A new method based on Picard’s approach to Ordinary Differential Equations (ODEs) initial-value problems is proposed to solve first-order quasilinear Partial Differential Equations (PDEs) with matrix-valued unknowns, in particular, the recently discovered variational Partial Differential Equations for the missing boundary values in Hamilton equations of optimal control. The application is based on (n+1)- dimensional Partial Differential Equations associated with the n-dimensional finite-horizon time-variant linear-quadratic problem, the LQR plays key role in freedom control strategies at two-degrees- for nonlinear systems with generalized costs.
The proposed methodology is based on Picard’s iterations, and is able to provide accurate solutions to the problems.The fact is that, the Picard’s method always converges in the ODE context (assuming the existence of a unique solution and precise calculations at each iteration step) was used to prove convergence of the scheme for the Partial Differential Equations under study.
This type of convergence gives a guarantee that the accuracy of final results can be improved with respect to non-iterative methods. Analytical bounds for the rate of convergence have also been checked in a case-study whose explicit solution is known.This method has potential applications in the hyper-sensitivity of Hamilton Canonical Equations (HCEs), whose boundary values are obtained from the variational Partial Differential Equations solutions, and therefore need we need highly accurate results.
The post Variational Partial Differential Equations Arising in Optimal Control Theory appeared first on Science Blog.