The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods
The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods
Algorithm 755: ADOL-C: a package for the automatic differentiation of algorithms written in C/C++
ACM Transactions on Mathematical Software (TOMS)
Evaluating derivatives: principles and techniques of algorithmic differentiation
Evaluating derivatives: principles and techniques of algorithmic differentiation
ACM Transactions on Mathematical Software (TOMS)
Journal of Computational and Applied Mathematics - Special issue on SQP-based direct discretization methods for practical optimal control problems
Practical methods for optimal control using nonlinear programming
Practical methods for optimal control using nonlinear programming
Automatic differentiation of algorithms: from simulation to optimization
Automatic differentiation of algorithms: from simulation to optimization
Continuous optimal control sensitivity analysis with AD
Automatic differentiation of algorithms
Application of automatic diffentiation to race car performance optimisation
Automatic differentiation of algorithms
Second-Order Runge--Kutta Approximations in Control Constrained Optimal Control
SIAM Journal on Numerical Analysis
On the discrete adjoints of adaptive time stepping algorithms
Journal of Computational and Applied Mathematics
Inexact Restoration for Runge-Kutta Discretization of Optimal Control Problems
SIAM Journal on Numerical Analysis
SIAM Journal on Scientific Computing
On the properties of runge-kutta discrete adjoints
ICCS'06 Proceedings of the 6th international conference on Computational Science - Volume Part IV
Convergence of the forward-backward sweep method in optimal control
Computational Optimization and Applications
Stability and consistency of discrete adjoint implicit peer methods
Journal of Computational and Applied Mathematics
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This paper considers the numerical solution of optimal control problems based on ODEs. We assume that an explicit Runge-Kutta method is applied to integrate the state equation in the context of a recursive discretization approach. To compute the gradient of the cost function, one may employ Automatic Differentiation (AD). This paper presents the integration schemes that are automatically generated when differentiating the discretization of the state equation using AD. We show that they can be seen as discretization methods for the sensitivity and adjoint differential equation of the underlying control problem. Furthermore, we prove that the convergence rate of the scheme automatically derived for the sensitivity equation coincides with the convergence rate of the integration scheme for the state equation. Under mild additional assumptions on the coefficients of the integration scheme for the state equation, we show a similar result for the scheme automatically derived for the adjoint equation. Numerical results illustrate the presented theoretical results.