Abstract
Partial and full sensitivity relations are obtained for nonauto-nomous optimal control problems with infinite horizon subject to state constraints, assuming the associated value function to be locally Lipschitz in the state. Sufficient structural conditions are given to ensure such a Lipschitz regularity in presence of a positive discount factor, as it is typical of macroeconomics models.
Highlights
Assume (H1), let g : Rn → R be a locally Lipschitz continuous function and consider the problem M (g, τ ) with τ > 0
Consider the infinite horizon optimal control problem B∞ ∞minimize L(t, x(t), u(t)) dt (1)t0 over all the trajectory-control pairs subject to the state constrained control system x (t) = f (t, x(t), u(t)) a.e. t ∈ [t0, ∞) x(t0) = x0 (2)u(t) ∈ U (t) x(t) ∈ A t ∈ [t0, ∞)where f : [0, ∞) × Rn × Rm → Rn and L : [0, ∞) × Rn × Rm → R are given, A is a nonempty closed subset of Rn, U : [0, ∞) ⇒ Rm is a Lebesgue measurable set valued map with closed nonempty images and (t0, x0) ∈ [0, ∞) × A is the initial datum
Every trajectory-control pair (x(·), u(·)) that satisfies the state constrained control system (2) is called feasible. We refer to such x(·) as a feasible trajectory
Summary
Assume (H1), let g : Rn → R be a locally Lipschitz continuous function and consider the problem M (g, τ ) with τ > 0. A family G of R-valued functions defined on E ⊂ Rk is uniformly locally Lipschitz continuous on E if for all R 0 there exists LR 0 such that
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