The Global Control of Nonlinear Diffusion Equations

The boundary control of a nonlinear difussion equation with an integral performance criterion and a fixed final state is considered. By means of a process of embedding used by the author and others for finite- dimensional systems, this problem is replaced by one in which a linear form is minimized over a set of pairs of positive measures satisfying linear constraints. The advantages of this formulation are: (i) There is an automatic existence theory. (ii) There exists the possibility of using linear functional analysis to develop the theory. (iii) The minimization is global. The final state is only reached, however, in an asymptotic fashion, as the number of constraints being considered tends to infinity. A theory of controllability and reachability is developed, as well as a computational method using an infinite-dimensional simplex method. An example is given.