CONICET | Buscador de Institutos y Recursos Humanos

Partial differential equations for the unknown final state and initial costatearising in the Hamiltonian formulation of regular optimal control problems with aquadratic final penalty are found. It is shown that the missing boundary conditionsfor Hamilton?s canonical ordinary differential equations satisfy a system of firstorderquasilinear vector partial differential equations (PDEs), when the functionaldependence of the H-optimal control in phase-space variables is explicitly known.Their solutions are computed in the context of nonlinear systems with Rn-valuedstates. No special restrictions are imposed on the form of the Lagrangian cost term.Having calculated the initial values of the costates, the optimal control can then beconstructed from on-line integration of the corresponding 2n-dimensional Hamiltonordinary differential equations (ODEs). The off-line procedure requires finding twoauxiliary n × n matrices that generalize those appearing in the solution of the differentialRiccati equation (DRE) associated with the linear-quadratic regulator (LQR)problem. In all equations, the independent variables are the finite time-horizon durationT and the final-penalty matrix coefficient S, so their solutions give informationon a whole two-parameter family of control problems, which can be used for designpurposes. The mathematical treatment takes advantage from the symplectic structureof the Hamiltonian formalism, which allows one to reformulate Bellman?s conjecturesconcerning the ?invariant-embedding? methodology for two-point boundaryvalueproblems. Results for LQR problems are tested against solutions of the associated differential Riccati equation, and the attributes of the two approaches are illustratedand discussed. Also, nonlinear problems are numerically solved and comparedagainst those obtained by using shooting techniques.