An algebraic solution method for the unsteady Hamilton-Jacobi equation

The unsteady Hamilton-Jacobi equation (HJE) plays an important role in the analysis and control of nonlinear systems and is very difficult to solve for general nonlinear systems. In this paper, the unsteady HJE for a Hamiltonian with coefficients belonging to meromorphic functions of time and rational functions of the state is considered, and its solutions with algebraic gradients are characterized in terms of commutative algebra. It is shown that there exists a solution with an algebraic gradient if and only if an H-invariant and involutive maximal ideal exists in a polynomial ring over the meromorphic functions of time and the rational functions of the state. If such an ideal is found, an algebraic gradient can be obtained by only solving a set of algebraic equations.