Volume 12, Issue 2

In this paper we study the inverse problem for the acoustic equation. The boundary control method (BC-method) and Gel'fand—Levitan—Krein method (GLK-method) are investigated and compared. In Section 1 inverse acoustic problem is formulated. In Section 2 we consider the scheme of BC-method. In Section 3 the multidimensional analog of Gel'fand—Levitan—Krein equation is obtained for inverse acoustic problem. In Section 4 we investigate discrete analogs of main relations of both methods. In Section 5 we consider the numerical results.

The study is motivated by the problem of stabilizing the concentration of atmospheric carbon, which is widely discussed in the context of global warming nowadays. A key difficulty in the design of stabilization strategies is the uncertainty of the underlying physical model. In the present paper, a general problem setting is suggested and a relevant alanytic framework elaborated. Analysis employs specific qualitative features of an uncertain dynamics, including automatic stabilization of the trajectories in the absence of input disturbances. An asymptotic version of Krasovskii's extremal shift control principle is developed and model-robust strategies stabilizing a state coordinate at a prescribed level are constructed.

We consider a general non-conservative Schrödinger equation defined on an open bounded domain Ω in , with C 2 -boundary subject to (Dirichlet and, as a main focus, to) Neumann boundary conditions on the entire boundary Γ. Here, Γ 0 and Γ 1 are the unobserved (or uncontrolled) and observed (or controlled) parts of the boundary, respectively, both being relatively open in Γ. The Schrödinger equation includes energy-level ( H 1 (Ω)-level) terms, which accordingly may be viewed as unbounded perturbations. The first goal of the paper is to provide Carleman-type inequalities at the H 1 -level, which do not contain lower-order terms; this is a distinguishing feature over most of the literature. This goal is accomplished in a few steps: the paper obtains first pointwise Carleman estimates for C 2 -solutions; and next, it turns these pointwise estimates into integral-type Carleman estimates with no lower-order terms, originally for H 2 -solutions, and ultimately for H 1 -solutions. The passage from H 2 - to H 1 -solutions is readily accomplished in the case of Dirichlet B.C., but it requires a delicate regularization argument in the case of Neumann B.C. This is so since finite energy solutions are known to have L 2 -normal traces in the case of Dirichlet B.C., but by contrast do not produce H 1 -traces in the case of Neumann B.C. From Carleman-type inequalities with no lower-order terms, one then obtains the sought-after benefits. These consist of deducing, in one shot, as a part of the same flow of arguments, two important implications: (i) global uniqueness results for H 1 -solutions satisfying over-determined boundary conditions, and—above all—(ii) continuous observability (or stabilization) inequalities with an explicit constant. The more demanding purely Neumann boundary conditions requires the same geometrical conditions on the triple {Ω,Γ 0 ,Γ 1 } that arise in the corresponding problems for second-order hyperbolic equations. The most general result, with weakest geometrical conditions, is, in fact, deferred to Section 9. Sections 1 through 8 provide the main body of our treatment with one vector field under a preliminary working geometrical condition, which is then removed in Section 9, by use of two suitable vector fields. The second and final goal of this paper is to shift the Carleman estimates (Hence, the continuous observability/stabilization inequalities) by one unit downward to the lower L 2 (Ω)-level. This is accomplished in Section 10 by means of pseudo-differential analysis, and accordingly, it contains lower-order terms. Applications of these L 2 (Ω)-Carleman estimate includes a new uniform stabilization of the conservative Schrödinger equation in the state space L 2 (Ω), by an attractive boundary feedback.