Insensitizing controls with one vanishing component for the Navier–Stokes system

In this paper we prove the existence of insensitizing controls, having one vanishing component, for the local L²$L^2$-norm of the solutions of the Navier–Stokes system. This problem can be recast as a null controllability problem for a nonlinear cascade system. We first prove a controllability result, with controls having one vanishing component, for a linear problem. Then, by means of an inverse mapping theorem, we deduce the controllability for the cascade system.     

On the controllability of the heat equation with nonlinear boundary Fourier conditions

In this paper we analyze the approximate and null controllability of the classical heat equation with nonlinear boundary conditions of the form and distributed controls, with support in a small set. We show that, when the function f is globally Lipschitz-continuous, the system is approximately controllable. We also show that the system is locally null controllable and null controllable for large time when f is regular enough and f(0)=0. For the proofs of these assertions, we use controllability results for similar linear problems and appropriate fixed point arguments. In the case of the local and large time null controllability results, the arguments are rather technical, since they need (among other things) Hölder estimates for the control and the state.     

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.     

Null controllability of viscous Hamilton–Jacobi equations

We study the problem of null controllability for viscous Hamilton–Jacobi equations in bounded domains of the Euclidean space in any space dimension and with controls localized in an arbitrary open nonempty subset of the domain where the equation holds. We prove the null controllability of the system in the sense that, every bounded (and in some cases uniformly continuous) initial datum can be driven to the null state in a sufficiently large time. The proof combines decay properties of the solutions of the uncontrolled system and local null controllability results for small data obtained by means of Carleman inequalities. We also show that there exists a waiting time so that the time of control needs to be large enough, as a function of the norm of the initial data, for the controllability property to hold. We give sharp asymptotic lower and upper bounds on this waiting time both as the size of the data tends to zero and infinity. These results also establish a limit on the growth of nonlinearities that can be controlled uniformly on a time independent of the initial data.     

On the null controllability of a one-dimensional fluid–solid interaction model

We analyze the null controllability of a one-dimensional nonlinear system which models the interaction of a fluid and a particle. The fluid is governed by the Burgers equation and the control is exerted on the boundary points. We present two main results: the global null controllability of a linearized system and the local null controllability of the nonlinear original model. The proofs rely on appropriate global Carleman inequalities and fixed point arguments. To cite this article: A. Doubova, E. Fernández-Cara, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Linear control systems: Controllability with constrained controls

We study the null controllability with bounded controls of an ordinary control process in ⁿ . To prove controllability conditions for the nonautonomous case, we need to generalize the definition of characteristic exponents of a linear system. Some theorems on characteristic exponents are stated.This work follows the program of the Gruppo Nazionale per l'Analisi Funzionale e le Sue Applicazioni, Consiglio Nazionale delle Ricerche, Rome, Italy.     

Local exact controllability to the trajectories of the 1-D Kuramoto-Sivashinsky equation

We are concerned with the boundary controllability to the trajectories of the Kuramoto-Sivashinsky equation. By using a Carleman estimate, we obtain the null controllability of the linearized equation around a given solution. From a local inversion theorem we get the local controllability to the trajectories of the nonlinear system.     

Exact controllability in projections for three-dimensional Navier–Stokes equations

The paper is devoted to studying controllability properties for 3D Navier–Stokes equations in a bounded domain. We establish a sufficient condition under which the problem in question is exactly controllable in any finite-dimensional projection. Our sufficient condition is verified for any torus in R³${\mathbb{R}}^3$. The proofs are based on a development of a general approach introduced by Agrachev and Sarychev in the 2D case. As a simple consequence of the result on controllability, we show that the Cauchy problem for the 3D Navier–Stokes system has a unique strong solution for any initial function and a large class of external forces.     

Exact null controllability of the Lobesia botrana model with diffusion

This paper is devoted to analyze the exact null controllability of the diffusive Lobesia botrana model with nonlocal boundary condition. We study the null controllability of butterfly population by acting on eggs, larvas and female moths in a small age interval. We assume that the control of female moths is in a certain given region of the vineyard. The main result is established by combining some estimations and the Carleman inequality for the backward system related to an optimal control problem. A fixed point theorem is then used to draw the conclusion.     

Null Controllability in Unbounded Domains for the Semilinear Heat Equation with Nonlinearities Involving Gradient Terms

We consider the null controllability problem for the semilinear heat equation with nonlinearities involving gradient terms in an unbounded domain of ^N with Dirichlet boundary conditions. The control is assumed to be distributed along a subdomain such that the uncontrolled region \ is bounded. Using Carleman inequalities, we prove first the null controllability of the linearized equation. Then, by a fixed-point method, we obtain the main result for the semilinear case. This result asserts that, when the nonlinearity is ^C1 and globally Lipschitz, the system is null controllable.     

On the minimal time function for distributed control systems in Banach spaces

In this paper, it is shown that the minimal time function is locally Lipschitz continuous for the control systemx=Ax+u in a Banach spadeE, under either of two conditions:A is linear and generates aC ₀-semigroup of bounded linear operators; orA is nonlinear, possibly multivalued, and dissipative. The main tool used for the nonlinear case is a result of Barbu concerning the null controllability of the system.     

