Equivalent Differential Equations in Problems of Control Theory and the Theory of Hamiltonian Systems

Abstract

New approaches are proposed in the problem of constructing equivalent scalar differential equations for multidimensional nonlinear systems of control theory, as well as in the problem of constructing equivalent Hamiltonian systems for nonlinear Lurie equations (scalar differential equations containing derivatives of only even orders). The conditions for the solvability of the corresponding problems are studied, and new formulas for the transition to equivalent equations and systems are proposed. For the Lurie equations, the proposed approaches are based on the transition from the linear part to the normal forms of the corresponding Hamiltonian systems with a subsequent transformation of the resulting system. Calculation formulas and algorithms are obtained, and their efficiency is illustrated by examples.