Linear estimates for trajectories of state-constrained differential inclusions and normality conditions in optimal control

We consider solutions to a differential inclusion (x) over circle is an element of F(x) constrained to a compact convex set Q. Here F is a compact, possibly non-convex valued, Lipschitz continuous multifunction, whose convex closure co F satisfies a strict inward pointing condition at every boundary point x epsilon partial derivative ohm. Given a reference trajectory x*(.) taking values in an E-neighborhood of,12, we prove the existence of a second (approximating) trajectory x : [0, 7] 1 ->ohm which satisfies the linear estimate parallel to x(.) x*(.)parallel to Ac([0,T]),<= K epsilon, if one of the following two cases occurs: (i) the initial datum x(0) = xo is given, but lies in a compact set containing only points where the boundary partial derivative ohm is smooth; (ii) the initial point x(0) E 12 of the approximating trajectory x(.) can be chosen arbitrarily. Subsequently we employ these linear AC-estimates to establish conditions for normality of the generalized Euler Lagrange condition for optimal control problems with state constraints, in which we have an integral term in the cost. We finally provide an illustrative example which underlines the fact that, if conditions (i) and (ii) above are not satisfied, then we can find a degenerate minimizer. (c) 2014 Elsevier Inc. All rights reserved.