## Optimal Control of One-Dimensional Partial
Differential Equations Applied to Transdermal
Diffusion of Substrates

M. Blatt, K. Schittkowski: Optimization Techniques and Applications, L. Caccetta, K.L. Teo, P.F. Siew,
Y.H. Leung, L.S. Jennings, V. Rehbock eds.,
School of Mathematics and Statistics, Curtin University of Technology, Perth,
Australia, Volume 1, 81 - 93 (1998)

**Abstract:**
We present an approach to compute optimal control functions in dynamic
models based on
one-dimensional partial differential equations.
By using the method of lines, the PDE is transformed into a large system of usually
stiff ordinary differential equations and integrated by standard methods.
The resulting nonlinear programming problem is solved by the
sequential quadratic programming code NLPQL.
Optimal control functions are approximated by piecewise constant, piecewise linear
or bang-bang functions.
Three different types of cost functions may be formulated.
The underlying PDE structure is quite flexible. We allow break points for
model changes, disjoint integration areas subject to spatial variable,
arbitrary boundary and transition conditions, coupled ordinary and algebraic
differential equations, partial algebraic equations,
and dynamic constraints for control and state functions.
The PDE is discretized by difference formulae, polynomial approximations with
arbitray degrees, and by special update formulae in case of hyperbolic
equations.
An application problem is outlined in detail. We present a
model for transdermal diffusion of drugs, where the diffusion speed is controlled
by an electric field.