## Optimal Control of One-Dimensional Partial
Differential Algebraic Equations with Applications

M. Blatt, K. Schittkowski: Annals of Operations Research, Vol. 98, 45-64 (2000)

**Abstract:**
We present an approach to compute optimal control functions in dynamic
models based on
one-dimensional partial differential algebraic equations.
By using the method of lines, the PDAE is transformed into a large system of usually
stiff ordinary differential algebraic 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 can be formulated.
The underlying model structure is quite flexible. We allow break points for
model changes, disjoint integration areas w.r.t. spatial variable,
arbitrary boundary and transition conditions, coupled ordinary and algebraic
differential equations, algebraic equations in time and space variables,
and dynamic constraints for control and state variables.
The PDAE is discretized by difference formulae, polynomial approximations with
arbitray degrees, and by special update formulae in case of hyperbolic
equations.

Two application problems are outlined in detail. We present a
model for optimal control of transdermal diffusion of drugs, where the
diffusion speed is controlled by an electric field, and a model
for the optimal control of the input feed of an acetylene reactor
given in form of a distributed parameter system.