 by Poddoskin.     

Transient growth in the asymptotic suction boundary layer

In the present experimental setup, the transient disturbance growth in a spatially invariant boundary layer flow, i.e., the asymptotic suction boundary layer (ASBL), has been investigated. The choice of the ASBL brings along several advantages compared with an ordinary spatially growing boundary layer. A unique feature of the ASBL is that the Reynolds number (Re) can be varied without changing the boundary layer thickness, which in turn allows for parameter variations not possible to carry out in traditional boundary layer flows. A spanwise array of discrete surface roughness elements was mounted on the surface to trigger modes with different spanwise wavenumbers (β). It is concluded that for each mode there exists a threshold roughness Reynolds number (Re _k ), below which no significant transient growth is present. The experimental data suggests that this threshold Re _k is both a function of β and Re. An interesting result is that the energy growth curves respond differently to a change in Re _k when caused by a change in roughness height k, implying that Re remains constant, compared with a change in the free-stream velocity U_￥U_\infty, which also affects the Re. The scaling of the energy growth curves both in level and the downstream direction is treated and appropriate scalings are found. The result shows a complex non-linear receptivity mechanism. Optimal perturbation theory, which has failed to predict the energy evolution in growing boundary layers, is tested for the ASBL and shows that it may satisfactorily predict the evolution of all transiently growing modes that are triggered by the roughness elements.     

Navier–Stokes Equations with Shear Thinning Viscosity. Regularity up to the Boundary

In this paper we consider a class of stationary Navier–Stokes equations with shear dependent viscosity, in the shear thinning case p < 2, under a non-slip boundary condition. We are interested in global (i.e., up to the boundary) regularity results, in dimension n = 3, for the second order derivatives of the velocity and the first order derivatives of the pressure. As far as we know, there are no previous global regularity results for the second order derivatives of the solution to the above boundary value problem. We consider a cubic domain and impose the non-slip boundary condition only on two opposite faces. On the other faces we assume periodicity, as a device to avoid effective boundary conditions. This choice is made so that we work in a bounded domain Ω and simultaneously with a flat boundary. The extension to non-flat boundaries is done in the forthcoming paper [7], by following ideas introduced by the author, for the case p > 2, in reference [5]. The results also hold in the presence of the classical convective term, provided that p is sufficiently close to the value 2.      

Measure-valued solutions of scalar conservation laws with boundary conditions

We define a solution concept for measure-valued solutions to scalar conservation laws with initial conditions and boundary conditions and prove a uniqueness theorem for such solutions. This result may be used to prove convergence, towards the unique solution, for approximate solutions which are uniformly bounded in L , weakly consistent with certain entropy inequalities and strongly consistent with the initial condition, i.e. without using derivative estimates. As an example convergence of a finite element method is demonstrated.     

Scattering of guided waves by through-thickness cavities with irregular shapes

This paper investigates the three-dimensional (3D) scattering of guided waves by a through-thickness cavity with irregular shape in an isotropic plate. The scattered field is decomposed on the basis of Lamb and SH waves (propagating and non-propagating), and the amplitude of the modes is calculated by writing the nullity of the total stress at the boundary of the cavity. In the boundary conditions, the functions depend on the through-thickness coordinate, z, but contrary to the case where the cavity has a circular shape, they also depend on the angular coordinate θ. This is dealt with by projecting the z-dependent functions onto a basis of orthogonal functions, and by expanding the θ-dependent functions in Fourier series. Examples include the scattering of the S₀, SH₀ and A₀ modes by elliptical cavities with different values of aspect ratio, and the scattering of the S₀ mode by a cavity with an arbitrary shape. Results obtained with this model are compared with results obtained with the finite element (FE) method, showing very good agreement.