Construction of a set of differerntial equations systems and stability in the vicinity of a program manifold

The problem of constructing a entire set of differential equation systems for a given program manifold is addressed. Necessary and sufficient conditions have been compiled to ensure that the program manifold is integral to the systems being developed. These conditions form a rectangular linear algebraic system in relation to the required functions. Using the property of a rectangularmatrix, the set of differential equation systems was constructed. Additionally, the problem of designing indirect automatic control systems with rigid feedback is explored. Since the given program is not always executed perfectly due to initial or ongoing disturbances, it is reasonable to require stability of the program manifold with respect to a certain function. This leads to the analysis of the stability of the system of equations in relation to the given program manifold. Through the construction of Lyapunov functions for the system in canonical form, sufficient conditions for the absolute stability of the program manifold are derived. These results can be applied to the design of stable automatic indirect control systems.