Value function, relaxation, and transversality conditions in infinite horizon optimal control

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:457

Article

Cannarsa, P.¹  Frankowska, H.² 

[1] Univ Roma Tor Vergata, Dipartimento Matemat, Via Ric Sci 1, I-00133 Rome, Italy

[2] UPMC Univ Paris 06, Sorbonne Univ, Inst Math Jussieu Paris Rive Gauche, CNRS,UMR 7586, Case 247,4 Pl Jussieu, F-75252 Paris, France

关键词: Infinite horizon problem;    Value function;    Relaxation theorem;    Sensitivity relation;    Maximum principle;   

DOI  :  10.1016/j.jmaa.2017.02.009

来源: Elsevier

【 摘 要 】

We investigate the value function V : R+ x R-n -> R+ boolean OR {+infinity} of the infinite horizon problem in optimal control for a general not necessarily discounted running cost and provide sufficient conditions for its lower semicontinuity, continuity, and local Lipschitz regularity. Then we use the continuity of V(t,.) to prove a relaxation theorem and to write the first order necessary optimality conditions in the form of a, possibly abnormal, maximum principle whose transversality condition uses limiting/horizontal supergradients of V(0,.) at the initial point. When V(0,.) is merely lower semicontinuous, then for a dense subset of initial conditions we obtain a normal maximum principle augmented by sensitivity relations involving the Frechet subdifferentia1s of V(t,.). Finally, when V is locally Lipschitz, we prove a normal maximum principle together with sensitivity relations involving generalized gradients of V for arbitrary initial conditions. Such relations simplify drastically the investigation of the limiting behavior at infinity of the adjoint state. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

Free