In this work we extend a variational method to study the approximate controllability and finite dimensional exact controllability ( finite-approximate controllability) for the semilinear evolution equations in Hilbert spaces. We state a useful characterization of the finite-approximate controllability for linear evolution equation in terms of resolvent-like operators. We also find a control so that, in addition to the approximate controllability requirement, it ensures finite dimensional exact controllability. Assuming the approximate controllability of the corresponding linearized equation we obtain sufficient conditions for the finite-approximate controllability of the semilinear evolution equation under natural conditions. The obtained results are generalization and continuation of the recent results on this issue. Applications to heat equations are treated.
In this paper, we intend to present some already known results about the internal controllability of the linear and nonlinear Schrödinger equation. After presenting the basic properties of the equation, we give a self contained proof of the controllability in dimension 1 using some propagation results. We then discuss how to obtain some similar results on a compact manifold where the zone of control satisfies the Geometric Control Condition. We also discuss some known results and open questions when this condition is not satisfied. Then, we present the links between the controllability and som…
This paper deals with the controllability of the second grade fluids, a class of non-Newtonian of differentiel type, on a two-dimensional torus. Using the method of Agrachev-Sarychev [1], [2] and of Sirikyan [26], we prove that the system of second grade fluids is approximately controllable by a finite-dimensional control force.
We consider the 3D Navier-Stokes system driven by an additive finite-dimensional control force. The purpose of this paper is to show how the approximate controllability of this system can be derived from the approximate controllability of the Euler system linearised around some suitable trajectory. The proof presented here is shorter than the previous ones obtained by Lie algebraic methods and gives some new information about the structure of the control. The dimension of the control space provided by this approach is larger, but it is still uniform with respect to the viscosity.
We consider the heat equation with dynamic bounary conditions involving gradient terms in a bounded domain. In this paper we study the cost of approximate controllability for this equation. Combining new developed Carleman estimates and some optimization techniques, we obtain explicit bounds of the minimal norm control. We consider the linear and the semilinear cases.
In this paper we study the approximate controllability of fractional partial differential equations associated with the so-called Hilfer type time fractional derivative and a non-negative selfadjoint operator $A$ with a compact resolvent on $L^2(Ω)$, where $Ω\subset\RR^N$ ($N\geq 1$) is an open set. More precisely, we show that if $0\leν\le 1$, $1<μ\le 2$ and $Ω\subset\RR^N$ is an open set, then the system \begin{equation*} \begin{cases} \D^{μ,ν}_tu+Au=fχ_ω\;\;&\mbox{ in }\;Ω\times(0,T),\\ (I_t^{(1-ν)(2-μ)}u)(\cdot,0)=u_0 &\mbox{ in }\;Ω,\\ (\partial_tI_t^{(1-ν)(2-μ)}u)(\cdot,0)=u_1 &\mbox{ in…
On a compact Riemannian manifold with boundary, we prove a spectral inequality for the bi-Laplace operator in the case of so-called "clamped" boundary conditions , that is, homogeneous Dirichlet and Neumann conditions simultaneously. We also prove a resolvent estimate for the generator of the damped plate semigroup associated with these boundary conditions. The spectral inequality allows one to observe finite sums of eigenfunctions for this fourth-order elliptic operator, from an arbitrary open subset of the manifold. Moreover, the constant that appears in the inequality grows as exp(C$μ$ 1/4)…
In the Eulerian approach, the motion of an incompressible fluid is usually described by the velocity field which is given by the Navier--Stokes system. The velocity field generates a flow in the space of volume-preserving diffeomorphisms. The latter plays a central role in the Lagrangian description of a fluid, since it allows to identify the trajectories of individual particles. In this paper, we show that the velocity field of the fluid and the corresponding flow of diffeomorphisms can be simultaneously approximately controlled using a finite-dimensional external force. The proof is based on…
This paper is devoted to studying the null and approximate controllability of two linear coupled parabolic equations posed on a smooth domain of R^N (N>1) with coupling terms of zero and first orders and one control localized in some arbitrary nonempty open subset of the domain. We prove the null controllability under a new sufficient condition and we also provide the first example of a not approximately controllable system in the case where the support of one of the nontrivial first order coupling terms intersects the control domain.
