Algebraic Solutions to the Hamilton-Jacobi Equation with the Time-Varying Hamiltonian

Abstract

The Hamilton-Jacobi equation (HJE) with the time-varying Hamiltonian 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 HJE with coefficients belonging to meromorphic functions 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 zero-dimensional radical ideal exists in a polynomial ring over the meromorphic functions of the time and the state. If such an ideal is found, an algebraic gradient can be obtained simply by solving a set of algebraic equations.