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Journal of Inverse and Ill-posed Problems

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Inverse scattering problem for the wave equation with locally perturbed centrifugal potential

M. I. Belishev1 / A. F. Vakulenko2

1Saint-Petersburg Department of the Steklov Mathematical Institute (POMI), 27 Fontanka, St-Petersburg 191023, Russia. Email:

2Saint-Petersburg Department of the Steklov Mathematical Institute (POMI), 27 Fontanka, St-Petersburg 191023, Russia. Email:

Citation Information: Journal of Inverse and Ill-posed Problems. Volume 17, Issue 2, Pages 127–157, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: 10.1515/JIIP.2009.013, March 2009

Publication History

Received:
2008-06-09
Published Online:
2009-03-30

Abstract

The forward scattering problem is to find the solution u = uf (x, t) of the dynamical system

where l ≥ 0 is an integer, fL 2(0, ∞) is a control, q is a smooth perturbation of the potential obeying sup supp q < ∞. With the problem one associates a response operator R acting by the rule

where the kernel r is a smooth function obeying sup supp r < ∞. By hyperbolicity of the wave equation, the kernel depends locally on the potential: for any ξ > 0, its part r|[2ξ,∞) is determined by q|[ξ,∞). Our setup of the inverse problem takes into account such a locality: let ξ > 0 be fixed; given r|[2ξ,∞) to determine q|[ξ,∞).

The paper proposes a procedure determining q and provides the characterization of inverse data, i.e., the necessary and sufficient conditions for a function r to be the response kernel of a dynamical system of the above-mentioned type.

The boundary control method is in use and the character of controllability of the system plays a key role. It depends on l: for l = 0 the system is locally controllable, whereas for l ≥ 1 a certain lack of controllability occurs. To recover the controllability we extend the space of controls by adding a finite dimensional subspace of polynomials. Thereafter, the standard devices of the BC-method (M-transform, visualization of waves, etc) are applied for determination of the potential.

Key words.: 1D inverse scattering; locally perturbed centrifugal potential; lack of controllability; boundary control method

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[1]
M I Belishev and V S Mikhailov
Inverse Problems, 2014, Volume 30, Number 12, Page 125013

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