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.

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 |