Controlled polyhedral sweeping processes: existence, stability, and optimality conditions

TL;DR

This paper investigates controlled polyhedral sweeping processes in Hilbert spaces, establishing existence, stability, and optimality conditions through new theorems and finite-dimensional analysis.

Contribution

It introduces novel existence and uniqueness theorems, stability estimates, and optimality conditions for controlled sweeping processes with polyhedral moving sets.

Findings

New existence and uniqueness theorems for sweeping trajectories.

Quantitative stability estimates relating controls and trajectories.

Necessary optimality conditions derived via discrete approximations.

Abstract

This paper is mainly devoted to the study of controlled sweeping processes with polyhedral moving sets in Hilbert spaces. Based on a detailed analysis of truncated Hausdorff distances between moving polyhedra, we derive new existence and uniqueness theorems for sweeping trajectories corresponding to various classes of control functions acting in moving sets. Then we establish quantitative stability results, which provide efficient estimates on the sweeping trajectory dependence on controls and initial values. Our final topic, accomplished in finite-dimensional state spaces, is deriving new necessary optimality and suboptimality conditions for sweeping control systems with endpoint constrains by using constructive discrete approximations.