Interpolants for Runge-Kutta formulas
ACM Transactions on Mathematical Software (TOMS)
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
Second-Order Runge--Kutta Approximations in Control Constrained Optimal Control
SIAM Journal on Numerical Analysis
Approximate Gradient Projection Method with Runge-Kutta Schemes for Optimal Control Problems
Computational Optimization and Applications
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We consider an optimal control problem described by ordinary differential equations, with control and state constraints. The state equation is first discretized by a general explicit Runge-Kutta scheme and the controls are approximated by piecewise polynomial functions. We then propose approximate gradient and gradient projection methods, and their penalized versions, that construct sequences of discrete controls and progressively refine the discretization during the iterations. Instead of using the exact discrete cost derivative, which usually requires tedious calculations of composite functions, we use here an approximate derivative of the cost defined by discretizing the continuous adjoint equation by the same, but nonmatching, Runge-Kutta scheme backward and the integral involved by a Newton-Cotes integration rule. We show that strong accumulation points in L2 of sequences constructed by these methods satisfy the weak necessary conditions for optimality for the continuous problem. Finally, numerical examples are given.