Partial Differential Equations for Missing Boundary Conditions in the Linear-Quadratic Optimal Control Problem

VICENTE COSTANZA; CARLOS E. NEUMAN

LATIN AMERICAN APPLIED RESEARCH

LAAR

Lugar: Bahía Blanca; Año: 2009 vol. 39 p. 207 - 212

0327-0793

New equations involving the unknown nal states and initial costates corresponding to families of LQR problems are found, and their solutions are computed and validated. Having the initial values of the costates, the optimal control can then be constructed, for each particular problem, from the solution to the Hamiltonian equations, now achievable through on-line integration. The missing boundary conditions are obtained by solving (off-line) two uncoupled, rst-order, quasi-linear, partial differential equations for two auxiliary n n matrices, whose independent variables are the time-horizon duration whose independent variables are the time-horizon duration whose independent variables are the time-horizon duration whose independent variables are the time-horizon duration n n matrices, whose independent variables are the time-horizon duration T and the nal-penalty matrix S: The solutions to these PDEs give information on the behavior of the whole two-parameter family of control problems, which can be used for design purposes. The mathematical treatment takes advantage of the symplectic structure of the Hamiltonian formalism, which allows to reformulate one of Bellman’s conjectures related to the ‘invariant-imbedding’ methodology. Results are tested against solutions of the differential Riccati equations associated with these problems, and the attributes of the two approaches are illustrated and discussed. to these PDEs give information on the behavior of the whole two-parameter family of control problems, which can be used for design purposes. The mathematical treatment takes advantage of the symplectic structure of the Hamiltonian formalism, which allows to reformulate one of Bellman’s conjectures related to the ‘invariant-imbedding’ methodology. Results are tested against solutions of the differential Riccati equations associated with these problems, and the attributes of the two approaches are illustrated and discussed. to these PDEs give information on the behavior of the whole two-parameter family of control problems, which can be used for design purposes. The mathematical treatment takes advantage of the symplectic structure of the Hamiltonian formalism, which allows to reformulate one of Bellman’s conjectures related to the ‘invariant-imbedding’ methodology. Results are tested against solutions of the differential Riccati equations associated with these problems, and the attributes of the two approaches are illustrated and discussed. to these PDEs give information on the behavior of the whole two-parameter family of control problems, which can be used for design purposes. The mathematical treatment takes advantage of the symplectic structure of the Hamiltonian formalism, which allows to reformulate one of Bellman’s conjectures related to the ‘invariant-imbedding’ methodology. Results are tested against solutions of the differential Riccati equations associated with these problems, and the attributes of the two approaches are illustrated and discussed. and the nal-penalty matrix S: The solutions to these PDEs give information on the behavior of the whole two-parameter family of control problems, which can be used for design purposes. The mathematical treatment takes advantage of the symplectic structure of the Hamiltonian formalism, which allows to reformulate one of Bellman’s conjectures related to the ‘invariant-imbedding’ methodology. Results are tested against solutions of the differential Riccati equations associated with these problems, and the attributes of the two approaches are illustrated and discussed.