A New Look at the Weierstrass Condition in Optimal Control

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 [2]. 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 [8], 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.