Necessity of limiting co-state arcs in Bolza-type infinite horizon problem

Abstract

We investigate necessary conditions of optimality for the Bolza-type infinite horizon problem with free right end. The optimality is understood in the sense of weakly uniformly overtaking optimal control. No previous knowledge in the asymptotic behaviour of trajectories or adjoint variables is necessary. Following Seierstads idea, we obtain the necessary boundary condition at infinity in the form of a transversality condition for the maximum principle. Those transversality conditions may be expressed in the integral form through an Aseev-Kryazhimskii-type formulae for co-state arcs. The connection between these formulae and limiting gradients of payoff function at infinity is identified; several conditions under which it is possible to explicitly specify the co-state arc through those Aseev-Kryazhimskii-type formulae are found. For infinite horizon problem of Bolza type, an example is given to clarify the use of the Aseev-Kryazhimskii formula as explicit expression of the co-state arc.Keywords: Optimal control; Problem of Bolza type; Infinite horizon problem; transversality condition for infinity; Uniformly overtaking optimal control; Limiting subdifferential; Unbounded Cost; Shadow prices 49K15; 49J52; 91B62The first necessary conditions of optimality for infinite-horizon control problems were proved [28] on the verge of 1950-60s by L.S. Pontryagin and his associates (for the problems with the right end fixed at infinity). Only later [19] was the Maximum Principle proved for a reasonably broad class of problems, and yet the transversality-type conditions at infinity were not provided. A significant number [19,21,27,34,32] of such conditions was proposed. Thus, the Maximum Principle for infinite horizon was not complete, and the set of extremals obtained through it was too broad; see [14,19,27,33], [2, Sect. 6], [30, Example 10.2].The principal obstacle on the way to transversality conditions at infinity is the fact that it is necessary to find the asymptotic conditions on the adjoint equation (i.e., on the linear system) that would be satisfied by at least one but not by all of its solutions. It was first done in [6] for linear autonomous control system through passing to a functional space that allowed to extend 1

On Hamiltonian as Limiting Gradient in Infinite Horizon Problem

“…Assumptions under which this expression is a necessary condition of optimality that are relatively easy to check may be found in [2,3,16]. This formula may not point towards a solution of the PMP even if the integral converges in the Lebesgue sense, see [17]. For details on the other (the more general formulas), see [16].…”

“…For a similar analysis of the Aseev-Kryazhimskii formula, refer to [2]. Here, the Michel condition is used along with some limiting solution of the Pontryagin maximum principle (see [17]); the limiting solution may be considered without assumptions on the asymptotic behaviour of trajectories or adjoint variables. The idea of the limiting solution can be traced to paper [24]; see its connection with the Aseev-Kryazhimskii formula in [16].The general case of Bolza-type infinite horizon problem with free right end was studied in [17].…”

“…Here, the Michel condition is used along with some limiting solution of the Pontryagin maximum principle (see [17]); the limiting solution may be considered without assumptions on the asymptotic behaviour of trajectories or adjoint variables. The idea of the limiting solution can be traced to paper [24]; see its connection with the Aseev-Kryazhimskii formula in [16].The general case of Bolza-type infinite horizon problem with free right end was studied in [17]. In this paper, we prove the existence of a limiting solution of PMP that satisfies the Michel condition for uniformly overtaking optimal control; the arising transversality conditions are expressed in the form of limiting gradients of payoff function at infinity.…”

“…In this paper, we prove the existence of a limiting solution of PMP that satisfies the Michel condition for uniformly overtaking optimal control; the arising transversality conditions are expressed in the form of limiting gradients of payoff function at infinity. The proof itself combines the ideas from [19] with the proof of the Pontryagin Maximum Principle from [17].The paper is structured as follows. First we describe the problem statement, impose the general conditions and propositions; at the same section, we provide the required definitions from the smooth analysis.…”

“…The general case of Bolza-type infinite horizon problem with free right end was studied in [17]. In this paper, we prove the existence of a limiting solution of PMP that satisfies the Michel condition for uniformly overtaking optimal control; the arising transversality conditions are expressed in the form of limiting gradients of payoff function at infinity.…”