INEXACT RESTORATION FOR RUNGE—KUTTA DISCRETIZATION OF OPTIMAL CONTROL PROBLEMS

A numerical method is presented for Runge-Kutta discretization of unconstrained optimal control problems. First, general Runge-Kutta discretization is carried out to obtain a finitedimensional approximation of the continous-time optimal control problem. Then a recent optimization technique, the inex...

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Bibliographic Details
Published in:SIAM journal on numerical analysis 2010-01, Vol.48 (4), p.1492-1517
Main Author: KAYA, C. YALÇIN
Format: Article
Language:English
Subjects:
Applied mathematics
Approximation
Control
Convergence
Discretization
Infrared radiation
Lagrangian function
Mathematical analysis
Mathematical functions
Mathematical models
Mathematical problems
Mathematics
Necessary conditions for optimality
Newtons method
Optimal control
Optimization techniques
Perceptron convergence procedure
Runge-Kutta method
Studies
Sufficient conditions
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Summary:A numerical method is presented for Runge-Kutta discretization of unconstrained optimal control problems. First, general Runge-Kutta discretization is carried out to obtain a finitedimensional approximation of the continous-time optimal control problem. Then a recent optimization technique, the inexact restoration (IR) method, due to Martinez and coworkers [E. G. Birgin and J. M. Martinez, J. Optim. Theory Appl., 127 (2005), pp. 229-247; J. M. Martinez and E. A. Pilotta, J. Optim. Theory Appl, 104 (2000), pp. 135-163; J. M. Martinez, J. Optim. Theory Appl, 111 (2001), pp. 39-58], is applied to the discretized problem to find an approximate solution. It is proved that, for optimal control problems, a key sufficiency condition for convergence of the IR method is readily satisfied. Under reasonable assumptions, the IR method for optimal control problems is shown to converge to a solution of the discretized problem. Convergence of a solution of the discretized problem to a solution of the continuous-time problem is also shown. It turns out that optimality phase equations of the IR method emanate from an associated Hamiltonian system, and so general Runge-Kutta discretization induces a symplectic partitioned Runge-Kutta scheme. A computational algorithm is described, and numerical experiments are made to demonstrate the working of the method for optimal control of the van der Pol system, employing the three-stage (order 6) Gauss-Legendre discretization.
ISSN:0036-1429
1095-7170
DOI:10.1137/090766668