Interpolants for Runge-Kutta formulas
ACM Transactions on Mathematical Software (TOMS)
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Consistent Approximations for Optimal Control Problems Based on Runge--Kutta Integration
SIAM Journal on Control and Optimization
On L2 Sufficient Conditions and the Gradient Projection Method for Optimal Control Problems
SIAM Journal on Control and Optimization
An a priori Estimate for Discrete Approximations in Nonlinear Optimal Control
SIAM Journal on Control and Optimization
SIAM Journal on Numerical Analysis
Optimization: algorithms and consistent approximations
Optimization: algorithms and consistent approximations
On the Time-Discretization of Control Systems
SIAM Journal on Control and Optimization
Journal of Computational and Applied Mathematics - Special issue on SQP-based direct discretization methods for practical optimal control problems
Second-Order Runge--Kutta Approximations in Control Constrained Optimal Control
SIAM Journal on Numerical Analysis
Discretization methods for optimal control problems with state constraints
Journal of Computational and Applied Mathematics
Discretization methods for optimal control problems with state constraints
Journal of Computational and Applied Mathematics
Inexact Restoration for Runge-Kutta Discretization of Optimal Control Problems
SIAM Journal on Numerical Analysis
LSSC'05 Proceedings of the 5th international conference on Large-Scale Scientific Computing
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We consider an optimal control problem for systems governed by ordinary differential equations with control constraints. The state equation is discretized by the explicit fourth order Runge-Kutta scheme and the controls are approximated by discontinuous piecewise affine ones. We then propose an approximate gradient projection method that generates sequences of discrete controls and progressively refines the discretization during the iterations. Instead of using the exact discrete directional derivative, which is difficult to calculate, we use an approximate derivative of the cost functional defined by discretizing the continuous adjoint equation by the same Runge-Kutta scheme and the integral involved by Simpson's integration rule, both involving intermediate approximations. The main result is that accumulation points, if they exist, of sequences constructed by this method satisfy the weak necessary conditions for optimality for the continuous problem. Finally, numerical examples are given.