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We analyze some open questions that are associated with the
Direct Transcription (direct simultaneous or all-at-once)
approach for the solution of optimal control and parameter
estimation problems involving Differential-Algebraic
Equations (DAEs). Direct transcription refers to the
procedure of approximating the infinite dimensional problem
by a finite-dimensional one, which is then solved using a
Nonlinear Programming (NLP) solver tailored to solve
large-scale problems. Systems governed by partial
differential equations (PDEs) will also be considered. The
PDEs will be spatially discretized to convert them to a
system of DAEs. We focus particularly on the following
topics:
* Convergence rates for direct transcription of a
class of optimal control problems
* The role of controllability in the
convergence analysis
* Advantages of NLP-based methods for inequality path-constrained
optimal control problems
Directions for future work and some initial results will also
be discussed, including mathematical regularization for
singular optimal control problems and the use of MPECs for
modeling conditional statements with applications in
two-phase flow through porous media and optimal control.
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