Journal of Mathematical Sciences, Vol. 102, No. 4, 2000 T H E D Y N A M I C A L I N V E R S E P R O B L E M F OR A N O N - S E L F - A D J O I N T S T U R M - L I O U V I L L E O P E R A T O R S. A. A v d o n i n , M. I. Belishev, a n d Yu. S. R o z h k o v UDC 517.946 An approach to the inverse problem (the so-called BC-method) based on boundary-control theory is developed. A procedure of reconstructing a nonsymmetric matrix-function (a potential) given on a semiaxis by a d response operator is described. The results of numerical tests are presented. Bibliography: 6 titles. O. INTRODUCTION In the present paper, an approach toinverse problems (the so-called BC-method) based on bound- ary control theory [1, 2] is developed. It also gives a new interpretation of the local approach due to A. S. Blagoveshenskii [3]. The BC-method for a non-self-adjoint Sturm-Liouville operator is stated in (see also [6]). In the present paper, a version of this method most suitable for numerical realization is considered. The results of numerical experiments are discussed. 1. THE DIRECT PROBLEM. THE BOUNDARY-CONTROL PROBLEM 1.1. T h e d i r e c t p r o b l e m Let V(x), x > O, be a real N x N matrix-function with continuously differentiable elements. Consider the initial boundary-value problem (Problem 1) ~ , ~ - ~ = + v ( x ) ~ = o , (x,t) e a + x (0,T), T > 0 , (1) ~(~,0) = ~,(~,0) =0, (2) u(O, t) = f(t). (3) The solution of this problem is a vector-function u - uf(x, t) with values in R n. Sometimes, when using physical terminology, we call V, f, and u f a potential, a control, and a wave, respectively. Let a matrix-function w(x, t) be a solution of the Goursat problem wtt - wx~ + V ( x ) w = O, w(0,t) =0, O < x < t < T , w(~,~) = - ~ V(s)ds. (4) It is known that w(x, t) is twice continuously differentiable in the domain {(x, t) : 0 <_ x < t < T}. The following statement is easily verified. P r o p o s i t i o n 1.1. (a) If f e C2([O,T];R N) and f(O) = if(O) = O, then Problem 1 has a unique classical solution u = u f (x, t).In this case, the representation { /' J ( x , t ) = : ( t - ~) + w ( x , s ) f ( t - s)ds, for ~ < t, 0, for x > t, (5) is valid. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 7-21. Original article submitted 16, 1997. 1072-3374/00/1024-4139 $25.00 (~)2000 Kluwer Academic/Plenum Publishers 4139