In this paper, we consider a class of second-order evolution differential inclusions in Hilbert spaces. This paper deals with the approximate controllability for a class of second-order control systems. First, we establish a set of sufficient conditions for the approximate controllability for a class of second-order evolution differential inclusions in Hilbert spaces. We use Bohnenblust-Karlin's fixed point theorem to prove our main results. Further, we extend the result to study the approximate controllability concept with nonlocal conditions and extend the result to study the approximate con…
In this paper we consider a multidimensional semilinear reaction-diffusion equation and we obtain at any arbitrary time an approximate controllability result between nonnegative states using as control term the reaction coefficient, that is via multiplicative controls.
In this paper, we consider the well-known Fattorini's criterion for approximate controllability of infinite dimensional linear systems of type $y'=A y+Bu$. We precise the result proved by H. O. Fattorini in \cite{Fattorini1966} for bounded input $B$, in the case where $B$ can be unbounded or in the case of finite-dimensional controls. More precisely, we prove that if Fattorini's criterion is satisfied and if the set of geometric multiplicities of $A$ is bounded then approximate controllability can be achieved with finite dimensional controls. An important consequence of this result consists in…
We study the approximate and mean approximate controllability properties of fractional partial differential equations associated with the so-called Hilfer type time-fractional derivative and a non-negative selfadjoint operator $A_B$ with a compact resolvent on $L^2(Ω)$, where $Ω\subset\mathbb{R}^N$ ($N\ge 1$) is a bounded open set. More precisely, we show that if $0\leν\le 1$, $0<μ\le 1$ and $Ω\subset\mathbb R^N$ is a bounded open set, then the system $$\mathbb D_t^{μ,ν} u+A_Bu=f|_ω\;\; \mbox{ in }\; Ω\times (0,T),\,\, (\mathbb I_t^{(1-ν)(1-μ)}u)(\cdot,0)=u_0 \mbox{ in }\;Ω,$$ is approximately…
Global small-time approximate null and Lagrangian controllability of the viscous non-resistive MHD system in a $3D$ domain with Navier type boundary conditions
In this paper, we consider a class of fractional integro-differential inclusions in Hilbert spaces. This paper deals with the approximate controllability for a class of fractional integro-differential control systems. First, we establishes a set of sufficient conditions for the approximate controllability for a class of fractional semilinear integro-differential inclusions in Hilbert spaces. We use Bohnenblust-Karlin's fixed point theorem to prove our main results. Further, we extend the result to study the approximate controllability concept with nonlocal conditions. An example is also given …
An explicit saturating set consisting of eigenfunctions of Stokes operator in general 3D Cylinders is proposed. The existence of saturating sets implies the approximate controllability for Navier--Stokes equations in $\rm 3D$ Cylinders under Lions boundary conditions.
This analysis is concerned with the controllability of quantum systems in the case where the standard dipolar approximation, involving the permanent dipole moment of the system, is corrected with a polarizability term, involving the field induced dipole moment. Sufficient conditions for approximate controllability are given. For transfers between eigenstates of the free Hamiltonian, the control laws are explicitly given. The results apply also for unbounded or non-regular potentials.
This paper is devoted to the study of the null and approximate controllability for some classes of linear coupled parabolic systems with less controls than equations. More precisely, for a given bounded domain in R^N, we consider a system of m linear parabolic equations (m > 2) with coupling terms of first and zero order, and m-1 controls localized in some arbitrary nonempty open subset. In the case of constant coupling coefficients, we provide a necessary and sufficient condition to obtain the null or approximate controllability in arbitrary small time. In the case m = 2 and N = 1, we also gi…
An algebraic characterization of the property of approximate controllability is given, for behaviours of spatially invariant dynamical systems, consisting of distributional solutions, that are periodic in the spatial variables, to a system of homogeneous, linear, constant coefficient partial differential equations corresponding to a polynomial matrix.
In this article, we study the observability (or, equivalently, the controllability) of some subelliptic evolution equations depending on their step. This sheds light on the speed of propagation of these equations, notably in the "degenerated directions" of the subelliptic structure. First, for any $γ\geq 1$, we establish a resolvent estimate for the Baouendi-Grushin-type operator $Δ_γ=\partial_x^2+|x|^{2γ}\partial_y^2$, which has step $γ+1$. We then derive consequences for the observability of the Schr\''odinger type equation $i\partial_tu-(-Δ_γ)^{s}u=0$ where $s\in N$. We identify three diffe…
The main purpose of this article is to prove a logarithmic convexity estimate for the solution of a linear heat equation subject to dynamic boundary conditions in a bounded convex domain. As an application, we prove the impulsive null approximate controllability for an impulsive heat equation with dynamic boundary conditions.
