A New Look at the Weierstrass Condition in Optimal Control
Abstract
First order necessary conditions for optimal
control problems in which the dynamic constraint is
modeled as a differential inclusion have been know for
many years. A definitive version is provided by Clarke’s
2005 Memoirs. They key ingredients are the generalized
Euler Lagrange inclusion (replacing the costate
equation of classical optimal control), the transversality
condition and the Weierstrass (or Hamiltonian maximization
property) condition. Clarke and de Pinho’s
2010 paper [3] provided an important application of
this theory, using it as a starting point to derive necessary
conditions satisfied by minimizers for problems,
having a ‘controlled differential equation’ formulation
and involving mixed state/control constraints. In his 2019
paper, Ioffe added a refinement to the Weierstrass
condition, identifying a larger set of controls over which
the Hamiltonian must be maximized. In this paper,
we explore, through new theory and examples, the
significance of this refinement. We derive new necessary
conditions for mixed constraint problems involving
controlled differential equations, via a reduction to a
differential inclusion problem, that, for the first time,
incorporate Ioffe’s refinement. Two examples, concerning
differential inclusion problems and controlled differential
equations problems with mixed constraints, are presented
that show how the extra tests present in the refined
Weierstrass conditions can be used to identify extremals,
i.e. processes satisfying earlier necessary conditions, that
are not minimizers.