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In this paper, we consider the well-known Fattorini’s criterion for approximate controllability of infinite dimensional linear systems of type y′ = Ay + Bu. We precise the result proved by Fattorini in [H.O. Fattorini, SIAM J. Control 4 (1966) 686–694.] for bounded input B, in the case where B can be unbounded or in the case of finite-dimensional controls. More precisely, we prove that if Fattorini’s criterion is satisfied and if the set of geometric multiplicities of A is bounded then approximate controllability can be achieved with finite dimensional controls. An important consequence of this result consists in using the Fattorini’s criterion to obtain the feedback stabilizability of linear and nonlinear parabolic systems with feedback controls in a finite dimensional space. In particular, for systems described by partial differential equations, such a criterion reduces to a unique continuation theorem for a stationary system. We illustrate such a method by tackling some coupled Navier−Stokes type equations (MHD system and micropolar fluid system) and we sketch a systematic procedure relying on Fattorini’s criterion for checking stabilizability of such nonlinear systems. In that case, the unique continuation theorems rely on local Carleman inequalities for stationary Stokes type systems.
We study the motion of a rigid body with a cavity filled with a viscous liquid. The main objective is to investigate the well-posedness of the coupled system formed by the Navier–Stokes equations describing the motion of the fluid and the ordinary differential equations for the motion of the rigid part. To this end, appropriate function spaces and operators are introduced and analysed by considering a completely general three-dimensional bounded domain. We prove the existence of weak solutions using the Galerkin method. In particular, we show that if the initial velocity is orthogonal, in a certain sense, to all rigid velocities, then the velocity of the system decays exponentially to zero as time goes to infinity. Then, following a functional analytic approach inspired by Kato's scheme, we prove the existence and uniqueness of mild solutions. Finally, the functional analytic approach is extended to obtain the existence and uniqueness of strong solutions for regular data.
In this paper we study a controllability problem for a simplified one dimensional model for the motion of a rigid body in a viscous fluid. The control variable is the velocity of the fluid at one end. One of the novelties brought in with respect to the existing literature consists in the fact that we use a single scalar control. Moreover, we introduce a new methodology, which can be used for other nonlinear parabolic systems, independently of the techniques previously used for the linearized problem. This methodology is based on an abstract argument for the null controllability of parabolic equations in the presence of source terms and it avoids tackling linearized problems with time dependent coefficients.
The ferroelectric switching curve of vinylidene fluoride / trifluoroethylene copolymer exhibits a characteristic time evolution consisting of two processes; an initial gradual increase in proportion to t0 5 followed by a rapid increase according to an exponential function with particularly large exponent 6. Such a switching curve was analyzed by means of computer simulation based on a modified nucleation-growth mechanism. It was found that the initial gradual increase is attributed to generation of considerably large nuclei that grow according to a random-walk scheme. Once such nuclei gain a critical size, they start to grow automatically either one or two dimensionally. The time required to generate critical nuclei serves as an incubation time for the later growth process to result in the large exponent.
We propose a numerical scheme to compute the motion of a two-dimensional rigid body in a viscous fluid. Our method combines the method of characteristics with a finite element approximation to solve an ALE formulation of the problem. We derive error estimates implying the convergence of the scheme.
A wide variety of organic compounds have been found in carbonaceous chondrites and comets, which suggests that extraterrestrial organic compounds could have been an important source of the first terrestrial biosphere. In the Greenberg model, these organic compounds in the small bodies were originally formed in interstellar dusts (ISD) in dense clouds by the action of cosmic rays and ultraviolet light. We irradiated a frozen mixture of methanol, ammonia and water with high-energy heavy ions from an accelerator (“HIMAC” in NIRS, Japan) to simulate the action of cosmic rays in dense clouds. Racemic mixtures of amino acids were detected after hydrolysis of the irradiation products. A mixture of carbon monoxide, ammonia and water also gave such complex amino acid precursors with large molecular weights. When such amino acid precursors were irradiated with circular polarized UV light from a synchrotron, enantiomeric excesses were detected. The yield of amino acids was not largely changed between, before, and after CPL-irradiation. The present results suggest that the seed of homochirality of terrestrial amino acids were originally formed in interstellar space.
The topological sensitivity analysis consists in studying the behavior of a given shape functional when the topology of the domain is perturbed, typically by the nucleation of a small hole. This notion forms the basic ingredient of different topology optimization/reconstruction algorithms. From the theoretical viewpoint, the expression of the topological sensitivity is well-established in many situations where the governing p.d.e. system is of elliptic type. This paper focuses on the derivation of such formulas for parabolic and hyperbolic problems. Different kinds of cost functionals are considered.
We consider the approximation of a class of
exponentially stable infinite dimensional linear systems modelling
the damped vibrations of one dimensional vibrating systems or of
square plates. It is by now well known that the approximating
systems obtained by usual finite element or finite difference are
not, in general, uniformly stable with respect to the discretization
parameter. Our main result shows that, by adding a suitable
numerical viscosity term in the numerical scheme, our approximations
are uniformly exponentially stable. This result is then applied to
obtain strongly convergent approximations of the solutions of the
algebraic Riccati equations associated to an LQR optimal control
problem. We next give an application to a non-homogeneous string
equation. Finally we apply similar techniques for approximating the
equations of a damped square plate.
In this paper we investigate the motion of a rigid ball in an
incompressible perfect fluid occupying
We prove the global in time existence and the uniqueness of
the classical solution for this fluid-structure problem. The proof relies
mainly on weighted estimates for the vorticity associated with
the strong solution of a fluid-structure problem
obtained by incorporating some dissipation.
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