The aim of this work is to consider the controllability problem of the linear system associated to Korteweg-de Vries Burgers equation posed in the whole real line. We obtain a sort of exact controllability for solutions in $L^2_{loc}(\R^2)$ by deriving an internal observability inequality and a Global Carlemann estimate. Following the ideas contained in \cite{rosier2000}, the problem is reduced to prove an approximate theorem.
The three concepts of exact, null and approximate controllabilities are analyzed from the exterior of the Moore--Gibson--Thompson equation associated with the fractional Laplace operator subject to the nonhomogeneous Dirichlet type exterior condition.
In this paper, we study the global approximate multiplicative controllability for nonlinear degenerate parabolic Cauchy-Neumann problems. First, we will obtain embedding results for weighted Sobolev spaces, that have proved decisive in reaching well-posedness for nonlinear degenerate problems. Then, we show that the above systems can be steered in $L^2$ from any nonzero, nonnegative initial state into any neighborhood of any desirable nonnegative target-state by bilinear piecewise static controls. Moreover, we extend the above result relaxing the sign constraint on the initial date.
We study the controllability of a Partial Differential Equation of transport type, that arises in crowd models. We are interested in controlling it with a control being a vector field, representing a perturbation of the velocity, localized on a fixed control set. We prove that, for each initial and final configuration, one can steer approximately one to another with Lipschitz controls when the uncontrolled dynamics allows to cross the control set. We also show that the exact controllability only holds for controls with less regularity, for which one may lose uniqueness of the associated soluti…
We prove that the approximate null-controllability with uniform cost of the hypoelliptic Ornstein-Uhlenbeck equations posed on $\mathbb R^n$ is characterized by an integral thickness geometric condition on the control supports. We also provide associated quantitative weak observability estimates. This result for the hypoelliptic Ornstein-Uhlenbeck equations is deduced from the same study for a large class of non-autonomous elliptic equations from moving control supports. We generalize in particular results known for parabolic equations posed on $\mathbb R^n$, for which the approximate null-con…
In this paper, we study, in the semiclassical sense, the global approximate controllability in small time of the quantum density and quantum momentum of the 1-D semiclassical cubic Schrödinger equation with two controls between two states with positive quantum densities. We first control the asymptotic expansions of the zeroth and first order of the physical observables via Agrachev-Sarychev's method. Then we conclude the proof through techniques of semiclassical approximation of the nonlinear Schrödinger equation.
This paper is dedicated to approximate controllability for Grushin equation on the rectangle $(x,y) \in (-1,1) \times (0,1)$ with an inverse square potential. This model corresponds to the heat equation for the Laplace-Beltrami operator associated to the Grushin metric on $\mathbb{R}^2$, studied by Boscain and Laurent. The operator is both degenerate and singular on the line $\{ x=0 \}$. The approximate controllability is studied through unique continuation of the adjoint system. For the range of singularity under study, approximate controllability is proved to hold whatever the degeneracy is.…
This paper is devoted to the controllability of linear systems of two coupled parabolic equations when the coupling involves a space dependent first order term. This system is set on an bounded interval, and the first equation is controlled by a force supported in a subinterval of I or on the boundary. In the case where the intersection of the coupling and control domains is nonempty, we prove null controllability at any time. Otherwise, we provide a minimal time for null controllability. Finally we give a necessary and sufficient condition for the approximate controllability. The main technic…
In this paper, we consider a parabolic PDE on a torus of arbitrary dimension. The nonlinear term is a smooth function of polynomial growth of any degree. In this general setting, the corresponding Cauchy problem is not necessarily well posed. We show that the equation in question is approximately controllable by only a finite number of Fourier modes. This result is proved by using some ideas from the geometric control theory introduced by Agrachev and Sarychev.
We prove that the Schrödinger equation is approximately controllable in Sobolev spaces $H^s$, $s>0$ generically with respect to the potential. We give two applications of this result. First, in the case of one space dimension, combining our result with a local exact controllability property, we get the global exact controllability of the system in higher Sobolev spaces. Then we prove that the Schrödinger equation with a potential which has a random time-dependent amplitude admits at most one stationary measure on the unit sphere $S$ in $L^2 $.
