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

Authors
Costanza, Vicente; Neuman, C. E.
Publication Year
2009
Language
English
Format
article
Status
Published version
Description
New equations involving the unknown final 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 (offline) two uncoupled, first-order, quasi-linear, partial differential equations for two auxiliary n × n matrices, whose independent variables are the timehorizon duration T and the final-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 “invariantimbedding” 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.
Fil: Costanza, Vicente. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Desarrollo Tecnológico para la Industria Química. Universidad Nacional del Litoral. Instituto de Desarrollo Tecnológico para la Industria Química; Argentina
Fil: Neuman, C. E.. Universidad Nacional del Litoral; Argentina
Subject
OPTIMAL CONTROL
LINEAR-QUADRATIC PROBLEM
FIRST-ORDER PDES
BOUNDARY-VALUE PROBLEMS
Ingeniería Química
Ingeniería Química
INGENIERÍAS Y TECNOLOGÍAS
Access level
Open access
License
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
Repository
CONICET Digital (CONICET)
Institution
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identifier
oai:ri.conicet.gov.ar:11336/17096