"Solutions to the Hamilton-Jacobi Equation with Algebraic Gradients"
Prof. Toshiyuki Ohtsuka (Osaka University)
http://www-sc.sys.es.osaka-u.ac.jp/~ohtsuka/Abstract:
The Hamilton-Jacobi equation (HJE) is a fundamental equation in the analysis and control of nonlinear systems. However, the HJE cannot be solved explicitly in general. In this talk, the HJE with coefficients consisting of rational 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 ideal satisfying some conditions exists in a polynomial ring over the rational function field. If an appropriate ideal is found, the gradient of the solution is defined implicitly by a set of algebraic equations. Then, the gradient is determined by solving the set of algebraic equations pointwise without storing the solution over a domain in the state space. Thus, the so-called curse of dimensionality can be removed when a solution to the HJE with an algebraic gradient exists. A new class of explicit solutions for a nonlinear optimal regulator problem is given as an application of the present approach.
