Reconstruction of Source Terms in Evolution Equations by Exact Controllability

For xed = (x; t), we consider the solution u(f) to u (x; t) + Au(x; t) = f(x) (x; t); x 2 ; t > 0 u(x; 0) = u(x; 0) = 0; x 2 ; Bju(x; t) = 0; x 2 @ ; t > 0; 1 j m; where u = @u @t , u = @ u @t , R, r 1 is a bounded domain with smooth boundary, A is a uniformly symmetric elliptic di erential operator of order 2m with t-independent smooth coe cients, Bj , 1 j m, are t-independent boundary di erential operators such that the system fA;Bjg1 j m is well-posed. Let fCjg1 j m be complementary boundary di erential operators of fBjg1 j m. We consider a multidimensional linear inverse problem : for given @ , T > 0 and n 2 f1; :::;mg, determine f(x), x 2 from Cju(f)(x; t), x 2 , 0 < t < T , 1 j n. By exact controllability based on the Hilbert uniqueness method, we reduce our inverse problem to an equation of the second kind which gives reconstruction of f . Moreover under extra regularity assumptions on , we can prove that this equation is a Fredholm equation of the second kind. Our methodology is widely applicable to various equations in mathematical physics. x