Application of the singular differential equations in constrained function minimization problems (Q2753000)

A new approach to the investigation of conditional extremum (maximum or minimum) of a function by use of a system of ordinary differential equations is suggested. Restrictive conditions determining the conditional extremum are presented as singular manifolds of the system of differential equations. A trajectory of such a system moves consequently from one of the stable manifold to another one. Construction of concrete systems of the abovementioned type is the main goal of the article. Two kinds of problems are considered. 1) The problem with restrictions of the equality type: NEWLINE\[NEWLINE f(x) \to \inf, \quad x\in X= \{x\in R^n, g^i (x)= 0, \quad i=1,\dots , m\} . NEWLINE\]NEWLINE 2) The problem with restrictions of the inequality type: NEWLINE\[NEWLINE f(x) \to \inf, \quad x\in X= \{x\in R^n, g^i (x)\leq 0,\quad i=1,\dots ,l\}. NEWLINE\]NEWLINE Several theorems are proved.