skip to main content

Title: Improved Sensitivity Relations in State Constrained Optimal Control

Sensitivity relations in optimal control provide an interpretation of the costate trajectory and the Hamiltonian, evaluated along an optimal trajectory, in terms of gradients of the value function. While sensitivity relations are a straightforward consequence of standard transversality conditions for state constraint free optimal control problems formulated in terms of control-dependent differential equations with smooth data, their verification for problems with either pathwise state constraints, nonsmooth data, or for problems where the dynamic constraint takes the form of a differential inclusion, requires careful analysis. In this paper we establish validity of both ‘full’ and ‘partial’ sensitivity relations for an adjoint state of the maximum principle, for optimal control problems with pathwise state constraints, where the underlying control system is described by a differential inclusion. The partial sensitivity relation interprets the costate in terms of partial Clarke subgradients of the value function with respect to the state variable, while the full sensitivity relation interprets the couple, comprising the costate and Hamiltonian, as the Clarke subgradient of the value function with respect to both time and state variables. These relations are distinct because, for nonsmooth data, the partial Clarke subdifferential does not coincide with the projection of the (full) Clarke subdifferential onmore » the relevant coordinate space. We show for the first time (even for problems without state constraints) that a costate trajectory can be chosen to satisfy the partial and full sensitivity relations simultaneously. The partial sensitivity relation in this paper is new for state constraint problems, while the full sensitivity relation improves on earlier results in the literature (for optimal control problems formulated in terms of Lipschitz continuous multifunctions), because a less restrictive inward pointing hypothesis is invoked in the proof, and because it is validated for a stronger set of necessary conditions.« less
Authors:
 [1] ;  [2] ;  [3]
  1. Université de Bretagne Occidentale, Laboratoire de Mathematiques (France)
  2. Université Pierre et Marie Curie (Paris 6), CNRS and Institut de Mathématiques de Jussieu (France)
  3. Imperial College London, Department of Electrical and Electronic Engineering (United Kingdom)
Publication Date:
OSTI Identifier:
22469993
Resource Type:
Journal Article
Resource Relation:
Journal Name: Applied Mathematics and Optimization; Journal Volume: 71; Journal Issue: 2; Other Information: Copyright (c) 2015 Springer Science+Business Media New York; http://www.springer-ny.com; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CONTROL SYSTEMS; COORDINATES; DIFFERENTIAL EQUATIONS; FUNCTIONS; HAMILTONIANS; LIMITING VALUES; MATHEMATICAL SPACE; OPTIMAL CONTROL; SENSITIVITY