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

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