arXiv subjects

Yeoneung Kim

Publications and source records attributed to Yeoneung Kim.

24 records · Page 2Linked to original sources

arXiv2022

State-constraint static Hamilton-Jacobi equations in nested domains

We study state-constraint static Hamilton-Jacobi equations in a sequence of domains $\{Ω_k\}_{k \in \mathbb{N}}$ in $\mathbb{R}^n$ such that $Ω_k \subset Ω_{k+1}$ for all $k\in \mathbb{N}$. We obtain rates of convergence of $u_k$, the solution to the state-constraint problem in $Ω_k$, to $u$, the solution to the corresponding problem in $Ω= \bigcup_{k \in \mathbb{N}} Ω_k$. In many cases, the rates obtained are proven to be optimal. Various new examples and discussions are provided at the end of the paper.

math.AP↗

arXiv2022

Infusing model predictive control into meta-reinforcement learning for mobile robots in dynamic environments

The successful operation of mobile robots requires them to adapt rapidly to environmental changes. To develop an adaptive decision-making tool for mobile robots, we propose a novel algorithm that combines meta-reinforcement learning (meta-RL) with model predictive control (MPC). Our method employs an off-policy meta-RL algorithm as a baseline to train a policy using transition samples generated by MPC when the robot detects certain events that can be effectively handled by MPC, with its explicit use of robot dynamics. The key idea of our method is to switch between the meta-learned policy and the MPC controller in a randomized and event-triggered fashion to make up for suboptimal MPC actions caused by the limited prediction horizon. During meta-testing, the MPC module is deactivated to significantly reduce computation time in motion control. We further propose an online adaptation scheme that enables the robot to infer and adapt to a new task within a single trajectory. The performance of our method has been demonstrated through simulations using a nonlinear car-like vehicle model with (i) synthetic movements of obstacles, and (ii) real-world pedestrian motion data. The simulation results indicate that our method outperforms other algorithms in terms of learning efficiency and navigation quality.

cs.RO↗

arXiv2022

On representation formulas for optimal control: A Lagrangian perspective

In this paper, we study representation formulas for finite-horizon optimal control problems with or without state constraints, unifying two different viewpoints: the Lagrangian and dynamic programming (DP) frameworks. In a recent work [1], the generalized Lax formula is obtained via DP for optimal control problems with state constraints and nonlinear systems. We revisit the formula from the Lagrangian perspective to provide a unified framework for understanding and implementing the nontrivial representation of the value function. Our simple derivation makes direct use of the Lagrangian formula from the theory of Hamilton-Jacobi (HJ) equations. We also discuss a rigorous way to construct an optimal control using a $δ$-net, as well as a numerical scheme for controller synthesis via convex optimization.

math.OC↗

arXiv2021

Training Wasserstein GANs without gradient penalties

We propose a stable method to train Wasserstein generative adversarial networks. In order to enhance stability, we consider two objective functions using the $c$-transform based on Kantorovich duality which arises in the theory of optimal transport. We experimentally show that this algorithm can effectively enforce the Lipschitz constraint on the discriminator while other standard methods fail to do so. As a consequence, our method yields an accurate estimation for the optimal discriminator and also for the Wasserstein distance between the true distribution and the generated one. Our method requires no gradient penalties nor corresponding hyperparameter tuning and is computationally more efficient than other methods. At the same time, it yields competitive generators of synthetic images based on the MNIST, F-MNIST, and CIFAR-10 datasets.

cs.LG↗

arXiv2018

On the Uniqueness for One-Dimensional Constrained Hamilton-Jacobi Equations

The goal of this paper is to study uniqueness of a one-dimensional Hamilton-Jacobi equation \begin{equation*} \begin{cases} u_t=|u_x|^2+R(x,I(t)) &\text{in }\mathbb{R} \times (0,\infty), \max_{\mathbb{R}} u(\cdot,t)=0 &\text{on }[0,\infty), \end{cases} \end{equation*} with an initial condition $u_0(x,0)=u_0(x)$ on $\mathbb{R}$. A reaction term $R(x,I(t))$ is given while $I(t)$ is an unknown constraint (Lagrange multiplier) that forces maximum of $u$ to be always zero. In the paper, we prove uniqueness of a pair of unknowns (u,I) using dynamic programming principle in one dimensional space for some particular class of nonseparable reaction $R(x,I(t))$.

math.AP↗

arXiv2018

Well-posedness for constrained Hamilton-Jacobi equations

The goal of this paper is to study a Hamilton-Jacobi equation \begin{equation*} \begin{cases} u_t=H(Du)+R(x,I(t)) &\text{in }\mathbb{R}^n \times (0,\infty), \sup_{\mathbb{R}^n} u(\cdot,t)=0 &\text{on }[0,\infty), \end{cases} \end{equation*} with initial conditions $I(0)=0$, $u_0(x,0)=u_0(x)$ on $\mathbb{R}^n$. Here $(u,I)$ is a pair of unknowns and the Hamiltonian $H$ and the reaction $R$ are given. And $I(t)$ is an unknown constraint (Lagrange multiplier) that forces supremum of $u$ to be always zero. We construct a solution in the viscosity setting using a fixed point argument when the reaction term $R(x,I)$ is strictly decreasing in $I$. We also discuss both uniqueness and nonuniqueness. For uniqueness, a certain structural assumption on $R(x,I)$ is needed. We also provide an example with infinitely many solutions when the reaction term is not strictly decreasing in $I$.

math.AP↗