Optimality conditions for a controlled sweeping process with applications to the crowd motion model

Tan H. Cao; Boris S. Mordukhovich

doi:10.3934/dcdsb.2017014

Article Contents

2017, Volume 22, Issue 2: 267-306. Doi: 10.3934/dcdsb.2017014

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Optimality conditions for a controlled sweeping process with applications to the crowd motion model

Tan H. Cao^1, and
Boris S. Mordukhovich^2,

1.
Vietnamese-German University, Le Lai Street, Hoa Phu Ward, Thu Dau Mot City, Binh Duong Province, Vietnam
2.
Department of Mathematics, Wayne State University, Detroit, Michigan 48202, USA and RUDN University, Moscow 117198, Russia

Received: October 31, 2015

Revised: October 31, 2016

Published: December 2017

This research was partly supported by the National Science Foundation under grants DMS-1007132 and DMS-1512846, by the Air Force Office of Scientific Research grant #15RT0462, and by the Ministry of Education and Science of the Russian Federation (the Agreement number 02.a03.21.0008 of 24 June 2016).

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Abstract

The paper concerns the study and applications of a new class of optimal control problems governed by a perturbed sweeping process of the hysteresis type with control functions acting in both play-and-stop operator and additive perturbations. Such control problems can be reduced to optimization of discontinuous and unbounded differential inclusions with pointwise state constraints, which are immensely challenging in control theory and prevent employing conventional variation techniques to derive necessary optimality conditions. We develop the method of discrete approximations married with appropriate generalized differential tools of modern variational analysis to overcome principal difficulties in passing to the limit from optimality conditions for finite-difference systems. This approach leads us to nondegenerate necessary conditions for local minimizers of the controlled sweeping process expressed entirely via the problem data. Besides illustrative examples, we apply the obtained results to an optimal control problem associated with of the crowd motion model of traffic flow in a corridor, which is formulated in this paper. The derived optimality conditions allow us to develop an effective procedure to solve this problem in a general setting and completely calculate optimal solutions in particular situations.

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Mathematics Subject Classification: Primary:49M25, 47J40;Secondary:90C30, 49J53.

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Figure 1. Direction of optimal control

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Figure 2. Two-dimensional motion.

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Figure 3. Crowd motion model in a corridor

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Figure 4. Two participants out of contact for $t<t_1$.

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Figure 5. Two participants in contact for $t\ge t_1$.

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Figure 6. Out of contact situation for two adjacent participants when $t<t_1$

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Figure 7. All the participants in contact for $t\ge t_1$

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