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.     

Local exact controllability of the Navier–Stokes system

In this paper we deal with the local exact controllability of the Navier–Stokes system with distributed controls supported in small sets. In a first step, we present a new Carleman inequality for the linearized Navier–Stokes system, 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 Navier–Stokes system.     

Local controllability for autonomous nonlinear systems

This paper is concerned with the study of local controllability for nonlinear, analytic, autonomous, control systems. To this aim, we consider the control system as a set-valued differential system, and we show that the convexification of the set of velocities does not change the local controllability properties. Given a finite family of vector fields, we also give a necessary condition for local controllability at timet, for everyt.This work was performed under the auspices of the National Research Council of Italy (CNR).     

Exact Controllability with Internal Controls

For first-order quasilinear hyperbolic systems with zero eigenvalues, the author establishes the local exact controllability in a shorter time-period by means of internal controls acting on suitable domains. In particular, under certain special but reasonable hypotheses, the local exact controllability can be realized only by internal controls, and the control time can be arbitrarily small.     

Instant controllability of linear autonomous systems

In this paper, the complete controllability of linear autonomous control systems in any interval of time and the local controllability in any interval of time are investigated. Necessary and sufficient conditions are obtained for both types of controllability.     

Exact Controllability with Internal Controls for First-Order Quasilinear Hyperbolic Systems with Zero Eigenvalues

For first-order quasilinear hyperbolic systems with zero eigenvalues, the author establishes the local exact controllability in a shorter time-period by means of internal controls acting on suitable domains. In particular, under certain special but reasonable hypotheses, the local exact controllability can be realized only by internal controls, and the control time can be arbitrarily small.     

LINEAR CONTROL SYSTEMS WITH TIME-LAG:CONTROLLABILITY AND LOCAL PARAMETRICAL IDENTIFIABILITY WITH EXTENSIONS IN PRE-HILBERT SPACES

Conditions are given for controllability, output controllability and local identifiabitity of a parametrization in a subset of of linear differential systems with time-lags.     

Methods for computing the local null controllability time of linear control systems

We present methods for computing the local null controllability time in the class of linear control systems (not necessarily autonomous) whose families of null controllability sets are monotone with respect to inclusion.     

Exact Controllability of Semilinear Evolution Systems and Its Application

In this paper, we obtain several abstract results concerning the exact controllability of semilinear evolution systems. First, we prove the null local exact controllability of semilinear first-order systems by means of the contraction mapping principle; in this case, we do not assume any compactness. Next, we derive the global and/or local exact controllability of semilinear second-order systems by means of the Schauder fixed-point theorem; in this case, we assume only the embedding of the related spaces having some compactness, which is reasonable for many concrete problems. Our main result shows that the observability of the dual of the linearized system implies the exact controllability of the original semilinear system. Finally, we apply our abstract results to the exact controllability of the semilinear wave equation.     

Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit

In this paper, we deal with controllability properties of linear and nonlinear Korteweg-de Vries equations in a bounded interval. The main part of this paper is a result of uniform controllability of a linear KdV equation in the limit of zero-dispersion. Moreover, we establish a result of null controllability for the linear equation via the left Dirichlet boundary condition, and of exact controllability via both Dirichlet boundary conditions. As a consequence, we obtain some local exact controllability results for the nonlinear KdV equation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Controllability of nonlinear discrete systems without differentiability assumption

The paper gives another approach to studying controllability of nonlinear discrete systems, which is based on proving the support principle for some boundary problems involving locally Lipschitzian set-valued maps. The approach enables us to develop sufficient conditions for various types of local controllability of nonlinear non-stationary discrete systems without differentiability assumption. For linear discrete systems with constrained states and controls, necessary and sufficient conditions for local controllability are given.     

Local controllability to stationary trajectories of a Burgers equation with nonlocal viscosity

This article studies the local controllability to trajectories of a Burgers equation with nonlocal viscosity. By linearization we are led to an equation with a non local term whose controllability properties are analyzed by using Fourier decomposition and biorthogonal techniques. Once the existence of controls is proved and the dependence of their norms with respect to the time is established for the linearized model, a fixed point method allows us to deduce the result for the nonlinear initial problem.     

Exact Boundary Controllability for a Kind of Second-Order Quasilinear Hyperbolic Systems

Based on the theory of semi-global C ¹ solution and the local exact boundary controllability for first-order quasilinear hyperbolic systems, the local exact boundary controllability for a kind of second-order quasilinear hyperbolic systems is obtained by a constructive method.     

On the controllability of the Vlasov-Poisson system

In this paper, we study the controllability of the Vlasov-Poisson system in a periodic domain, by means of an interior control located in an spatial subdomain.The first result proves the local exact zero controllability in the two-dimensional torus between two small acceptable distribution functions, with an arbitrary control zone.A second result establishes the global exact controllability in arbitrary dimension, provided the control zone satisfies the condition that it contains a hyperplane of the torus.     

一类非线性chemotaxis系统的局部零能控性

本文研究了一类非线性chemotaxis系统的能控性问题.利用这类线性化结合不动点的方法,获得了非线性系统的局部零能控性结果,推广了非线性抛物-椭圆型方程能控性的结果.     

Exact controllability for the magnetohydrodynamic equations

We study the local exact controllability of the steady state solutions of the magnetohydrodynamic equations. The main result of the paper asserts that the steady state solutions of these equations are locally controllable if they are smooth enough. We reduce the local exact controllability of the steady state solutions of the magnetohydrodynamic equations to the global exact controllability of the null solution of the linearized magnetohydrodynamic system via a fixed‐point argument. The treatment of the reduced problem relies on two Carleman‐type inequalities for the backward adjoint system. © 2003 Wiley Periodicals, Inc.     

Exact controllability for 1‐D quasilinear wave equations with internal controls

For 1‐D quasilinear wave equations with different types of boundary conditions, based on the theory of the local exact boundary controllability, using an extension method, the author establishes the exact controllability in a shorter time by means of internal controls acting on suitable domains. In particular, the exact controllability can be realized only by internal controls, and the control time can be arbitrarily small. Copyright © 2016 John Wiley & Sons, Ltd.     

Global Null Controllability of the 1-Dimensional Nonlinear Slow Diffusion Equation

The authors prove the global null controllability for the 1-dimensional nonlinear slow diffusion equation by using both a boundary and an internal control. They assume that the internal control is only time dependent. The proof relies on the return method in combination with some local controllability results for nondegenerate equations and rescaling techniques.     

Optimal Control and Controllability of a Phase Field System with One Control Force

We investigate the relation between optimal control and controllability for a phase field system modeling the solidification process of pure materials in the case that only one control force is used. Such system is constituted of one energy balance equation, with a localized control associated to the density of heat sources and sinks to be determined, coupled with a phase field equation with the classical nonlinearity derived from the two-well potential. We prove that this system has a local controllability property and we establish that a sequence of solutions of certain optimal control problems converges to a solution of such controllability problem.     

Exact Local Controllability of a One-Control Reaction-Diffusion System

This work concerns the controllability of some parabolic systems with the control acting in one equation. The exact local controllability of a reaction-diffusion system with one control and the null controllability of a class of linear parabolic systems with one control are obtained. As an application, the existence of a time-optimal control for a reaction-diffusion system with one control is proved.Communicated by D. A. CarlsonThis work was supported by the New Century Excellent Teachers Plan of the Ministry of National Education of China, by the Key Laboratory- Optimal Control and Discrete Mathematics of Hubei Province, and by the National Science Foundation of China.