A BOUNDARY CONTROL PROBLEM WITH A NONLINEAR REACTION TERM

The authors study the problem ut = uxx au, 0 < x < 1, t > 0; u(x,0) = 0, and ux(0,t) = ux(1,t) = (t), where a = a(x,t,u), and (t) = 1 for t2k < t < t2k+1 and (t) = 0 for t2k+1 < t < t2k+2, k = 0,1,2,... with t0 = 0 and the sequence tk is determined by the equations R 1 0 u(x,tk)dx = M, for k = 1,3,5,..., and R 1 0 u(x,tk)dx = m, for k = 2,4,6,..., where 0 < m < M. Note that the switching points tk, are unknown. A maximum principal argument has been used to prove that the solution is positive under certain conditions. Existence and uniqueness are demonstrated. Theoretical estimates of the tk and tk+1 tk are obtained and numerical verifications of the estimates are presented.