In this paper, we obtain a quantitative estimate of unique continuation and an observability inequality from an equidistributed set for solutions of the diffusion equation in the whole space RN. This kind of observability indicates that the total energy of solutions can be controlled by the energy localized in a measurable subset, which is equidistributed over the whole space. The proof of our results is based on an interesting reduction method [18, 22], as well as the propagation of smallness for the gradient of solutions to elliptic equations [24].
This paper presents a unique continuation estimate for 2-D Stokes equations with the Naiver slip boundary condition in a bounded and simply connected domain. Consequently, an observability estimate for this equation from a subset of positive measure in time follows from the aforementioned unique continuation estimate and the new strategy developed in [16]. Several applications of the above-mentioned observability estimate to control problems of the Stokes equations are given.
Our objective is to study the observability and unique continuation property of the adjoint of a linearized compressible fluid structure interaction model in a 2D channel. Concerning the structure we will consider a damped Euler-Bernoulli beam located on a portion of the boundary. In the present article we establish an observability inequality for the adjoint of the linearized fluid structure interaction problem under consideration which in principle is equivalent with the null controllability of the linearized system. As a corollary of the derived observability inequality we also obtain a uni…
In this paper we establish an observability inequality for the heat equation with bounded potentials on the whole space. Roughly speaking, such a kind of inequality says that the total energy of solutions can be controlled by the energy localized in a subdomain, which is equidistributed over the whole space. The proof of this inequality is mainly adapted from the parabolic frequency function method, which plays an important role in proving the unique continuation property for solutions of parabolic equations. As an immediate application, we show that the null controllability holds for the heat…
We establish a unique continuation property for stochastic heat equations evolving in a bounded domain $G$. Our result shows that the value of the solution can be determined uniquely by means of its value on an arbitrary open subdomain of $G$ at any given positive time constant. Further, when $G$ is convex and bounded, we also give a quantitative version of the unique continuation property. As applications, we get an observability estimate for stochastic heat equations, an approximate result and a null controllability result for a backward stochastic heat equation.
In this paper, we study quantitative spatial analytic bounds and unique continuation inequalities of solutions for fractional heat equations with an analytic lower order term on the whole space. At first, we show that the solution has a uniform positive analytic radius for all time, and the solution enjoys a log-type ultra-analytic bound if the coefficient is ultra-analytic. Second, we prove a Hölder type interpolation inequality on a thick set, with an explicit dependence on the analytic radius of coefficient. Finally, by the telescoping series method, we establish an observability inequality…
We consider elliptic differential operators on either the entire Euclidean space $\mathbb{R}^d$ or on subsets consisting of a cube $Λ_L$ of integer length $L$. For eigenfunctions of the operator, and more general solutions of elliptic differential quations, we derive several quantitative unique continuation results. The first result is of local nature and estimates the vanishing order of a solution. The second is a sampling result and compares the $L^2$-norm of a solution over a union of equidistributed $δ$-balls in space with the $L^2$-norm on the entire space. In the case where the space $\m…
We prove a scale-free, quantitative unique continuation principle for functions in the range of the spectral projector $χ_{(-\infty,E]}(H_L)$ of a Schrödinger operator $H_L$ on a cube of side $L\in \mathbb{N}$, with bounded potential. Such estimates are also called, depending on the context, uncertainty principles, observability estimates, or spectral inequalities. We apply it to (i) prove a Wegner estimate for random Schrödinger operators with non-linear parameter-dependence and to (ii) exhibit the dependence of the control cost on geometric model parameters for the heat equation in a multi-s…
We prove in this Note an observation estimate at one point in time for the Kolmogorov equation in the whole space. Such estimate implies the observability and the null controllability for the Kolmogorov equation with a control region which is sufficiently spread out throughout the whole space.
Fractal Turán-Nazarov Inequality and Observability for Schrödinger Equations
School of Mathematics and Statistics, Huazhong University of Science and Technology（华中科技大学数学与统计学院）；School of Mathematics (Zhuhai), Sun Yat-sen University（中山大学数学学院（珠海））
In this paper, we establish a globally quantitative estimate of unique continuation at one time point for solutions of parabolic equations with Neumann boundary conditions in bounded domains. Our proof is mainly based on Carleman commutator estimates and a global frequency function argument, which is motivated from a recent work [5]. As an application, we obtain an observability inequality from measurable sets in time for all solutions of the above equations.