In the paper, problems of controllability, approximate controllability, reachability and approximate reachability are studied for the control system $w_t=w_{xx}$, $w(0,\cdot)=u$, $x>0$, $t\in(0,T)$, where $u\in L^\infty(0,T)$ is a control. It is proved that each end state of this system is approximately reachable in a given time $T$, and each its initial state is approximately controllable in a given time $T$. A necessary and sufficient condition for reachability in a given time $T$ is obtained in terms of solvability a Markov power moment problem. It is also shown that there is no initial sta…
We prove that the thickness property is a necessary and sufficient geometric condition that ensures the (rapid) stabilization or the approximate null-controllability with uniform cost of a large class of evolution equations posed on the whole space $\mathbb R^n$. These equations are associated with operators of the form $F(\vert D_x\vert)$, the function $F:[0,+\infty)\rightarrow\mathbb R$ being continuous and bounded from below. We also provide explicit feedbacks and constants associated with these stabilization properties. The notion of thickness is known to be a necessary and sufficient cond…
We study the controllability of the Bloch equation, for an ensemble of non interacting half-spins, in a static magnetic field, with dispersion in the Larmor frequency. This system may be seen as a prototype for infinite dimensional bilinear systems with continuous spectrum, whose controllability is not well understood. We provide several mathematical answers, with discrimination between approximate and exact controllability, and between finite time or infinite time controllability: this system is not exactly controllable in finite time $T$ with bounded controls in $L^2(0,T)$, but it is approxi…
We consider a controlled Schrödinger equation with a dipolar and a polarizability term, used when the dipolar approximation is not valid. The control is the amplitude of the external electric field, it acts non linearly on the state. We extend in this infinite dimensional framework previous techniques used by Coron, Grigoriu, Lefter and Turinici for stabilization in finite dimension. We consider a highly oscillating control and prove the semi-global weak $H^2$ stabilization of the averaged system using a Lyapunov function introduced by Nersesyan. Then it is proved that the solutions of the Sch…
The paper is devoted to studying the 1D viscous Burgers equation controlled by an external force. It is assumed that the initial state is essentially bounded, with no decay condition at infinity, and the control is a trigonometric polynomial of low degree with respect to the space variable. We construct explicitly a control space of dimension 11 that enables one to steer the system to any neighbourhood of a given final state in local topologies. The proof of this result is based on an adaptation of the Agrachev-Sarychev approach to the case of an unbounded domain.
In this paper, we study the local exact boundary controllability of entropy solutions to a class linearly degenerate hyperbolic systems of conservation laws with constant multiplicity. The authors prove the two-sided boundary controllability, one-sided boundary controllability and two-sided controllability with less controls, by applying the strategy used originally for classical solutions with essential modifications. Our constructive method is based on the well-posedness of semi-global solutions constructed by the limit of $ \e $-approximate front tracking solutions to the mixed initial-boun…
Approximate null-controllability of discrete heat equations with potentials on lattices
Facultad de Ciencia y Tecnología, Universidad del País Vasco /Euskal Herriko Unibertsitatea (UPV/EHU)（巴斯克大学）；Univ. Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251（波尔多大学）；Louisiana State University（路易斯安那州立大学）
We consider the approximate control of solitons in generalized Korteweg-de Vries equations. By introducing a suitable internal bilinear control on the equation, we prove that any soliton is locally null controllable, and moreover, any soliton can be accelerated to any particular positive velocity, after a suitable large amount of time. Precise estimates on the error terms and the rate of decay in the approximate null controllability result are also given. Our method introduces a new insight on the control of nonlinear objects, from the point of view of interaction and collision problems for no…
In the paper, the problems of controllability and approximate controllability are studied for the control system $w_t=\frac{1}ρ\left(kw_x\right)_x+γw$, $\left.\left(\sqrt{\frac{k}ρ}w_x\right)\right|_{x=0}=u$, $x>0$, $t\in(0,T)$, where $u$ is a control, $u\in L^\infty(0,T)$. It is proved that each initial state of the control system is approximately controllable to any target state in a given time $T>0$. To obtain this result, the transformation operator generated by the equation data $ρ$, $k$, $γ$ is applied. The results are illustrated by examples.
