Abstract
Necessary conditions for optimality are proved for smooth infinite horizon optimal control problems with unilateral state constraints (pathwise constraints) and with terminal conditions on the states at the infinite horizon. The aim of the paper is to obtain strong necessary conditions including transversality conditions at infinity, which in many cases lead to a set of candidates for optimality containing only a few elements, similar to what is the case in finite horizon problems. However, strong growth conditions are needed for the results to hold.
Highlights
The aim of this paper is, in a control problem with unilateral state constraints and terminal conditions at infinity, to obtain necessary conditions, with a full set of transversality conditions at infinity, which frequently make it possible to narrow down the set of candidates for optimality to only a few, or sometimes a single one
The limited types of transversality conditions mentioned are in problems with several
With strong growth conditions there exist necessary conditions, with a full set of transversality conditions at infinity, which in many cases make it possible to narrow down the set of candidates to only a few, or sometimes a single one, see Theorem 16, p. 2441 in [5]
Summary
The aim of this paper is, in a control problem with unilateral state constraints and terminal conditions at infinity, to obtain necessary conditions, with a full set of transversality conditions at infinity, which frequently make it possible to narrow down the set of candidates for optimality to only a few, or sometimes a single one. For Theorem 2 to hold, we can weaken (7) and the basic assumptions on f i x and g j x as follows: the derivatives g j x and f i x exist at t, x∗ (t) for all t and the three conditions on g j below are satisfied: For all N > 0. In addition to p0 ∈{0,1}, p∞ , and μ j ≥ 0 satisfying (20), (23), and (24), j ≤ j∗, there exist bounded nonnegative finitely additive set functions μ j , j > j∗, vanishing on Lebesgue null sets, such that (22) holds for u ∈U (t ) , summing over j = 1, , j∗∗. In this case, for a.e. t, for all u ∈U ( )( ) ( ) Hu t, x∗ (t ),u∗ (t ), p (t ) u − u∗ (t ) ∑ + j> j∗guj (t ) μ j (t ) u − u∗ (t ) ≤ 0
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