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

This paper investigates the necessary conditions of optimality for uniformly overtaking optimal control on infinite horizon with free right endpoint. Clarke’s form of the Pontryagin Maximum Principle is proved without the assumption 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 in [1],[2] 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 nonconvergent improper integral), unbounded constraint on the control, and uniformly sporadically catching up optimal control.