We investigate the value function V:R+×Rn→R+∪{+∞}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 Fréchet subdifferentials of V(t, ·). Finally, when Vis locally Lipschitz, we prove a normal maximum principle together with sensitivity relations involving generalized gradients of Vfor arbitrary initial conditions. Such relations simplify drastically the investigation of the limiting behaviorat infinity of the adjoint state.

Cannarsa, P., & Frankowska, H. (2018). Value function, relaxation, and transversality conditions in infinite horizon optimal control. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 457(2), 1188-1217 [10.1016/j.jmaa.2017.02.009].

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

Cannarsa P.;
2018

Abstract

We investigate the value function V:R+×Rn→R+∪{+∞}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 Fréchet subdifferentials of V(t, ·). Finally, when Vis locally Lipschitz, we prove a normal maximum principle together with sensitivity relations involving generalized gradients of Vfor arbitrary initial conditions. Such relations simplify drastically the investigation of the limiting behaviorat infinity of the adjoint state.
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/05 - Analisi Matematica
English
Infinite horizon problem; value function; relaxation theorem; sensitivity relation; maximum principle
Cannarsa, P., & Frankowska, H. (2018). Value function, relaxation, and transversality conditions in infinite horizon optimal control. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 457(2), 1188-1217 [10.1016/j.jmaa.2017.02.009].
Cannarsa, P; Frankowska, H
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2108/191796
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