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

We consider solutions to a differential inclusion x ˙ ∈ F ( x ) constrained to a compact convex set Ω. 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 ∈ ∂ Ω . Given a reference trajectory x ⁎ ( . ) taking values in an ε-neighborhood of Ω, we prove the existence of a second (approximating) trajectory x : [ 0 , T ] ↦ Ω which satisfies the linear estimate ‖ x ( . ) − x ⁎ ( . ) ‖ AC ( [ 0 , T ] ) ⩽ K ε , if one of the following two cases occurs: (i) the initial datum x ( 0 ) = x 0 is given, but lies in a compact set containing only points where the boundary ∂Ω is smooth; (ii) the initial point x ( 0 ) ∈ Ω 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.