On Trajectories Satisfying a State Constraint: $W^{1,1}$ Estimates and Counterexamples

On relations of the adjoint state to the value function for optimal control problems with state constraints

“…For F merely continuous with respect to time, this last condition is not sufficient for having NFT estimates (see [4] for a counter-example). Earlier counter-examples to NFT estimates with F depending only on time in a discontinuous way can be found in [3]. It became clear then that a different inward pointing condition should be found.…”

Hölder estimates for trajectories of differential inclusions and HJB equations with state constraints

“…In this case, Bressan and Facchi [6] have shown that, if F does not depend on t and is Lipschitz continuous in the space variable x, roughly the same conclusions as in [14] are still valid. In [3][4][5] the case when F depends on t is analyzed. In particular, in [5] it is shown that there exist dynamics F, measurable with respect to t and Lipschitz in x, Ω ⊂ R n and an F -trajectory x * , with x * (0) ∈ Ω and 0 < max t∈ [0,T ] dist(x * (t), Ω) .…”

Trajectories of differential inclusions with state constraints

“…In this case, (1.9) is equivalent to (1.6). Estimates such as (1.8)-(1.9) play a key role in determining the regularity of the value function, for optimal control problems with state constraints [3][4][5]11,12,15,14,16].…”