Necessary conditions for optimal control problems with state constraints

Abstract

Abstract. Necessary conditions of optimality are derived for optimal control problems with pathwise state constraints, in which the dynamic constraint is modelled as a differential inclusion. The novel feature of the conditions is the unrestrictive nature of the hypotheses under which these conditions are shown to be valid. An Euler Lagrange type condition is obtained for problems where the multifunction associated with the dynamic constraint has values possibly unbounded, nonconvex sets and satisfies a mild 'one-sided' Lipschitz continuity hypothesis. We recover as a special case the sharpest available necessary conditions for state constraint free problems proved in a recent paper by Ioffe. For problems where the multifunction is convex valued it is shown that the necessary conditions are still valid when the one-sided Lipschitz hypothesis is replaced by a milder, local hypothesis. A recent 'dualization' theorem permits us to infer a strengthened form of the Hamiltonian inclusion from the Euler Lagrange condition. The necessary conditions for state constrained problems with convex valued multifunctions are derived under hypotheses on the dynamics which are significantly weaker than those invoked by Loewen and Rockafellar to achieve related necessary conditions for state constrained problems, and improve on available results in certain respects even when specialized to the state constraint free case.Proofs make use of recent 'decoupling' ideas of the authors, which reduce the optimization problem to one to which Pontryagin's maximum principle is applicable, and a refined penalization technique to deal with the dynamic constraint.

Participation constraints in discontinuous adverse selection models

“…Section 2 presents our main result: A theorem that characterizes solutions to optimal control problems in which the objective function is only required to be upper semicontinuous. This theorem builds on earlier work by Vinter and Zheng (1998), but refines its application to the case of quasilinear objectives, which is prevalent in contract theory. While Vinter and Zheng (1998) focus on necessary conditions for optimality, we prove that these conditions are also sufficient in our context.…”

“…This theorem builds on earlier work by Vinter and Zheng (1998), but refines its application to the case of quasilinear objectives, which is prevalent in contract theory. While Vinter and Zheng (1998) focus on necessary conditions for optimality, we prove that these conditions are also sufficient in our context. We also discuss to what extent this theorem extends the existing literature and especially the work by Jullien (2000) under much weaker conditions.…”

“…Preliminaries for Nonsmooth Analysis. We draw heavily from Vinter and Zheng (1998) in the following presentation. A complete treatment can be found in the monograph of Vinter (2000).…”

“…To illustrate, provided that is absolutely continuous, one can add a new control a.e., and impose the constraint . While this introduces multiple dimensions to the state space, we note that our analysis can readily be extended to multidimensional settings because Theorem 3 from Vinter and Zheng (1998), used in Appendix B in our one‐dimensional setting, applies to multidimensional state variables. That said, this approach has limits.…”

“…2222 We choose to state the theorem using as an arbitrary normalization rather than , which is the normalization chosen in Vinter and Zheng (1998). Later, by setting , we will succeed in normalizing μ to a probability measure, which is a more familiar object. …”