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A geometric approach is pursued
that analyzes the structure of a local field of extremals around a reference trajectory
with boundary arcs (an arc of the trajectory or its graph on the state
constraint) if the state constraints are control invariant submanifolds of
relative degree 1. In this case, following Maurer's reasoning, it can
be shown that the measures associated with the state constraints are
absolutely continuous with respect to Lebesgue measure and formulas for the
associated Radon-Nikodym derivative are easily computed.
Strengthened versions of the necessary conditions for optimality guarantee
the local embedding of the reference trajectory into a local field of
extremals. The local embedding of a boundary arc clarifies the roles of the
measures and provides the link between boundary arcs and the trajectories
away from the constraint. It is different from the classical local imbeddings
for unconstrained problems in the sense that, because of the presence of the
constraint, this local field necessarily contains small pieces of
trajectories which when propagated backward are not close to the reference trajectory.
This, however, does not effect the memoryless properties required for a
synthesis forward in time and strong local optimality of the reference
trajectory follows from the existence of the field. In the papers below both
concatenations between boundary arcs and bang arcs (corresponding to constant
controls) and singular arcs are considered in a nondegenerate setting.
In the proof of optimality a local solution to the Hamiton-Jacobi-Bellman
equation is constructed by adapting the classical method of characteristics
to the optimal control problem. Then perturbation arguments are used to
handle the state-space constraints and points where the value is not
differentiable.
Presentations:
These results were presented at a series of three lectures at an RTN (research training network) workshop on "Evolution Equations" at
the Technical University
in Vienna, Austria, June 5 - 8, 2006 under the title
“Optimal
Control for Systems with Order 1 State Space Constraints”
Part 1: The Maximum Principle and an
Application to a Problem in Electronics
Part 2: Local Fields of
Bang-Boundary-Bang Extremals
Part 3: Sufficiency of the
Construction
Selected Recent
Publications:
- A
local field of extremals near boundary-arc interior-arc junctions, Proceedings of the 44th IEEE Conference on
Decision and Control (CDC), Sevilla,
Spain,
December 2005, pp. 945-950
- U. Ledzewicz
and H.Sch., A local field of extremals for single-input systems near
state constraints of relative degree 1, Proceedings
of the 43rd IEEE Conference on Decision and Control (CDC),
Nassau, The Bahamas, December 2004,
pp. 923-928
Motivation for this research
comes from the problem of minimizing the base transit time in semiconductor
devices that was studied jointly with P Rinaldi.
- P.
Rinaldi and H. Sch., On optimal control problems with state space
constraints arising in the design of bipolar transistors, Proceedings of the 41st IEEE Conference on
Decision and Control (CDC), Las Vegas, Nevada, December 2002, pp. 4722-4727
Background Information on the Method of Charateristics in Optimal Control
Theory:
- John
Noble and H. Sch., Sufficient conditions for relative minima of broken
extremals in optimal control theory, J.
of Mathematical Analysis and Applications, 269, 2002, pp. 98-128
- Matthew
Kiefer and H. Sch., Parametrized families of extremals and singularities
in solutions to the Hamilton-Jacobi-Bellman equation, SIAM J. on Control and Optimization, 37,
No. 5, 1999, pp. 1346-1371
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