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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. Thetransversality condition for adjoint variable is shown to become necessary ifthe adjoint variable is partially Lyapunov stable. The modifications of thiscondition are proposed for the case of unbounded adjoint variable. TheCauchy-type formula for the adjoint variable proposed by S. M. Aseev and A. V.Kryazhimskii is shown to complement relations of the Pontryagin MaximumPrinciple up to the complete set of necessary conditions of optimality if theimproper integral in the formula converges conditionally and continuouslydepends on the original position. The results are extended to an unboundedobjective functional (described by a non- convergent improper integral),unbounded constraint on the control, and uniformly sporadically catching upoptimal control.

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