I know that Hamilton- Jacobi equation (HJ) is a partial differential equation that has several applications in different fields, including optimal control. Currently, I am interested to use some open source HJ numerical solvers in order to compute reachable sets of a certain non-linear dynamical system. I found an open source solver that solves for reachability and optimal control based on HJ equation. Now, I'd like to know that procedure of formulating certain non-linear dynamical system in terms of HJ equation. Any help is highly appreciated.
NOTE: In case you are interested, the solver can be found here:
http://itn-sadco.inria.fr/software/ROC-HJ
Thanks.
In the simplest case, if you have a Hamiltonian dynamical system with n degrees of freedom having the Hamiltonian H=H(q_1,..,q_n,p_1,..,p_n), the Hamilton-Jacobi equation is simply S_t+H (q_1,..,q_n,S_{q_1},..,S_{q_n})=0. Here S_t and S_{q_j} denote partial derivatives of the unknown function S=S(t,q_1,...,q_n) with respect to the variables t and q_j respectively. Once you know a complete integral of the Hamilton--Jacobi equation, i.e., roughly speaking, a solution of the Hamilton--Jacobi equation that depends on n arbitrary constants in an essential fashion, you can obtain the general solutions for the equations of motion of your original system. The details can be found in many advanced books on classical mechanics, see e.g. E. DiBenedetto, Classical Mechanics, Springer, 2011 (http://link.springer.com/book/10.1007/978-0-8176-4648-6) or G. Goldstein, Classical Mechanics (3rd ed.). Addison Wesley, 2002.
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