Null Controllability Result Research Articles

Near viability for fully nonlinear differential inclusions

AbstractWe consider the nonlinear differential inclusion x′(t) ∈ Ax(t) + F(x(t)), where A is an m-dissipative operator on a separable Banach space X and F is a multi-function. We establish a viability result under Lipschitz hypothesis on F, that consists in proving the existence of solutions of the differential inclusion above, starting from a given set, which remain arbitrarily close to that set, if a tangency condition holds. To this end, we establish a kind of set-valued Gronwall’s lemma and a compactness theorem, which are extensions to the nonlinear case of similar results for semilinear differential inclusions. As an application, we give an approximate null controllability result.

Carleman estimates for semi-discrete parabolic operators with a discontinuous diffusion coefficient and applications to controllability

In the discrete setting of one-dimensional finite-differences we prove a Carleman estimate for a semi-discretization of the parabolic operator $\partial_t-\partial_x (c\partial_x )$ where the diffusion coefficient $c$ has a jump. As a consequence of this Carleman estimate, we deduce consistent null-controllability results for classes of semi-linear parabolic equations.

Insensitizing exact controls for the scalar wave equation and exact controllability of $$2$$ -coupled cascade systems of PDE’s by a single control

We study the exact controllability, by a reduced number of controls, of coupled cascade systems of PDE’s and the existence of exact insensitizing controls for the scalar wave equation. We give a necessary and sufficient condition for the observability of abstract-coupled cascade hyperbolic systems by a single observation, the observation operator being either bounded or unbounded. Our proof extends the two-level energy method introduced in Alabau-Boussouira (Siam J Control Opt 42:871–906, 2003) and Alabau-Boussouira and Léautaud (J Math Pures Appl 99:544–576, 2013) for symmetric coupled systems, to cascade systems which are examples of non-symmetric coupled systems. In particular, we prove the observability of two coupled wave equations in cascade if the observation and coupling regions both satisfy the Geometric Control Condition (GCC) of Bardos et al. (SIAM J Control Opt 30:1024–1065, 1992). By duality, this solves the exact controllability, by a single control, of \(2\)-coupled abstract cascade hyperbolic systems. Using transmutation, we give null-controllability results for the multidimensional heat and Schrödinger \(2\)-coupled cascade systems under GCC and for any positive time. By our method, we can treat cases where the control and coupling coefficients have disjoint supports, partially solving an open question raised by de Teresa (CPDE 25:39–72, 2000). Moreover we answer the question of the existence of exact insensitizing locally distributed as well as boundary controls of scalar multidimensional wave equations, raised by Lions (Actas del Congreso de Ecuaciones Diferenciales y Aplicaciones (CEDYA), Universidad de Málaga, pp 43–54, 1989) and later on by Dáger (Siam J Control Opt 45:1758–1768, 2006) and Tebou (C R Acad Sci Paris 346(Sér I):407–412, 2008).

Control of coupled parabolic systems and Diophantine approximations

In this paper we consider the boundary null controllability problem of two coupled one dimensional parabolic equations. We show that in the case in which each equation have different diffusion constant, lets say \(\nu \) and 1, the null controllability result depends on the nature of the diffusion constant \(\nu \). Using rational approximations of the number \(\nu \), we show that null controllability is possible or not. In this sense the result is a number theoretical result.

Null controllability of degenerate parabolic equations in non divergence form via Carleman estimates

We prove null controllability results for the one dimensional degenerate heat equation in non divergence form with a drift term and Neumann boundary conditions. To this aim we prove Carleman estimates for the associated adjoint problem. Some linear extensions are considered.

OBSERVABILITY ESTIMATE AND NULL CONTROLLABILITY FOR ONE-DIMENSIONAL FOURTH ORDER PARABOLIC EQUATION

This paper studies the observability and null controllability for a class of one-dimensional fourth order parabolic equation. By means of establishing the global Carleman estimates, we derive the observability inequalities for one-dimensional linear fourth order parabolic equation with potentials. The null controllability results for one-dimensional fourth order semilinear equation are also presented.

