The utilization of total mass to determine the switching points in the symmetric boundary control problem with a linear reaction term

The authors study the problem ut = uxx − u, 0 < x < 1, t >0; u(x, 0) = 0, and u(0, t) = u(1, t) = ψ(t), where ψ(t) = u0 for t2k < t < t2k+1 and ψ(t) = 0 for t2k+1 < t < t2k+2, k = 0, 1, 2, . . . , with t0 = 0 and the sequence tk is determined by the equations 1 0 u(x, tk)dx = M, for k = 1, 3, 5, . . ., and 1 0 u(x, tk)dx = m, for k = 2, 4, 6, . . . and where 0 < m < M. Note that the switching points tk, k = 1, 2, 3, . . . , are unknown. Existence and uniqueness are demonstrated. Theoretical estimates of the tk and tk+1−tk are obtained and numerical verifications of the estimates are presented. The case of ux (0, t) = ux (1, t) = ψ(t) is also considered and analyzed.  2005 Published by Elsevier Inc.