A Boundary Control Problem with a Nonlinear Reaction Term
by
M. Salman
Department of Mathematics, Tuskegee University
Coauthors: John R. Cannon

The authors study the problem u_t=u_xx-au,  0 < x < 1,  t > 0;  u(x, 0)=0, and -u_x(0, t)=u_x(1, t)=f(t), where a=a(x, t, u), and f(t)=1 for t_2k < t < t_2k+1 and f(t)=0 for t_2k+1 < t < t_2k+2,  k=0, 1, 2, ... with t₀=0 and the sequence t_k is determined by the equations ∫₀¹u(x, t_k)dx = M, for k=1, 3, 5, ..., and ∫₀¹ u(x, t_k)dx = m, for k=2, 4, 6, ... and where 0 < m < M. Note that the switching points t_k,     k=1, 2, 3, ... 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 t_k and t_k+1-t_k are obtained and numerical verifications of the estimates are presented.

PDF

Date received: August 31, 2007