Sergei A. Avdonin

Solving the dynamical inverse problem for the Schrödinger equation by the boundary control method

We consider the inverse problem of determining the potential in the one-dimensional Schrodinger equation from dynamical boundary observations, which are the range values of the Neumann-to-Dirichlet map. Dynamical boundary data have not been used in the inverse problem for the Schrodinger equation, since the traditional Gelfand–Levitan–Marchenko approach reconstructs the potential from spectral or scattering data. Here we show that one can completely recover the spectral data from the dynamical boundary data. The construction of the spectral data uses new results on exact and spectral controllability for the Schrodinger equation, which we obtain by using the properties of exponential Riesz bases (nonharmonic Fourier series). From the spectral data, we solve the inverse problem using the boundary control method, which—unlike other identification methods based on control and optimization—is consistently linear and, in principle, independent of dimensionality.

Determining the potential in the Schrödinger equation from the Dirichlet to Neumann map by the boundary control method

We consider the problem of identifying the potential in the one-dimensional Schrödinger equation with input Dirichlet data, from measured Neumann data. Knowledge of the Dirichlet to Neumann map together with spectral controllability results for the Schrödinger equation obtained using properties of exponential Riesz bases allow recovery of the spectral data. Once the the spectral data is recovered, we use the Boundary Control method to solve the identification problem.

Iterative Methods for Solving a Nonlinear Boundary Inverse Problem in Glaciology

Abstract We address a Cauchy problem for a nonlinear elliptic PDE arising in glaciology. After recasting the Cauchy problem as an ill-posed operator equation, we prove (for values of a certain parameter allowing Hilbert space techniques) differentiability properties of the associated operator. We also suggest iterative methods which can be applied to solve the operator problem.

The boundary control approach to inverse spectral theory

We establish connections between four approaches to inverse spectral problems: the classical Gelfand–Levitan theory, the Simon theory, the approach proposed by Remling, and the Boundary Control method. We show that the Boundary Control approach provides simple and physically motivated proofs of the central results of other theories. We demonstrate also the connections between the dynamical and spectral data and derive the local version of the classical Gelfand–Levitan equations. In this paper we consider the Schrödinger operator (0.1) H = −∂2 x + q (x) on L (R+) ,R+ := [0,∞), with a real-valued locally integrable potential q and Dirichlet boundary condition at x = 0. Let dρ(λ) be the spectral measure corresponding to H, and m(z) be the (principal or Dirichlet) Titchmarsh-Weyl mfunction. 1. Three approaches to inverse spectral theory In this section we give a brief review of three different approaches to inverse problems for the operator (0.1): the Gelfand–Levitan theory, the Simon theory and the Remling approach. In the next section we describe the Boundary Control method and its connections with the other approaches. 1.1. Gelfand–Levitan theory. Determining the potential q from the spectral measure is the main result of the seminal paper by Gelfand and Levitan [16]. To formulate the result let us define the following functions: σ(λ) = { ρ(λ)− 2 3π λ 3 2 , λ > 0, ρ(λ), λ < 0 (1.1) F (x, t) = ∫ ∞ −∞ sin √ λx sin √ λt λ dσ(λ). (1.2) Let φ(x, λ) be a solution to the equation −φ′′ + q(x)φ = λφ, x > 0, (1.3) 1991 Mathematics Subject Classification. 34B20, 34E05, 34L25, 34E40, 47B20, 81Q10.

Construction of Sampling and Interpolating Sequences for Multi-Band Signals. The Two-Band Case

Construction of Sampling and Interpolating Sequences for Multi-Band Signals. The Two-Band Case Recently several papers have related the production of sampling and interpolating sequences for multi-band signals to the solution of certain kinds of Wiener-Hopf equations. Our approach is based on connections between exponential Riesz bases and the controllability of distributed parameter systems. For the case of two-band signals we derive an operator whose invertibility is equivalent to the existence of a sampling and interpolating sequence, and prove the invertibility of this operator.

Temperature and heat flux dependence/independence for heat equations with memory

We present and extend our recent results on the relations between temperature and flux for heat equations with memory. The key observation is that we can interpret “independence” as a kind of “controllability” and this suggests the study of controllability of the pair heat-flux in an appropriate functional space.

Boundary Control approach to the spectral estimation problem: the case of multiple poles

There exist many methods for solving the spectral estimation problem. This paper proposes a new approach to this problem based on the Boundary Control method. We show that the problem of decomposition of a signal modeled by a sum of exponentials with polynomial coefficients can be reduced to an identification problem for a discrete time linear dynamical system. It follows that values of exponentials can be found solving a generalized eigenvalue problem as in the Matrix Pencil method. We also give exact formulas for the polynomial amplitudes.

Dynamical inverse problem on a metric tree

We consider the problem of reconstruction of the potential for the wave equation on a specific star graph using the dynamical Dirichlet-to-Neumann map. Our algorithm is based on the boundary control method. We reduce the problem of reconstruction to a second kind Fredholm integral equation, the kernel and the right-hand side of which arise from an auxiliary second kind Volterra integral equation. A second-order accurate numerical method for the equations is described and implemented. Then several numerical examples demonstrate that the algorithm works and can be used to reconstruct an unknown potential accurately.

Controllability in a filled domain for the multidimensional wave equation with a singular boundary control

In accordance with the demands of the so-called local approach to inverse problems, the set of “waves” uf (·, T) is studied, where uf (x,t) is the solution of the initial boundary-value problem utt−Δu=0 in Ω×(0,T), u|t<0=0, u|∂Ω×(0,T)=f, and the (singular) control f runs over the class L2((0,T); H−m (∂Ω)) (m>0). The following result is established. Let ΩT={x ∈ Ω : dist(x, ∂Ω)<T)} be a subdomain of Ω ⊂ ℝn (diam Ω<∞) filled with waves by a final instant of time t=T, let T*=inf{T : ΩT=Ω} be the time of filling the whole domain Ω. We introduce the notation Dm=Dom((−Δ)m/2), where (−Δ) is the Laplace operator, Dom(−Δ)=H2(Ω)∩H01(Ω);D−m=(Dm)′;D−m(ΩT)={y∈D−m:supp y ⋐ ΩT. If T<T., then the reachable set RmT={ut(·, T): f ∈ L2((0,T), H−m (∂Ω))} (∀m>0), which is dense in D−m(ΩT), does not contain the class C0∞(ΩT). Examples of a ∈ C0∞, a ∈ RmT, are presented.

Boundary control method and coefficient identification in the presence of boundary dissipation

We apply the boundary control method to the identification of coefficients in a wave equation with dissipative boundary conditions. This problem is suggested by the closed loop obtained when a stabilizing feedback is applied to a wave equation.