On relations of the adjoint state to the value function for optimal control problems with state constraints

Abstract

Abstract.We consider an optimal control problem under state constraints and show that to every optimal solution corresponds an adjoint state satisfying the first order necessary optimality conditions in the form of a maximum principle and sensitivity relations involving the value function. Such sensitivity relations were recently investigated by P. Bettiol and R.B. Vinter for state constraints with smooth boundary. In the difference with their work, our setting concerns differential inclusions and nonsmooth state constraints. To obtain our result we derive neighboring feasible trajectory estimates using a novel generalization of the so-called inward pointing condition.Mathematics Subject Classification (2010). Primary 49K15, 49Q12.

Improved Sensitivity Relations in State Constrained Optimal Control

“…to be merely measurable with respect to the time variable, however.) We also observe that the sensitivity relations in this paper are established with reference to a stronger set of necessary conditions (namely the partially convexified Euler Lagrange condition) than those featuring in [9] (namely the Hamiltonian inclusion). The novel idea in our proofs is to replace the Mayer optimal control problem by a Bolza problem with three additional state variables (see problem (Q ) in Section 4).…”

“…2.2 was previously derived in [9] under the modified inward pointing condition (IPC) . (See also [2] for such a relation, in the context of state constrained optimal control problems involving control dependent differential equations, when (IPC) is strengthened to require A to have smooth boundary.)…”

“…(See also [2] for such a relation, in the context of state constrained optimal control problems involving control dependent differential equations, when (IPC) is strengthened to require A to have smooth boundary.) Thm 2.1 improves on [9] because, in addition to adjoining the partial sensitivity relation, it asserts the co-state trajectory can be chosen to satisfy the partially convexified Euler Lagrange in place of the weaker Hamiltonian inclusion. (See comment (e).)…”

“…Recently, in [2] and [9] (in a controlled-dependent differential equation and a differential inclusion setting respectively), analogous sensitivity relations have been derived for optimal control problems with state constraints, involving velocity sets which are measurable with respect to t and Lipschitz with respect to x. Now, they take the form…”

“…or, more generally, F (., . )'s which are absolutely continuous from the left with respect to the time variable), by invoking, for the class of control systems considered, a less restrictive inward pointing condition than that of [9], and not requiring A to be a set with smooth boundary, as in [2]. ( [2] and [9] permit F (., .)…”