Conditions implying the vanishing of the Hamiltonian at infinity in optimal control problems

Abstract

In an infinite horizon optimal control problem the Hamiltonian vanishes at infinity when the differential equation is autonomous and the integrand in the criterion satisfies some weak integrability conditions. A generalization of Michel's result (in Econometrica 50:975-985, 1982) is obtained.Keywords Optimal control · Infinite horizon ∞ 0 e rt g(x, u) dt, r a given real number. In problems with one state variable, this condition, together with the other conditions of the maximum principle, often determine a unique solution candidate. The main purpose of this note is to generalize 1 There is a misprint in the theorem. Please replace the inequality b > k by b > (n − m)k.

On Hamiltonian as Limiting Gradient in Infinite Horizon Problem

“…Since ( 23b ) and ( 25 ) imply ρ(α * , α n , τ n ) < γ n < 1/2 < dist( y n (0), bd S), and y n → y * , γ n → 0 as n → ∞ , we know that Lemma 2 guarantees κ( y n (τ n ), τ n ) → y * (0).…”

“…In [31], the necessity was proved without assuming the dynamics to be smooth; in [15], it was studied in the calculus of variations setting; see [20] for infinite horizon control problem with state constraints; in [25] it was proved for the general statement, including the problems with fixed right end; in [2], under sufficiently weak assumptions on the summability, the connection of this condition with the Aseev-Kryazhimskii formula was studied. The assumptions used in this paper could not be embedded into assumptions of the above-mentioned papers; in particular, in contrast with [2,15,20,25], here, as well as in [19], the case of λ * = 0 is not generally excluded.Note that the Michel condition, if convenient, is only one-dimensional and, therefore, this condition, together with the core conditions of the maximum principle, can determine a unique solution candidate only for the problems with one state variable. In view of that, it is important to know not only when this condition is necessary but also when it is consistent with other transversality conditions.…”

Another view of the maximum principle for infinite-horizon optimal control problems in economics

“…The complementary character of condition (9) is demonstrated in [9,Example 6.6]. A generalization of Michel's result to the case when the instantaneous utility f 0 (•, •, •) depends on the variable t in more general way was developed in [58], using a slightly modified argument. A normal form version of the maximum principle with adjoint variable ψ(•) having all positive components which involves condition (25) was developed also in [46] under some monotonicity type assumptions.…”

Infinite Horizon Optimal Control of Non-Convex Problems Under State Constraints

“…The above setting subsumes the classical infinite horizon optimal control problem when f and U are time independent, L(t, x, u) = e −λt (x, u) for some mapping : R n × R m → R + and λ > 0, t 0 = 0. Infinite horizon problems exhibit many phenomena not arising in the context of finite horizon problems and their study is still going on, even in the absence of state constraints, see [1,2,3,4,5,6,33,36,37,38,40,43,44] and their bibliographies. Among such phenomena let us recall that already in 70ies Halkin, see [32] and also [36], observed that in the necessary optimality conditions for an infinite horizon problem it may happen that the co-state of the maximum principle is different from zero at infinity and that only abnormal maximum principles hold true (even for problems without state constraints).…”