Transformation  of degenerate indirect control systems in the vicinity of a program manifold

We consider one of the classes of implicit differential systems, systems of ordinary differential equations that are not resolved with respect to the highest derivative. Such equations are often found in everyday life in mechanics, physics, economics, biology, etc. The problems of constructing automatic control systems according to a given smooth program manifold also come down to such equations. This is the case when the dimension of the system of equations under construction is greater than the dimension of the program manifold. Then systems of algebraic equations with a rectangular matrix arise. We consider a system with a square matrix,the discriminant of which is zero. The general problem of constructing systems of differential equations for a given manifold is considered.The necessary and sufficient condition is drawn that the manifold is integral to the system of equations. The Yerugin function is linear with respect to the manifold. Then an indirect control system is built, taking into account that a given manifold is integral to it under certain conditions. In general, the Jacobi matrix is rectangular. The case is investigated when the matrix is quadratic and has zero roots. The manifold is assumed to be linear with respect to the desired variable. A degenerate indirect control system is obtained, unresolved with respect to the highest derivative. equivalent to a certain system is established, the matrices of which are constant and have a special structure. Lyapunov transformation matrices are found. It is shown that the considered control systems can be reduced to a central canonical form. A brief overview is provided.