Today, the next seminar of the Department of Optimal Control was held. Professor Telman Malikov, Senior Scientific Researcher of the “Optimal Control” Department, talked about “Necessary conditions for an extremum in nonsmooth problems of the calculus of variations”.
The paper proposes two approaches to determine strong and weak extrema in two nonsmooth variation problems in a vector form: in the simplest problem of calculus of variations and in the variational problem with higher derivatives. The essence of the proposed approaches consists of introducing various special variations: (a) Weierstrass variations in a new modified form; (b) variations expressed by Legendre polynomials. As a result, necessary conditions for strong and weak optimalities are obtained. These conditions give rise to obtaining the Weierstrass and Legendre conditions. In the case that the Legendre condition degenerates, the necessary conditions of the form equality and inequality for a weak local minimum are obtained.
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