Discrete approximation of relaxed optimal control problems
Journal of Optimization Theory and Applications
Convergent computational method for relaxed optimal control problems
Journal of Optimization Theory and Applications
Consistent Approximations for Optimal Control Problems Based on Runge--Kutta Integration
SIAM Journal on Control and Optimization
Mixed Frank-Wolfe penalty method with applications to nonconvex optimal control problems
Journal of Optimization Theory and Applications
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
Galerkin Finite Element Methods for Parabolic Problems (Springer Series in Computational Mathematics)
Discretization-optimization methods for optimal control problems
SMO'05 Proceedings of the 5th WSEAS international conference on Simulation, modelling and optimization
Classical and relaxed optimization methods for nonlinear parabolic optimal control problems
LSSC'09 Proceedings of the 7th international conference on Large-Scale Scientific Computing
LSSC'11 Proceedings of the 8th international conference on Large-Scale Scientific Computing
Convergence of the forward-backward sweep method in optimal control
Computational Optimization and Applications
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We consider an optimal control problem described by nonlinear ordinary differential equations, with control and state constraints, including pointwise state constraints. Since this problem may have no classical solutions, it is also formulated in relaxed form. The classical control problem is then discretized by using the implicit midpoint scheme, while the controls are approximated by (not necessarily continuous) piecewise linear classical controls. We first study the behavior in the limit of properties of discrete optimality, and of discrete admissibility and extremality. We then apply a penalized gradient projection method to each discrete classical problem, and also a corresponding progressively refining combined discretization-optimization method to the continuous classical problem, thus reducing computing time and memory. We prove that accumulation points of sequences generated by these methods are admissible and extremal in some sense for the corresponding discrete or continuous, classical or relaxed, problem. For nonconvex problems whose solutions are nonclassical, we show that we can apply the above methods to the problem formulated in Gamkrelidze relaxed form. Finally, numerical examples are given.