Maximum principle in problems with mixed constraints under weak assumptions of regularity

Abstract

In the present work, optimal control problems with mixed constraints are investigated. A novel weakening of the conventional regularity assumptions on mixed constraints is introduced. A maximum principle is derived in which the maximum condition is of nonstandard type: the maximum is taken over the closure of the set of regular points, but not over the whole feasible set.

Keywords:

• optimal control
• maximum principle
• mixed constraints

AMS Subject Classification::

• 49J15

Acknowledgements

We are thankful for the referee's useful remarks. The research was supported by RFBR (Russia), projects 09-01-00619, by the grant of the President of the Russian Federation MK-119.2009.1 by FCT (Portugal), projects PTDC/EEA-ACR/75242/2006, SFRH/BPD/26231/2006 by the grant of the President of the Russian Federation MK-119.2009.1.

Notes

Notes

1. Say that the RCQ holds at a point (x, u) ∈ 𝒳 × 𝒰 such that G(x, u) ∈ C _𝒴 with respect to the mapping G(x, ·) and closed convex set C _𝒴 provided

where G _u is Freshet derivative.

2. Here span {y} = {αy, α ∈ ℝ¹} denotes the line generated by a vector. Note that the inclusion in (13) may be strict only when .