1,138 research outputs found
An a posteriori error estimate for Symplectic Euler approximation of optimal control problems
This work focuses on numerical solutions of optimal control problems. A time
discretization error representation is derived for the approximation of the
associated value function. It concerns Symplectic Euler solutions of the
Hamiltonian system connected with the optimal control problem. The error
representation has a leading order term consisting of an error density that is
computable from Symplectic Euler solutions. Under an assumption of the pathwise
convergence of the approximate dual function as the maximum time step goes to
zero, we prove that the remainder is of higher order than the leading error
density part in the error representation. With the error representation, it is
possible to perform adaptive time stepping. We apply an adaptive algorithm
originally developed for ordinary differential equations. The performance is
illustrated by numerical tests
Discrete mechanics and optimal control: An analysis
The optimal control of a mechanical system is of crucial importance in many application areas. Typical examples are the determination of a time-minimal path in vehicle dynamics, a minimal energy trajectory in space mission design, or optimal motion sequences in robotics and biomechanics. In most cases, some sort of discretization of the original, infinite-dimensional optimization problem has to be performed in order to make the problem amenable to computations. The approach proposed in this paper is to directly discretize the variational description of the system's motion. The resulting optimization algorithm lets the discrete solution directly inherit characteristic structural properties from the continuous one like symmetries and integrals of the motion. We show that the DMOC (Discrete Mechanics and Optimal Control) approach is equivalent to a finite difference discretization of Hamilton's equations by a symplectic partitioned Runge-Kutta scheme and employ this fact in order to give a proof of convergence. The numerical performance of DMOC and its relationship to other existing optimal control methods are investigated
Model Order Reduction for Nonlinear Schr\"odinger Equation
We apply the proper orthogonal decomposition (POD) to the nonlinear
Schr\"odinger (NLS) equation to derive a reduced order model. The NLS equation
is discretized in space by finite differences and is solved in time by
structure preserving symplectic mid-point rule. A priori error estimates are
derived for the POD reduced dynamical system. Numerical results for one and two
dimensional NLS equations, coupled NLS equation with soliton solutions show
that the low-dimensional approximations obtained by POD reproduce very well the
characteristic dynamics of the system, such as preservation of energy and the
solutions
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