Null controllability for the parabolic equation with a complex principal part

The paper is devoted to a study of the null controllability for the semilinear parabolic equation with a complex principal part. For this purpose, we establish a key weighted identity for partial differential operators (with real functions α and β), by which we develop a universal approach, based on global Carleman estimate, to deduce not only the desired explicit observability estimate for the linearized complex Ginzburg-Landau equation, but also all the known controllability/observability results for the parabolic, hyperbolic, Schrödinger and plate equations that are derived via Carleman estimates.     

Lipschitz Stability in Inverse Source Problems for Singular Parabolic Equations

We consider 2-D Klein-Gordon equation with quadratic nonlinearity and prove Strichartz type dispersive estimates for the global solution with small initial data in the Sobolev space H ^1+?.     

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.     

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.     

Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients

We derive global Carleman estimates for one-dimensional linear parabolic equations t_∂±x_∂(cx_∂) with a coefficient of bounded variations. These estimates are obtained by approximating c by piecewise constant coefficients, cε, and passing to the limit in the Carleman estimates associated to the operators defined with cε. Such estimates yields observability inequalities for the considered linear parabolic equation, which, in turn, yield controllability results for classes of semilinear equations.     

Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem

We study the observability and some of its consequences (controllability, identification of diffusion coefficients) for one-dimensional heat equations with discontinuous coefficients (piecewise C¹). The observability, for a linear equation, is obtained by a Carleman-type estimate. This kind of observability inequality yields controllability results for a semi-linear equation as well as a stability result for the identification of the diffusion coefficient.     

Null controllability for a semilinear parabolic equation with gradient quadratic growth

This article is concerned with the null controllability of a semilinear parabolic equation with the nonlinear term involving the gradient quadratic term. The technique in this paper is a combination of Cole–Hopf transformation and some methods from [A.Y. Khapalov, Controllability of the semilinear parabolic equation governed by a multiplicative control in the reaction term: A qualitative approach, SIAM J. Control Optim. 41 (2003) 1886–1900].     

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.     

Null controllability and global blowup controllability of ordinary differential equations with feedback controls

This paper concerns the problem of feedback null controllability and blowup controllability with feedback controls for ordinary differential equations. First, we study the feedback null controllability on a time-varying ordinary differential system by unbounded feedback operators. Then, the global exact blowup controllability with feedback controls is derived on a time-invariant ordinary differential system. Finally, we obtain the approximate null controllability by bounded feedback operators, and get the approximate blowup controllability with feedback controls for ordinary differential equations.     

Null Controllability for Some Systems of Two Backward Stochastic Heat Equations with One Control Force

The authors establish the null controllability for some systems coupled by two backward stochastic heat equations. The desired controllability result is obtained by means of proving a suitable observability estimate for the dual system of the controlled system.     

Some controllability results for the 2D Kolmogorov equation

In this article, we prove the null controllability of the 2D Kolmogorov equation both in the whole space and in the square. The control is a source term in the right-hand side of the equation, located on a subdomain, that acts linearly on the state. In the first case, it is the complementary of a strip with axis x and in the second one, it is a strip with axis x.The proof relies on two ingredients. The first one is an explicit decay rate for the Fourier components of the solution in the free system. The second one is an explicit bound for the cost of the null controllability of the heat equation with potential that the Fourier components solve. This bound is derived by means of a new Carleman inequality.     

Observability Inequalities by Internal Observation and Their Applications

By means of a Carleman estimate, we obtain several observability inequalities by internal observation for hyperbolic equations with lower-order terms. Also, we apply our observability results to an exact controllability problem and an inverse wave source problem.     

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.     

Global Uniqueness and Stability for an Inverse Plate Problem

This paper deals with the global uniqueness and stability of an inverse problem for the plate equation. By means of a suitable transformation, the problem is reduced to the observability estimate for the plate equations with memory. The main tool that we employ is a global Carleman-type estimate. This work was supported by FANEDD of China Project 200119 and by NSFC under Grant 10371084. The author thanks Professor X. Zhang for help.