An a priori Estimate for Discrete Approximations in Nonlinear Optimal Control
SIAM Journal on Control and Optimization
On the Time-Discretization of Control Systems
SIAM Journal on Control and Optimization
Inexact-restoration algorithm for constrained optimization
Journal of Optimization Theory and Applications
The Euler approximation in state constrained optimal control
Mathematics of Computation
Iterative Dynamic Programming
Uniform Convergence and Mesh Independence of Newton's Method for Discretized Variational Problems
SIAM Journal on Control and Optimization
Consistent Approximations and Approximate Functions and Gradients in Optimal Control
SIAM Journal on Control and Optimization
Second-Order Runge--Kutta Approximations in Control Constrained Optimal Control
SIAM Journal on Numerical Analysis
Second Order Sufficient Conditions for Time-Optimal Bang-Bang Control
SIAM Journal on Control and Optimization
Approximate Gradient Projection Method with Runge-Kutta Schemes for Optimal Control Problems
Computational Optimization and Applications
Automatic differentiation of explicit Runge-Kutta methods for optimal control
Computational Optimization and Applications
Computational Optimization and Applications
SIAM Journal on Numerical Analysis
An inexact-restoration method for nonlinear bilevel programming problems
Computational Optimization and Applications
SIAM Journal on Scientific Computing
Metric Regularity of Newton's Iteration
SIAM Journal on Control and Optimization
Convergence of the forward-backward sweep method in optimal control
Computational Optimization and Applications
Journal of Computational and Applied Mathematics
A numerical method for nonconvex multi-objective optimal control problems
Computational Optimization and Applications
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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 finite-dimensional approximation of the continous-time optimal control problem. Then a recent optimization technique, the inexact restoration (IR) method, due to Martínez and coworkers [E. G. Birgin and J. M. Martínez, J. Optim. Theory Appl., 127 (2005), pp. 229-247; J. M. Martínez and E. A. Pilotta, J. Optim. Theory Appl., 104 (2000), pp. 135-163; J. M. Martínez, 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.