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

Abstract

We present a recently developed complete version of the Pontryagin maximum principle for a class of infinite-horizon optimal control problems arising in economics. The peculiarity of the result is that the adjoint variable is explicitly specified by a formula which resembles the Cauchy formula for solutions of linear differential systems. In certain situations this formula implies the "standard" transversality conditions at infinity. Moreover, it can serve as their alternative. We provide examples demonstrating the advantage of the suggested version of the maximum principle. In particular, we consider its applications to Halkin's example, to Ramsey's optimal growth model and to a basic model of optimal extraction of a non-renewable resource. An economic interpretation of the developed characterization of the adjoint variable is also presented.

Optimal R&D investment with learning‐by‐doing: Multiple steady states and thresholds

“…Using q (i, t) = 1/(r + ) gives g(i, t) = (i)/(r + ) and, thus, Equation (15). The first 2 relations together with n (t) = u(t) yield Equation (13).…”

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

“…Another way to deal with this issue is to modify the very definition of optimal solution, cf. [3,4,17,32,46]. However, the notions like overtaking (or weakly overtaking) optimal controls do not have appropriate existence theory.…”

“…In the present paper, we consider the nonautonomous infinite horizon optimal control problem (1) V (t 0 , x 0 ) = sup x (t) = f (t, x(t), u(t)), u(t) ∈ U (t) for a.e. t ≥ t 0 x(t 0 ) = x 0 , satisfying the state constraint (3) x(t) ∈ K for all t ≥ t 0 ,…”

“…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).…”

On Necessary Optimality Conditions for Ramsey-Type Problems

“…Note that Arrow's sufficiency theorem contains a condition that follows from (1.4), which one can expect for problems, where the maximum principle provides both necessary and sufficient conditions of optimality. In [10][11][12][13][14], the authors determine the adjoint variable uniquely by a Cauchy-type formula, that solves the adjoint equation with transversality conditions in the following form:…”