In this article, we give a completely constructive proof of the observability/controllability of the wave equation on a compact manifold under optimal geometric conditions. This contrasts with the original proof of Bardos-Lebeau-Rauch, which contains two non-constructive arguments. Our method is based on the Dehman-Lebeau Egorov approach to treat the high-frequencies, and the optimal unique continuation stability result of the authors for the low-frequencies. As an application, we first give estimates of the blowup of the observability constant when the time tends to the limit geometric contro…
We develop a new abstract derivation of the observability inequalities at two points in time for Schrödinger type equations. Our approach consists of two steps. In the first step we prove a Nazarov type uncertainty principle associated with a non-negative self-adjoint operator $H$ on $L^2(\mathbb{R}^n)$. In the second step we use results on asymptotic behavior of $e^{-itH}$, in particular, minimal velocity estimates introduced by Sigal and Soffer. Such observability inequalities are closely related to unique continuation problems as well as controllability for the Schrödinger equation.
We study uncertainty principles for function classes on the torus. The classes are defined in terms of spectral subspaces of the energy or the momentum, respectively. In our main theorems, the support of the Fourier transform of the considered functions is allowed to be supported in a (finite number of) parallelepipeds. The estimates we obtain do not depend on the size of the torus and the position of the parallelepipeds, but only on their size and number, and the density and scale of the observability set. Our results are on the one hand closely related to unique continuation for linear combi…
We prove a quantitative unique continuation principle for infinite dimensional spectral subspaces of Schrödinger operators. Let $Λ_L = (-L/2,L/2)^d$ and $H_L = -Δ_L + V_L$ be a Schrödinger operator on $L^2 (Λ_L)$ with a bounded potential $V_L : Λ_L \to \mathbb{R}^d$ and Dirichlet, Neumann, or periodic boundary conditions. Our main result is of the type \[ \int_{Λ_L} \lvert ϕ\rvert^2 \leq C_{\mathrm{sfuc}} \int_{W_δ(L)} \lvert ϕ\rvert^2, \] where $ϕ$ is an infinite complex linear combination of eigenfunctions of $H_L$ with exponentially decaying coefficients, $W_δ(L)$ is some union of equidistr…
We consider the quantitative uniqueness properties for a parabolic type equation $ u_t-Δu = w(x,t) \nabla u + v(x,t) u$, when $v \in L^{p_2}_{t} L^{p_1}_x$ and $w \in L^{q_2}_{t} L^{q_1}_x$, with a suitable range for exponents $p_1$, $p_2$, $q_1$, and $q_2$. We prove a strong unique continuation property and provide a pointwise in time observability estimate.
In this paper, we study the simultaneous controllability of wave systems in an open domain of R d , d $\in$ N *. We obtain a partial controllability result on a co-finite dimensional space for wave equations coupled by a single control function. We use microlocal defect measures and the unique continuation property of eigenfunctions to prove that an appropriate observability inequality holds for wave equations with space varying and different speeds coupled by a single control function. For the unique continuation property of eigenfunctions, we construct a counterexample to show that in some m…
In this paper we study the inverse problem of identifying a source or an initial state in a time-fractional diffusion equation from the knowledge of a single boundary measurement. We derive logarithmic stability estimates for both inversions. These results show that the ill-posedness increases exponentially when the fractional derivative order tends to zero, while it exponentially decreases when the regularity of the source or the initial state becomes larger. The stability estimate concerning the problem of recovering the initial state can be considered as a weak observability inequality in c…
This paper is concerned with the decay estimate of solutions to the semilinear wave equation subject to two localized dampings in a bounded domain. The first one is of the nonlinear Kelvin-Voigt type and is distributed around a neighborhood of the boundary according to the Geometric Control Condition. While the second one is a frictional damping and we consider it hurting the geometric condition of control. We show uniform decay rate results of the corresponding energy for all initial data taken in bounded sets of finite energy phase-space. The proof is based on obtaining an observability ineq…
