## Abstract

The forward scattering problem is to find the solution *u* = *u ^{f}* (

*x, t*) of the dynamical system

where *l* ≥ 0 is an integer, *f* ∈ *L*
_{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.

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