The transversality conditions in infinite horizon problems and the stability of adjoint variable

TL;DR

This paper advances the understanding of necessary optimality conditions in infinite horizon control problems, especially regarding the transversality condition and the stability of the adjoint variable, without requiring bounded variation.

Contribution

It proves Clarke's Pontryagin Maximum Principle without assuming bounded total variation of the adjoint, and extends conditions for unbounded adjoint variables and objectives.

Findings

Transversality condition becomes necessary under Lyapunov stability.

Modified conditions for unbounded adjoint variables are proposed.

Cauchy-type formula complements the maximum principle with convergence conditions.

Abstract

This paper investigates the necessary conditions of optimality for uni- formly overtaking optimal control on infinite horizon with free right endpoint. Clarke's form of the Pontryagin Maximum Principle is proved without the as- sumption on boundedness of total variation of adjoint variable. The transversality condition for adjoint variable is shown to become necessary if the adjoint variable is partially Lyapunov stable. The modifications of this condition are proposed for the case of unbounded adjoint variable. The Cauchy-type formula for the adjoint variable proposed by S. M. Aseev and A. V. Kryazhimskii is shown to complement relations of the Pontryagin Maximum Principle up to the complete set of necessary conditions of optimality if the improper integral in the formula converges conditionally and continuously depends on the original position. The results are extended to an unbounded objective functional (described by a non- convergent improper integral), unbounded constraint on the control, and uniformly sporadically catching up optimal control.