arXiv subjects

Daniel Veldman

Publications and source records attributed to Daniel Veldman.

3 recordsLinked to original sources

arXiv2025

A Random Batch Method for the Efficient Simulation and Optimal Control of Networked 1-D Wave Equations

In this paper, a stochastic algorithm for the efficient simulation and optimal control of networked wave equations based on the random batch method is proposed and analyzed. The random approximation is constructed by dividing the time interval into subintervals and restricting the dynamics to a randomly chosen subnetwork during each of these subintervals. It is proven that the solution for this randomized system converges in expectation to the solution on the original network when the length of the subintervals approaches zero. Furthermore, the optimal controls for the randomized system converge (in $H^2$ and in expectation) to the optimal controls for the original system. The computational advantage of the proposed method is demonstrated in two numerical examples.

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arXiv2022

Local Stability and Convergence of Unconstrained Model Predictive Control

The local stability and convergence for Model Predictive Control (MPC) of unconstrained nonlinear dynamics based on a linear time-invariant plant model is studied. Based on the long-time behavior of the solution of the Riccati Differential Equation (RDE), explicit error estimates are derived that clearly demonstrate the influence of the two critical parameters in MPC: the prediction horizon $T$ and the control horizon $\tau$. In particular, if the MPC-controller has access to an exact (linear) plant model, the MPC-controls and the corresponding optimal state trajectories converge exponentially to the solution of an infinite-horizon optimal control problem when $T-\tau \rightarrow \infty$. When the difference between the linear model and the nonlinear plant is sufficiently small in a neighborhood of the origin, the MPC strategy is locally stabilizing and the influence of modeling errors can be reduced by choosing the control horizon $\tau$ smaller. The obtained convergence rates are validated in numerical simulations.

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arXiv2021

A framework for randomized time-splitting in linear-quadratic optimal control

Inspired by the successes of stochastic algorithms in the training of deep neural networks and the simulation of interacting particle systems, we propose and analyze a framework for randomized time-splitting in linear-quadratic optimal control. In our proposed framework, the linear dynamics of the original problem is replaced by a randomized dynamics. To obtain the randomized dynamics, the system matrix is split into simpler submatrices and the time interval of interest is split into subintervals. The randomized dynamics is then found by selecting randomly one or more submatrices in each subinterval. We show that the dynamics, the minimal values of the cost functional, and the optimal control obtained with the proposed randomized time-splitting method converge in expectation to their analogues in the original problem when the time grid is refined. The derived convergence rates are validated in several numerical experiments. Our numerical results also indicate that the proposed method can lead to a reduction in computational cost for the simulation and optimal control of large-scale linear dynamical systems.

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