Insensitizing controls for the Navier–Stokes equations

In this paper, we deal with the existence of insensitizing controls for the Navier–Stokes equations in a bounded domain with Dirichlet boundary conditions. We prove that there exist controls insensitizing the L2-norm of the observation of the solution in an open subset O of the domain, under suitable assumptions on the data. This problem is equivalent to an exact controllability result for a cascade system. First we prove a global Carleman inequality for the linearized Navier–Stokes system with right-hand side, which leads to the null controllability at any time T>0. Then, we deduce a local null controllability result for the cascade system.

Indirect controllability of locally coupled wave-type systems and applications

We consider symmetric systems of two wave-type equations only one of them being controlled. The two equations are coupled by zero order terms, localized in part of the domain. We prove an internal and a boundary controllability result in any space dimension, provided that both the coupling and the control regions satisfy the Geometric Control Condition. We deduce similar null-controllability results in any positive time for parabolic systems and Schrödinger-type systems under the same geometric conditions on the coupling and the control regions. This includes several examples in which these two regions have an empty intersection.

OBSERVABILITY ESTIMATE AND NULL CONTROLLABILITY FOR ONE-DIMENSIONAL FOURTH ORDER PARABOLIC EQUATION

This paper studies the observability and null controllability for a class of one-dimensional fourth order parabolic equation. By means of establishing the global Carleman estimates, we derive the observability inequalities for one-dimensional linear fourth order parabolic equation with potentials. The null controllability results for one-dimensional fourth order semilinear equation are also presented.

Exact Null Controllability for Fractional Nonlocal Integrodifferential Equations via Implicit Evolution System

We introduce a new concept called implicit evolution system to establish the existence results of mild and strong solutions of a class of fractional nonlocal nonlinear integrodifferential system, then we prove the exact null controllability result of a class of fractional evolution nonlocal integrodifferential control system in Banach space. As an application that illustrates the abstract results, two examples are provided.

Null controllability for a parabolic equation with nonlocal nonlinearities

In this paper, we establish a local null controllability result for a nonlinear parabolic PDE with nonlocal nonlinearities. The result relies on the (global) null controllability of similar linear equations and a fixed point argument. We also analyze other similar controllability problems and we present several open questions.

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.

Lipschitz Stability in Inverse Source Problems for Singular Parabolic Equations

We address the question of Lipschitz stability results in inverse source problems for the heat equation perturbed by a singular inverse-square potential − μ/|x|2. Since the works of Baras and Goldstein [2], the existence of a critical parameter μ☆(N) = (N − 2)2/4 (the optimal constant appearing in the so-called Hardy inequality) is by now well-known. Recently specific Carleman inequalities (involving suitable weight functions to take into account the singularity) have been developed by Vancostenoble and Zuazua [42] and Ervedoza [20] in the sub-critical and critical cases. They have been used to deduce observability and null controllability results. In this paper, we are interested in inverse source problems with a locally distributed observation, such as those studied by Immanuvilov and Yamamoto [25]: using the classical Carleman estimates by Fursikov and Immanuvilov [21], they obtain Lipschitz stability results in the case of the classical heat equation. Here we study the case of the heat equation with inverse-square potentials. In this purpose, the specific Carleman inequalities by Vancostenoble and Zuazua and Ervedoza are not sufficient to conclude. Hence we first complete them by deriving sharper Carleman estimates. Major steps in the proof rely on several improved forms of Hardy-Poincaré inequalities. Next we prove Lipschitz stability results in the case of the heat equation with sub-critical or critical inverse-square potentials that extend the results by Immanuvilov-Yamamoto.

Indirect controllability of locally coupled systems under geometric conditions

We consider systems of two wave/heat/Schrödinger-type equations coupled by a zero order term, only one of them being controlled. We prove an internal and a boundary null-controllability result in any space dimension, provided that both the coupling and the control regions satisfy the Geometric Control Condition. This includes several examples in which these two regions have an empty intersection.

