Solving Hamilton-Jacobi-Bellman equations by a modified method of characteristics
Continuing previous work, the authors propose yet another numerical procedure for solving fixed time-interval optimal control problems (without end-point constraints) which consist in minimizing cost functionals of the form: \[ C(u(.))=h(x(1))+\int_{0}^{1}L(t,x(t),u(t)) dt \] subject to: \[ x'(t)=f(t,x(t),u(t)), \;u(t)\in U \subset R^m, \text{ a.e. }([0,1]), \;x(0)=y\in \Omega \subset R^n . \] Mentioning certain existing results in the theory of viscosity solutions in [\textit{M. Bardi} and \textit{I. Capuzzo-Dolcetta}, ``Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations'' (1997; Zbl 0890.49011)], the authors propose a numerical procedure for solving the ``viscosity approximating equation'': \[ \varepsilon \nabla^2v-{{\partial v}\over {\partial t}}+H(t,x,-\nabla_x v)=0, \;v(1,x)=h(x), \;x\in \Omega,\leqno (1) \] defined by the Hamiltonian of the problem: \[ H(t,x,p) =\sup_{u\in U}{\mathcal H}(t,x,p,u), \;{\mathcal H}(t,x,p,u)=\langle p,f(t,x,u)\rangle- L(t,x,u), \] \[ u^*(t,x,p) = \text{argmax}_{u\in U} {\mathcal H}(t,x,p,u) \] and to which one adds certain ``artificial'' Dirichlet and Neumann boundary conditions. The numerical algorithm proposed by the authors uses an adaptation of the ``modified method of characteristics'' in [\textit{J. Douglas jun.} and \textit{T. F. Russell}, SIAM J. Numer. Anal. 19, 871-885 (1982; Zbl 0492.65051)] for the ``convection-diffusion'' equation in (1) in which the ``marginal multifunction'' \(u^*(.,.,.)\) seems to be considered a (single-valued) function. The largest part of the paper (some 8 pages out of 15) is dedicated to the reports (Tables and Figures) of numerical experiments on 5 examples.
scientific article; zbMATH DE number 1472219
scientific article