General scheme for constructing the determining function in a control problem for a dynamical system with partial derivatives of different orders

For a control system with partial derivatives, a criterion for the complete controllability is derived by using the cascade decomposition method based on the transition from the original system to reduced systems in subspaces. We obtain a function, which belongs to a subspace of minimal dimension and determines the type of solution of the program control problem, i.e., the state and control functions in the analytical form. Necessary and sufficient conditions for the existence of the determining function are established and a scheme of its construction is given. Necessary and sufficient conditions for the existence of a determining function in the polynomial, exponential, and fractional-rational forms are found; formulas for constructing such functions are proposed. For the original system, a solution of the program control problem is constructed.