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Optimal Control of Nonconvex Sweeping Processes with Variable Time via Finite-Difference Approximations

Tan H. Cao, Boris S. Mordukhovich, Dao Nguyen, Trang Nguyen, Nguyen N. Thieu

TL;DR

This work advances optimal control for nonconvex sweeping processes with variable time by introducing and validating discrete-approximation methods that strongly converge to continuous-time optima. It develops a comprehensive variational-analysis toolkit, including second-order subdifferentials and coderivative calculus, to obtain necessary optimality conditions for discrete approximations and their continuous-time counterparts. The methodology is then demonstrated on motion-model applications, enabling computation of optimal times and controls and revealing interaction patterns such as pre-contact acceleration and post-contact synchronization. The results offer a rigorous pathway from discrete, tractable problems to complex free-time sweeping-control problems with nonpolyhedral moving sets, with potential impact on robotics, traffic, and nanotechnology modeling.

Abstract

The paper is devoted to the study of a new class of optimal control problems for nonsmooth dynamical systems governed by nonconvex discontinuous differential inclusions of the sweeping type with involving variable time into optimization. We develop a novel version of the method of discrete approximations of its own qualitative and numerical importance with establishing its well-posedness and strong convergence to optimal solutions of the controlled sweeping process. Using advanced tools of variational analysis and generalized differentiation leads us to deriving new necessary conditions for optimal solutions to discrete approximation problems, which serve as suboptimality conditions for the original continuous-time controlled sweeping process. The obtained results are applied to a class of motion models of practical interest, where the established necessary conditions are used to investigate the agents' interactions and to develop an algorithm for calculating optimal solutions.

Optimal Control of Nonconvex Sweeping Processes with Variable Time via Finite-Difference Approximations

TL;DR

This work advances optimal control for nonconvex sweeping processes with variable time by introducing and validating discrete-approximation methods that strongly converge to continuous-time optima. It develops a comprehensive variational-analysis toolkit, including second-order subdifferentials and coderivative calculus, to obtain necessary optimality conditions for discrete approximations and their continuous-time counterparts. The methodology is then demonstrated on motion-model applications, enabling computation of optimal times and controls and revealing interaction patterns such as pre-contact acceleration and post-contact synchronization. The results offer a rigorous pathway from discrete, tractable problems to complex free-time sweeping-control problems with nonpolyhedral moving sets, with potential impact on robotics, traffic, and nanotechnology modeling.

Abstract

The paper is devoted to the study of a new class of optimal control problems for nonsmooth dynamical systems governed by nonconvex discontinuous differential inclusions of the sweeping type with involving variable time into optimization. We develop a novel version of the method of discrete approximations of its own qualitative and numerical importance with establishing its well-posedness and strong convergence to optimal solutions of the controlled sweeping process. Using advanced tools of variational analysis and generalized differentiation leads us to deriving new necessary conditions for optimal solutions to discrete approximation problems, which serve as suboptimality conditions for the original continuous-time controlled sweeping process. The obtained results are applied to a class of motion models of practical interest, where the established necessary conditions are used to investigate the agents' interactions and to develop an algorithm for calculating optimal solutions.

Paper Structure

This paper contains 8 sections, 11 theorems, 202 equations, 1 figure.

Table of Contents

  1. Introduction and Problem Formulation
  2. Preliminaries and Standing Assumptions
  3. Discrete Approximations of Constrained Sweeping Dynamics
  4. Existence of Optimal Solutions and Local Minimizers
  5. Discrete Approximations of Local Minimizers
  6. Second-Order Subdifferential Evaluations
  7. Necessary Optimality Conditions via Discrete Approximations
  8. Optimal Control of Motion Models

Key Result

Proposition 2.2

Under the assumptions in (H2.2) and (H2.3), the moving set $C_u(t)$, $t\in[0,T]$, in e:SP and e:MS is $\eta$-prox-regular with $\eta=\frac{\alpha}{M_3\beta}$.

Figures (1)

  • Figure 1: Crowd motion model in a corridor before the contacting time.

Theorems & Definitions (14)

  • Definition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Theorem 3.1
  • Proposition 4.1
  • Theorem 4.2
  • Definition 4.3
  • Proposition 5.1
  • proof
  • Theorem 5.2
  • ...and 4 more