Stability, reconstruction formula and regularization for an inverse source hyperbolic problem by a control method

Let u(f) be the solution to a hyperbolic equation in a bounded domain Omega subset R': u"(x, t) = Delta u(x,t) + sigma (t)f(x) (x in Omega, 0 < t < T) u(x,0) = 0    u'(x,0) = 0 (x in Omega) u(x,t) = 0 (x in partial Omega, 0 < t < T). We assume that sigma in C¹[0,T] is a known function, sigma(0) not= 0, and f in L²(Omega) is unknown, and Gamma subset partial Omega is given. We consider an inverse problem of determining f(x)(x in Omega) from [partial u(f)/partial n](x,t)(x in Gamma, 0 < t < T). For a sufficiently large T > 0, we will show the stability estimate of ||f||_L²(Omega) by ||partial u(f)/partial n||_{H¹(0,T;L²(Gamma))}, a reconstruction formula of f from partial u(f)/partial n and a Tikhonov regularization. Our methodology is based on exact boundary controllability and a Volterra integral equation of the first kind with kernel sigma.