Abstract
Conditions are investigated for the coincidence of the programmed maximin and the value of a positional differential game /1–5/. In connection with this the infinitesimal form of the stability property, derived in /6/, is used. It is shown that under the fulfillment of the well-known regularity conditions /3–5/ the programmed maximin function is directionally differentiable and the stability property holds for it. The paper's main results are the necessary and sufficient condition for the coincidence of the programmed maximin and the positional differential game's value when the controlled system's right-hand side is differentiable with respect to the phase variable, and a sufficient regularity condition when the controlled system's right-hand side satisfies a Lipschitz condition in the phase variable. These conditions generalize the previously known /3–5/ regularity conditions. The paper abuts the investigations in /1–10/.
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