TY - JOUR

T1 - Regularity properties of optimal controls for problems with time-varying state and control constraints

AU - Shvartsman, Ilya A.

AU - Vinter, Richard B.

PY - 2006/7/15

Y1 - 2006/7/15

N2 - In this paper we report new results on the regularity of optimal controls for dynamic optimization problems with functional inequality state constraints, a convex time-dependent control constraint and a coercive cost function. Recently, it has been shown that the linear independence condition on active state constraints, present in the earlier literature, can be replaced by a less restrictive, positive linear independence condition, that requires linear independence merely with respect to non-negative weighting parameters, provided the control constraint set is independent of the time variable. We show that, if the control constraint set, regarded as a time-dependent multifunction, is merely Lipschitz continuous with respect to the time variable, then optimal controls can fail to be Lipschitz continuous. In these circumstances, however, a weaker Hölder continuity-like regularity property can be established. On the other hand, Lipschitz continuity of optimal controls is guaranteed for time-varying control sets under a positive linear independence hypothesis, when the control constraint sets are described, at each time, by a finite collection of functional inequalities.

AB - In this paper we report new results on the regularity of optimal controls for dynamic optimization problems with functional inequality state constraints, a convex time-dependent control constraint and a coercive cost function. Recently, it has been shown that the linear independence condition on active state constraints, present in the earlier literature, can be replaced by a less restrictive, positive linear independence condition, that requires linear independence merely with respect to non-negative weighting parameters, provided the control constraint set is independent of the time variable. We show that, if the control constraint set, regarded as a time-dependent multifunction, is merely Lipschitz continuous with respect to the time variable, then optimal controls can fail to be Lipschitz continuous. In these circumstances, however, a weaker Hölder continuity-like regularity property can be established. On the other hand, Lipschitz continuity of optimal controls is guaranteed for time-varying control sets under a positive linear independence hypothesis, when the control constraint sets are described, at each time, by a finite collection of functional inequalities.

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U2 - 10.1016/j.na.2005.07.041

DO - 10.1016/j.na.2005.07.041

M3 - Article

AN - SCOPUS:33646743755

VL - 65

SP - 448

EP - 474

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

SN - 0362-546X

IS - 2

ER -