Null controllability of a system of viscoelasticity with a moving control

In this paper, we consider the wave equation with both viscous Kelvin–Voigt and frictional damping as a model of viscoelasticity in which we incorporate an internal control with a moving support. We prove the null controllability when the control region, driven by the flow of an ODE, covers all the domain. The proof is based upon the interpretation of the system as, roughly, the coupling of a heat equation with an ordinary differential equation (ODE). The presence of the ODE for which there is no propagation along the space variable makes the controllability of the system impossible when the control is confined into a subset in space that does not move. The null controllability of the system with a moving control is established in using the observability of the adjoint system and some Carleman estimates for a coupled system of a parabolic equation and an ODE with the same singular weight, adapted to the geometry of the moving support of the control. This extends to the multi-dimensional case the results by P. Martin et al. in the one-dimensional case, employing 1-d Fourier analysis techniques.     

Controllability for systems governed by semilinear evolution inclusions without compactness

In this paper we study the controllability for a class of semilinear differential inclusions in Banach spaces. Since we assume the regularity of the nonlinear part with respect to the weak topology, we do not require the compactness of the evolution operator generated by the linear part. As well we are not posing any conditions on the multivalued nonlinearity expressed in terms of measures of noncompactness. We are considering the usual assumption on the controllability of the associated linear problem. Notice that, in infinite dimensional spaces, the above mentioned compactness of the evolution operator and linear controllability condition are in contradiction with each other. We suppose that the nonlinear term has convex, closed, bounded values and a weakly sequentially closed graph when restricted to its second argument. This regularity setting allows us to solve controllability problem under various growth conditions. As application, a controllability result for hyperbolic integro-differential equations and inclusions is obtained. In particular, we consider controllability of a system arising in a model of nonlocal spatial population dispersal and a system governed by the second order one-dimensional telegraph equation.     

The profile of blowing-up solutions to a nonlinear system of fractional differential equations

We investigate the profile of the blowing-up solutions to the nonlinear nonlocal system (FDS):

     

Bifurcation and multiplicity results for critical fractional p‐Laplacian problems

We prove a bifurcation and multiplicity result for a critical fractional p‐Laplacian problem that is the analog of the Brézis‐Nirenberg problem for the nonlocal quasilinear case. This extends a result in the literature for the semilinear case to all , in particular, it gives a new existence result. When , the nonlinear operator , has no linear eigenspaces, so our extension is nontrivial and requires a new abstract critical point theorem that is not based on linear subspaces. We prove a new abstract result based on a pseudo‐index related to the ‐cohomological index that is applicable here.     

Multidimensional constant linear systems

A continuous resp. discrete r-dimensional (r1) system is the solution space of a system of linear partial differential resp. difference equations with constant coefficients for a vector of functions or distributions in r variables resp. of r-fold indexed sequences. Although such linear systems, both multidimensional and multivariable, have been used and studied in analysis and algebra for a long time, for instance by Ehrenpreis et al. thirty years ago, these systems have only recently been recognized as objects of special significance for system theory and for technical applications. Their introduction in this context in the discrete one-dimensional (r=1) case is due to J. C. Willems. The main duality theorem of this paper establishes a categorical duality between these multidimensional systems and finitely generated modules over the polynomial algebra in r indeterminates by making use of deep results in the areas of partial differential equations, several complex variables and algebra. This duality theorem makes many notions and theorems from algebra available for system theoretic considerations. This strategy is pursued here in several directions and is similar to the use of polynomial algebra in the standard one-dimensional theory, but mathematically more difficult. The following subjects are treated: input-output structures of systems and their transfer matrix, signal flow spaces and graphs of systems and block diagrams, transfer equivalence and (minimal) realizations, controllability and observability, rank singularities and their connection with the integral respresentation theorem, invertible systems, the constructive solution of the Cauchy problem and convolutional transfer operators for discrete systems. Several constructions on the basis of the Gröbner basis algorithms are executed. The connections with other approaches to multidimensional systems are established as far as possible (to the author).Partially supported by US Air Force Grant AFOSR-87-0249 and by Office of Naval Research Grant N 00014-86-K-0538 through the Center for Mathematical System Theory, University of Florida, Gainesville, Florida, U.S.A.     

Some results on controllability of a nonlinear degenerate parabolic system by bilinear control

In this paper, we discuss the controllability of a nonlinear degenerate parabolic system with bilinear control. Based on the shrinking property of the solutions, we prove that the system is not globally approximately controllable. Furthermore, we give an approximate null controllability result. We also prove that the system is not globally exactly null controllable by a comparison principle.     

One-sided exact boundary null controllability of entropy solutions to a class of hyperbolic systems of conservation laws

In this paper, the one-sided exact boundary null controllability of entropy solutions is studied for a class of general strictly hyperbolic systems of conservation laws, whose negative (or positive) characteristic families are all linearly degenerate. The authors first prove the well-posedness of semi-global solutions constructed as the limit of ε-approximate front tracking solutions to the mixed initial-boundary value problem with general nonlinear boundary conditions and they establish various properties of both the ε-approximate front tracking solutions and such solutions. By means of essential modifications of the strategy suggested by the first author in [17] originally for the local exact boundary controllability in the framework of classical solutions, the one-sided local exact boundary null controllability of entropy solutions can then be realized via boundary controls acting on one side of the boundary, where the incoming characteristics are all linearly degenerate.     

带极大单调图的半线性抛物型方程的零能控性(英文)

本文讨论非线性项带一个极大单调图的半线性抛物型方程系统的零能控性.文中利用Kakutani不动点定理和线性抛物型方程的能控性得到该系统是零能控的,如果控制作用在内部区域上.由此,还得到该系统是零能控的,如果控制施加在一部分边界上.