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.     

EQUIVALENCE BETWEEN EXACT INTERNAL CONTROLLABILITY OF THE KIRCHHOFF PLATE-LIKE EQUATION AND THEWAVE EQUATION

1.IntroductionConsiderthefollowingillitial-boundaryvalueproblemLetD(A)={vEHI(fl)ItvEL'(fl)},V~Ha(~~),H~L'(fl).Thenthefiniteenergystatespacesofthesystems(1.1),(1.2)are,respectively,uk~D(A)xV,U.~VxH.ConsiderthegeneralizedfunctionspaceH-1~V'DHDV.ForbEC'(fl)anduEH-',wedefinebaEH-1by(bu)(()~u(bo,V(6HJ(n).(1.3)WehaveusedtheSobolevspaces.Wereferthereadersto[1]fortheknowledgeofthosespaces.Definition1.1.Thesystem(1.1)issaidtobeexactlycontrollableinukbyH-1-controlsifforevery(yo,yi)6uk,th…     

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.     

Null Controllability in a Heat–Solid Structure Model

A model representing a coupling between a heat conducting medium and a solid structure is considered. We establish a Carleman inequality for this model. Next we deduce a null controllability result with an internal control in the conducting medium and there is no control in the solid part.     

Null controllability of a pseudo-parabolic equation with moving control

In this paper, we study the internal controllability of the pseudo-parabolic equation on the one-dimensional torus. Our control function is acting on a moving small interval with a constant velocity. With this moving distributed control, we obtain the system is null controllable for given data with certain regularity.     

Exact Null Controllability of a Nonlinear Thermoelastic Contact Problem

We study the controllability properties of a nonlinear parabolic system that models the temperature evolution of a one-dimensional thermoelastic rod that may come into contact with a rigid obstacle. Basically the system dynamics is described by a one-dimensional nonlocal heat equation with a nonlinear and nonlocal boundary condition of Newmann type. We focus on the control problem and treat the case when the control is distributed over the whole space domain. In this case the system is proved to be exactly null controllable provided the parameters of the system are smooth. The proof is based on changing the control variable and using Aubins Compactness Lemma to obtain an invariant set for the linearized controllability map. Then, by proving that the found solution is sufficiently smooth, we get the null controllability for the original system.     

Null controllability of the complex Ginzburg–Landau equation

The paper investigates the boundary controllability, as well as the internal controllability, of the complex Ginzburg–Landau equation. Zero-controllability results are derived from a new Carleman estimate and an analysis based upon the theory of sectorial operators.     

关于耗散波动方程精确能控性的奇异极限的一个注记

本文在比文献［１］更一般的几何控制条件下，分析了具齐次Ｄｉｒｉｃｈｌｅｔ边界条件的耗散波动方程精确能控性的奇异摄动问题．结论是由这类波动方程的精确能控性可得到热传导方程的精确零能控性．     

Some results on the controllability of forward stochastic heat equations with control on the drift

In this paper, we establish the null/approximate controllability for forward stochastic heat equations with control on the drift. The null controllability is obtained by a time iteration method and an observability estimate on partial sums of eigenfunctions for elliptic operators. As a consequence of the null controllability, we obtain the observability estimate for backward stochastic heat equations, which leads to a unique continuation property for backward stochastic heat equations, and hence the desired approximate controllability for forward stochastic heat equations. It deserves to point out that one needs to introduce a little stronger assumption on the controller for the approximate controllability of forward stochastic heat equations than that for the null controllability. This is a new phenomenon arising in the study of the controllability problem for stochastic heat equations.     

Remarks on global controllability for the Burgers equation with two control forces

In this paper we deal with the viscous Burgers equation. We study the exact controllability properties of this equation with general initial condition when the boundary control is acting at both endpoints of the interval. In a first result, we prove that the global exact null controllability does not hold for small time. In a second one, we prove that the exact controllability result does not hold even for large time.     

