DETERMINATION OF A CONTROL PARAMETER IN A PARABOLIC PARTIAL-DIFFERENTIAL EQUATION

The authors consider in this paper the inverse problem of finding a pair of functions (u, p) such that u(t) = u(xx) + pu + F(x, t), 0 < x < 1, 0 < t less-than-or-equal-to T, u(x, 0) = f(x), 0 less-than-or-equal-to x less-than-or-equal-to 1, alpha-1(t)u(x)(0, t) + beta-1(t)u(0, t) + gamma-1(t)u(1, t) = g1(t), 0 < t less-than-or-equal-to T, alpha-2(t)u(x)(1, t) + beta-2(t)u(0, t) + gamma-2(t)u(1, t) = g2(t), 0 < t less-than-or-equal-to T, integral 0s(t)u(x, t) dx = E(t), 0 less-than-or-equal-to t, less-than-or-equal-to T, 0 < s(t) less-than-or-equal-to 1. where F, f, E, s, alpha-i, beta-i, gamma-i, g(i), i = 1, 2, are given functions. The existence and uniqueness of a smooth global solution pair (u, p) which depends continuously upon the data are demonstrated under certain assumptions on the data.