Agrachev, A. A.; Gamkrelidze, R. V. The index of extremality and quasiextremal controls. (English. Russian original) Zbl 0587.49018 Sov. Math., Dokl. 32, 478-481 (1985); translation from Dokl. Akad. Nauk SSSR 284, 777-781 (1985). We consider the extremal problem for a functional \(\phi_ 0:Z\to {\mathbb{R}}\) under constraints \(\phi_ i(0)=0\) for \(i=1,...,m\). Let \(z_ 0\in Z\) and \(\ell <0\); we assume that the index of extremality at \(z_ 0\) is greater than \(\ell\) if the point \(z_ 0\) can be made extremal on adding (-\(\ell)\) new constraints in a ”stable manner” (stability here meaning that if the new constraints are changed slightly, \(z_ 0\) remains extremal). Second, suppose that \(z_ 0\in Z\) is an extremal point, and \(0\leq k\leq m\); we assume that the index of extremality at \(z_ 0\) is greater than k if k of the constraints can be omitted in a ”stable manner” while retaining extremality of \(z_ 0.\) We shall actually use a more geometric approach, in which the functional is not considered separately from the constraints: instead of treating a functional \(\phi_ 0\) and constraints \(\phi_ 1,...,\phi_ m\) we shall consider the vector-valued function \(\Phi =(\phi_ 0,\phi_ 1,...,\phi_ m)^ T\), and extremal values will be the boundary points of the image im \(\Phi\). The concept of extremality index is then modified appropriately. Further, we shall not treat quite arbitrary mappings \(\Phi\), but restrict ourselves to control systems. The quasiextremality index of a given control is the largest extremality index at the corresponding ”point” that can be achieved by an arbitrarily small change of the system. Cited in 7 Documents MSC: 49J99 Existence theories in calculus of variations and optimal control 58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals 93C10 Nonlinear systems in control theory 93C15 Control/observation systems governed by ordinary differential equations 34H05 Control problems involving ordinary differential equations Keywords:index of extremality; quasiextremality PDFBibTeX XMLCite \textit{A. A. Agrachev} and \textit{R. V. Gamkrelidze}, Sov. Math., Dokl. 32, 478--481 (1985; Zbl 0587.49018); translation from Dokl. Akad. Nauk SSSR 284, 777--781 (1985)