Numerical Solution of the Variational PDEs arising in optimal control theory

VICENTE COSTANZA; MARÍA INÉS TROPAREVSKY; PABLO S. RIVADENEIRA

Rosario

Congreso; RPIC 2009 - XIII Reunión de Trabajo para el procesamiento de la Información y Control; 2009

Universidad de Rosario - RPIC

The recently discovered variational PDEs for the missing boundaryconditions in Hamilton equations of optimal control are solved by a newconvergent iterative method similar to Picard's approach to ODEs'initial-value problems. \ It is shown that this new method actually appliesto first-order quasilinear PDEs with matrix-valued unknowns. \ Asillustrations the iterative numerical solutions to the PDEs are checkedagainst: (i) the analytical solutions to examples in dimension one andhigher,\ some arising from optimal control problems for nonlinear systemsand regular Lagrangians, and (ii) the numerical solution obtained fromstraightforward use of standard mathematical software. \ It is speciallydiscussed an application to the $(n+1)$-dimensional variational PDEsassociated with the $n$-dimensional finite-horizon time-variantlinear-quadratic problem, due to the key role the LQR plays intwo-degrees-of freedom control strategies for nonlinear systems withgeneralized costs.