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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.

##### 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