The Generalized Euler-Lagrange and Hamiltonian Inclusion Conditions

The subject matter of this chapter is necessary conditions of optimality for differential inclusion problems, by which we mean dynamic optimization problems in which the dynamic constraint takes the form of a differential inclusion. We provide two sets of conditions. These are the generalized Euler Lagrange condition and a later refinement of a set of conditions known as Clarke’s Hamiltonian inclusion. As their names imply, these conditions result from reformulating the differential inclusion problem as a generalized Bolza problem in the calculus of variations, in which indicator functions taking account of the dynamic constraint appear in an extended valued Lagrangian, and deriving analogues of the similarly named classical conditions, valid in this more general setting. The conditions, expressed in terms of limiting subgradients and limiting normal cones, are inherently non-smooth. We bring fully up to date the generalized Euler Lagrange condition, which has been significantly improved, since the time it was first derived in the 1980s under a convexity hypothesis on velocity sets. The generalized Euler Lagrange condition covered in this chapter, for problems with non-convex, unbounded velocity sets, builds on Clarke’s stratified conditions. It also incorporates a recent improvement in the Weierstrass condition, which we refer to as the Ioffe refinement. The first necessary condition to be proved for differential inclusion problems with convex velocity sets, was Clarke’s Hamiltonian inclusion. This chapter includes a subsequent refinement of this condition, the so-called partially convexified Hamiltonian inclusion. It is shown, by means of a duality theorem linking Euler Lagrange and Hamiltonian inclusions, that the partially convexified Hamiltonian inclusion is not an independent condition, but is in fact a consequence of the generalized Euler Lagrange condition. Questions regarding validity of the Hamiltonian inclusion, for differential inclusion problems with possibly non-convex velocity sets, was the subject of speculation for many years. Current understanding of these issues is conveyed in this chapter. Clarke’s (fully convexified) Hamiltonian inclusion for problems with non-convex velocity sets is now known to hold, at local minimizers w.r.t. certain topologies. But the condition may not be valid for local minimizers w.r.t. other topologies.