Improved Hardy-Poincaré inequalities and sharp Carleman estimates for degenerate/singular parabolic problems

We consider the following class of degenerate/singular parabolic operators:$Pu=u_t-(x^\a u_x)_x-$λ$ u$/($x^$β) , $x\in (0,1)$,associated to homogeneous boundary conditions of Dirichlet and/or Neumann type. Under optimal conditions on the parameters$\a\geq 0$, β, λ$ \in \mathbb R$, we derive sharp global Carleman estimates. As an application, we deduce observability and null controllability results for the corresponding evolution problem. A key step in the proof of Carleman estimates is the correct choice of the weight functions and a key ingredient in the proof takes the form of special Hardy-Poincaré inequalities

Recent progress on controllability/observability for systems governed by partial differential equations

The main purpose of this paper is to overview some recent methods and results on controllability/observability problems for systems governed by partial differential equations. First, the authors review the theory for linear partial differential equations, including the iteration method for the null controllability of the time-invariant heat equation and the Rellich-type multiplier method for the exact controllability of the time-invariant wave equation, and especially a unified controllability/observability theory for parabolic and hyperbolic equations based on a global Carleman estimate. Then, the authors present sharp global controllability results for both semi-linear parabolic and hyperbolic equations, based on linearization approach, sharp observability estimates for the corresponding linearized systems and the fixed point argument. Finally, the authors survey the local null controllability result for a class of quasilinear parabolic equations based on the global Carleman estimate, and the local exact controllability result for general hyperbolic equations based on a new unbounded perturbation technique.

Discrete Carleman Estimates for Elliptic Operators in Arbitrary Dimension and Applications

In arbitrary dimension, we consider the semidiscrete elliptic operator $-\partial_t^2+\mathcal{A}^{\scriptscriptstyle\mathfrak{M}}$, where $\mathcal{A}^{\scriptscriptstyle\mathfrak{M}}$ is a finite-difference approximation of the operator $-\nabla_x(\Gamma(x)\nabla_x)$. For this operator we derive a global Carleman estimate, in which the usual large parameter is connected to the discretization step-size. We address discretizations on some families of smoothly varying meshes. We present consequences of this estimate, such as a partial spectral inequality of the form of that proven by G. Lebeau and L. Robbiano for $\mathcal{A}^{\scriptscriptstyle\mathfrak{M}}$ and a null-controllability result for the parabolic operator $\partial_t+\mathcal{A}^{\scriptscriptstyle\mathfrak{M}}$ for the lower part of the spectrum of $\mathcal{A}^{\scriptscriptstyle\mathfrak{M}}$. With the control function that we construct (whose norm is uniformly bounded) we prove that the $L^2$-norm of the final state converges to zero exponentially, as the step-size of the discretization goes to zero. A relaxed observability estimate is then deduced.

Null controllability in a fluid–solid structure model

We consider a system coupling the Stokes equations in a two-dimensional domain with a structure equation which is a system of ordinary differential equations corresponding to a finite dimensional approximation of equations modeling deformations of an elastic body or vibrations of a rigid body. For that system we establish a null controllability result for localized distributed controls acting only in the fluid equations and there is no control in the solid part. This controllability result follows from a Carleman inequality that we prove for the adjoint system.

Discrete Carleman estimates for elliptic operators and uniform controllability of semi-discretized parabolic equations

We derive a semi-discrete two-dimensional elliptic global Carleman estimate, in which the usual large parameter is connected to the one-dimensional discretization step-size. The discretizations we address are some families of smoothly varying meshes. As a consequence of the Carleman estimate, we derive a partial spectral inequality of the form of that proven by G. Lebeau and L. Robbiano, in the case of a discrete elliptic operator in one dimension. Here, this inequality concerns the lower part of the discrete spectrum. The range of eigenvalues/eigenfunctions we treat is however quasi-optimal and represents a constant portion of the discrete spectrum. For the associated parabolic problem, we then obtain a uniform null controllability result for this lower part of the spectrum. Moreover, with the control function that we construct, the L 2 -norm of the final state converges to zero super-algebraically as the step-size of the discretization goes to zero. A relaxed observability estimate is then deduced.

Controllability results for a class of one-dimensional degenerate parabolic problems in nondivergence form

We give null controllability results for some degenerate parabolic equations in non divergence form on a bounded interval. In particular, the coefficient of the second order term degenerates at the extreme points of the domain. For this reason, we obtain an observability inequality for the adjoint problem. Then we prove Carleman estimates for such a problem. Finally, in a standard way, we deduce null controllability also for semilinear equations.