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Optimal Control of Several Motion Models

Tan H. Cao, Nilson Chapagain, Haejoon Lee, Phung Ngoc Thi, Nguyen Nang Thieu

TL;DR

The paper tackles dynamic optimization of planar crowd motion by modeling it as a perturbed nonconvex sweeping process with obstacles. It develops a rigorous variational-analytic framework, deriving necessary optimality conditions that introduce multipliers, adjoints, and measures, and uses proximal normal cones and uniform prox-regularity to manage nonconvex constraints. Building from a single-agent model to multi-agent systems in corridors, it formulates Bolza-type costs with energy terms and designs algorithms that leverage the structure of the conditions to compute optimal controls for two and three agents, supported by illustrative simulations. The results provide a principled, mathematically grounded approach to energy-efficient, collision-avoiding crowd navigation in constrained environments and offer a foundation for future extensions to more complex settings and real-world deployments.

Abstract

This paper is devoted to the study of the dynamic optimization of several controlled crowd motion models in the general planar settings, which is an application of a class of optimal control problems involving a general nonconvex sweeping process with perturbations. A set of necessary optimality conditions for such optimal control problems involving the crowd motion models with multiple agents and obstacles is obtained and analyzed. Several effective algorithms based on such necessary optimality conditions are proposed and various nontrivial illustrative examples together with their simulations are also presented. The implementation of all the considered motion models can be found via the link: https://github.com/tancao1128/Optimal_Control_of_Several_Motion_Models with the instruction and demonstration video uploaded at https://www.youtube.com/watch?v=B8DQ0wvCtIQ.

Optimal Control of Several Motion Models

TL;DR

The paper tackles dynamic optimization of planar crowd motion by modeling it as a perturbed nonconvex sweeping process with obstacles. It develops a rigorous variational-analytic framework, deriving necessary optimality conditions that introduce multipliers, adjoints, and measures, and uses proximal normal cones and uniform prox-regularity to manage nonconvex constraints. Building from a single-agent model to multi-agent systems in corridors, it formulates Bolza-type costs with energy terms and designs algorithms that leverage the structure of the conditions to compute optimal controls for two and three agents, supported by illustrative simulations. The results provide a principled, mathematically grounded approach to energy-efficient, collision-avoiding crowd navigation in constrained environments and offer a foundation for future extensions to more complex settings and real-world deployments.

Abstract

This paper is devoted to the study of the dynamic optimization of several controlled crowd motion models in the general planar settings, which is an application of a class of optimal control problems involving a general nonconvex sweeping process with perturbations. A set of necessary optimality conditions for such optimal control problems involving the crowd motion models with multiple agents and obstacles is obtained and analyzed. Several effective algorithms based on such necessary optimality conditions are proposed and various nontrivial illustrative examples together with their simulations are also presented. The implementation of all the considered motion models can be found via the link: https://github.com/tancao1128/Optimal_Control_of_Several_Motion_Models with the instruction and demonstration video uploaded at https://www.youtube.com/watch?v=B8DQ0wvCtIQ.
Paper Structure (10 sections, 2 theorems, 103 equations, 4 figures)

This paper contains 10 sections, 2 theorems, 103 equations, 4 figures.

Table of Contents

  1. Introduction and Problem Formulations
  2. Preliminaries
  3. Controlled Motion Models with Single Agent
  4. Necessary Optimality Conditions
  5. Controlled Crowd Motion Models with Multiple Agents and Multiple Obstacles
  6. The Controlled Crowd Motion Models with Multiple Agents in a Corridor
  7. The Crowd Motion Model with Two Agents
  8. The Crowd Motion Model with Three Agents
  9. Concluding Remarks and Future Research
  10. Algorithms

Key Result

Theorem 3.2

(Necessary optimality conditions for optimization of controlled crowd motions with obstacles) Let $(\overline{\bf x}(\cdot), \overline a(\cdot)) \in W^{2,\infty}([0,T];\mathbb{R}^{2}\times \mathbb{R})$ be a strong local minimizer of the crowd motion problem in e5 with $\tau=1$. There exist some dual

Figures (4)

  • Figure 1: Illustration for Example \ref{['eg51']} at $t=0$ and $t=T$
  • Figure 2: Illustration for Example \ref{['eg52']} at $t=0$ and $t=T$
  • Figure 3: Three Agents in a Corridor
  • Figure 4: Illustration for Example \ref{['eg53']} at $t=0$ and $t=T$

Theorems & Definitions (9)

  • Definition 2.1
  • Definition 3.1
  • Theorem 3.2
  • Theorem 4.1
  • Example 4.2
  • Example 4.3
  • Remark 4.4
  • Example 4.5
  • Remark 4.6