The Euler-Lagrange and Hamiltonian Inclusion Conditions in the Presence of State Constraints

This chapter concerns necessary conditions of optimality for dynamic optimization problems with pathwise state constraints, when the dynamic constraint takes the form of a differential inclusion. Here, the pathwise constraint is expressed as a time-dependent scalar functional inequality constraint. Through redefinition of the state constraint function, which is merely required to be upper semicontinuous and Lipschitz continuous w.r.t. the state variable, we can subsume within this framework other formulations of the pathwise state constraint (vector inequality constraints, combined equality/inequality constraints, set inclusion constraints, etc.) We have seen in Chap. 10 , which concerned necessary conditions for path-constrained dynamic optimization problems involving controlled differential inclusions, how it is not possible, in general, to accommodate the state constraint by means of an absolutely continuous Lagrange multiplier; instead we must introduce a measure multiplier. The same is true when, as in this chapter, we substitute a differential inclusion for a controlled differential equation in the problem formulation. Once again, the necessary conditions involve a costate trajectory that is discontinuous, though we hide this fact, expressing the conditions in terms of a modified, absolutely continuous, co-state trajectory, obtained by subtracting off the singular component from the ‘true’ co-state trajectory. The necessary conditions for state constrained differential inclusion problems appearing in this chapter resemble the generalized Euler Lagrange condition of Chap. 8 , but now including a measure multiplier associated with the state constraint. They are derived under unrestrictive conditions that allow the velocity sets to be non-convex and unbounded. Under an additional hypothesis that the velocity sets are convex, we can prove, with the help of the duality theorem relating Euler Lagrange and Hamiltonian conditions, also a Hamiltonian version of the condition. The generalized Euler Lagrange condition for state constrained problems of this chapter incorporates the stratified conditions of Clarke and the Ioffe refinement. The proof technique mimics that used in Chap. 8 , where the full necessary conditions were built up in stages, starting with an application of the maximum principle to a simple version of the problem. The difference is that, now, we use the state-constrained maximum principle in the first stage. We also address the degeneracy issue associated with necessary conditions for pure state constraint problems. We showed in Chap. 10 that, for state constrained problems involving controlled differential equations, standard necessary conditions are sometimes degenerate when the state constraint is active at either end-time. The same is true when the dynamic constraint takes the form of a differential inclusion. This chapter also provides non-degenerate necessary conditions, under extra hypotheses, covering these situations. Formulations of the state constrained differential inclusion problem, in which the end-times are included among the choice variables, are also considered. The extra necessary conditions associated with the free end-times take, as usual, the form of boundary conditions on the Hamiltonian evaluated along the minimizing state and costate trajectories.