Global robust controllability of the triangular integro-differential Volterra systems

A solution of the global controllability problem for a class of nonlinear control systems of the Volterra integro-differential equations is presented. It is proven that there exists a family of continuous controls that solve the global controllability problem for this class. The constructed controls depend continuously on the initial and the terminal states. It makes possible to prove the global controllability of the uniformly bounded perturbations of these systems under the global Lipschitz condition for the unperturbed system with respect to the states and the controls.     

A Note on the Singular Limit of the Exact Controllability of Dissipative Wave Equations

Abstract In this note we analyze the exact controllability of singularly perturbed damped wave equations under more general geometric control condition than that of [1]. We show that the null controllability of the heat equation can be obtained as a singular limit of the exact controllability of such sorts of wave equations. The work of Yuping Tang was carried out when she visited the “School of Mathematics, Sichuan University”. The work of Xu Zhang was partially supported by NSF of China under Grant 19901024     

Global Approximately Controllability and Finite Dimensional Exact Controllability of Semilinear Heat Equation in <Emphasis Type="Italic">R</Emphasis><Superscript><Emphasis Type="Italic">N</Emphasis></Superscript>

We prove the approxomate controllability and finite dimensional exact controllability of semilinear heat equation in R^N with the same control by introducing the weighted Soblev spaces.     

LOCAL EXACT BOUNDARY CONTROLLABILITY FOR A CLASS OF QUASILINEAR HYPERBOLIC SYSTEMS

InroductionLet us consider the following irst order quaslllnear hyperbolic     

Insensitizing controls for a class of nonlinear Ginzburg-Landau equations

This paper shows the existence of insensitizing controls for a class of nonlinear complex Ginzburg-Landau equations with homogeneous Dirichlet boundary conditions and arbitrarily located internal controller. When the nonlinearity in the equation satisfies a suitable superlinear growth condition at infinity, the existence of insensitizing controls for the corresponding semilinear Ginzburg-Landau equation is proved. Meanwhile, if the nonlinearity in the equation is only a smooth function without any additional growth condition, a local result on insensitizing controls is obtained. As usual, the problem of insensitizing controls is transformed into a suitable controllability problem for a coupled system governed by a semilinear complex Ginzburg-Landau equation and a linear one through one control. The key is to establish an observability inequality for a coupled linear Ginzburg-Landau system with one observer.     

Topological methods for the global controllability of nonlinear systems

Sufficient conditions for the local and global controllability of general nonlinear systems, by means of controls belonging to a fixed finite-dimensional subspace of the space of all admissible controls, are established with the aid of topological methods, such as homotopy invariance principles. Some applications to certain classes of nonlinear control processes are given, and various known results on the controllability of perturbed linear systems are also derived as particular cases.     

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.     

Skew-product semi-flows and non-autonomous control systems

In this paper we give a necessary and sufficient conditions for exact and approximate controllability of a wide class of linear infinite-dimensional non-autonomous control systems. This is done by employing skew-product semi-flow technique. Finally, we apply these results to prove the controllability of a broad class of non-autonomous reaction diffusion equations in Hilbert spaces.     

Energy decay and control for a system of strings elastically connected in parallel

In the present paper, we study energy decay and exact boundary controllability for a system of n one-dimensional linear wave equations coupled in parallel. The control obtained is a square integrable of the Neuman type for initial data with finite energy. The controllability time is near optimal value. We treat the case of control in the whole boundary and also in part of it.     

Approximate controllability and homogenization of a semilinear elliptic problem

The L²- and H¹-approximate controllability and homogenization of a semilinear elliptic boundary-value problem is studied in this paper. The principal term of the state equation has rapidly oscillating coefficients and the control region is locally distributed. The observation region is a subset of codimension 1 in the case of L²-approximate controllability or is locally distributed in the case of H¹-approximate controllability. By using the classical Fenchel-Rockafellar's duality theory, the existence of an approximate control of minimal norm is established by means of a fixed point argument. We consider its asymptotic behavior as the rapidly oscillating coefficients H-converge. We prove its convergence to an approximate control of minimal norm for the homogenized problem.