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

2017 ◽  
Vol 22 (2) ◽  
pp. 267-306 ◽  
Author(s):  
Tan H. Cao ◽  
◽  
Boris S. Mordukhovich ◽  
Keyword(s):  
Optimality Conditions ◽  
Sweeping Process ◽  
Motion Model ◽  
Crowd Motion

scholarly journals Modeling Pedestrian Motion in Crowded Scenes Based on the Shortest Path Principle

2021 ◽  
Vol 12 (1) ◽  
pp. 381
Author(s):  
Yi Zou ◽  
Yuncai Liu
Keyword(s):  
Shortest Path ◽  
State Of The Art ◽  
Meaningful Work ◽  
Movement Behavior ◽  
Motion Model ◽  
Human Dynamics ◽  
Motion Trajectories ◽  
Crowd Motion ◽  
Crowded Scenes ◽  
Part Motion

In the computer vision field, understanding human dynamics is not only a great challenge but also very meaningful work, which plays an indispensable role in public safety. Despite the complexity of human dynamics, physicists have found that pedestrian motion in a crowd is governed by some internal rules, which can be formulated as a motion model, and an effective model is of great importance for understanding and reconstructing human dynamics in various scenes. In this paper, we revisit the related research in social psychology and propose a two-part motion model based on the shortest path principle. One part of the model seeks the origin and destination of a pedestrian, and the other part generates the movement path of the pedestrian. With the proposed motion model, we simulated the movement behavior of pedestrians and classified them into various patterns. We next reconstructed the crowd motions in a real-world scene. In addition, to evaluate the effectiveness of the model in crowd motion simulations, we created a new indicator to quantitatively measure the correlation between two groups of crowd motion trajectories. The experimental results show that our motion model outperformed the state-of-the-art model in the above applications.

A mathematical framework for a crowd motion model

2008 ◽  
Vol 346 (23-24) ◽  
pp. 1245-1250 ◽  
Author(s):  
Bertrand Maury ◽  
Juliette Venel
Keyword(s):  
Mathematical Framework ◽  
Motion Model ◽  
Crowd Motion

scholarly journals A MACROSCOPIC CROWD MOTION MODEL OF GRADIENT FLOW TYPE

2010 ◽  
Vol 20 (10) ◽  
pp. 1787-1821 ◽  
Author(s):  
BERTRAND MAURY ◽  
AUDE ROUDNEFF-CHUPIN ◽  
FILIPPO SANTAMBROGIO
Keyword(s):  
Ad Hoc ◽  
Gradient Flow ◽  
Emergency Evacuation ◽  
Probability Measures ◽  
Motion Model ◽  
Exit Door ◽  
Incompressibility Constraint ◽  
Macroscopic Approach ◽  
Crowd Motion ◽  
Wasserstein Space

A simple model to handle the flow of people in emergency evacuation situations is considered: at every point x, the velocity U (x) that individuals at x would like to realize is given. Yet, the incompressibility constraint prevents this velocity field to be realized and the actual velocity is the projection of the desired one onto the set of admissible velocities. Instead of looking at a microscopic setting (where individuals are represented by rigid discs), here the macroscopic approach is investigated, where the unknown is a density ρ(t,x). If a gradient structure is given, say U = -∇D where D is, for instance, the distance to the exit door, the problem is presented as a Gradient Flow in the Wasserstein space of probability measures. The functional which gives the Gradient Flow is neither finitely valued (since it takes into account the constraints on the density), nor geodesically convex, which requires for an ad hoc study of the convergence of a discrete scheme.

scholarly journals Modeling of crowds in regions with moving obstacles

2021 ◽  
Vol 41 (11) ◽  
pp. 5009
Author(s):  
Nadezhda Maltugueva ◽  
Nikolay Pogodaev
Keyword(s):  
Normal Cone ◽  
Optimization Problems ◽  
Well Posedness ◽  
Sweeping Process ◽  
Nonlinear Transport ◽  
Moving Obstacles ◽  
Nonlinear Transport Equation ◽  
Sweeping Processes ◽  
Crowd Motion ◽  
Regular Sets

<p style='text-indent:20px;'>We present a model of crowd motion in regions with moving obstacles, which is based on the notion of measure sweeping process. The obstacle is modeled by a set-valued map, whose values are complements to <inline-formula><tex-math id="M1">\begin{document}$ r $\end{document}</tex-math></inline-formula>-prox-regular sets. The crowd motion obeys a nonlinear transport equation outside the obstacle and a normal cone condition (similar to that of the classical sweeping processes theory) on the boundary. We prove the well-posedness of the model, give an application to environment optimization problems, and provide some results of numerical computations.</p>

Optimal control of higher order differential inclusions with functional constraints

2020 ◽  
Vol 26 ◽  
pp. 37 ◽  
Author(s):  
Elimhan N. Mahmudov
Keyword(s):  
Optimal Control ◽  
Optimality Conditions ◽  
Differential Inclusion ◽  
Differential Inclusions ◽  
Higher Order ◽  
Second Order ◽  
Sufficient Optimality Conditions ◽  
Functional Constraints ◽  
Complementary Slackness ◽  
Complementary Slackness Conditions

The present paper studies the Mayer problem with higher order evolution differential inclusions and functional constraints of optimal control theory (PFC); to this end first we use an interesting auxiliary problem with second order discrete-time and discrete approximate inclusions (PFD). Are proved necessary and sufficient conditions incorporating the Euler–Lagrange inclusion, the Hamiltonian inclusion, the transversality and complementary slackness conditions. The basic concept of obtaining optimal conditions is locally adjoint mappings and equivalence results. Then combining these results and passing to the limit in the discrete approximations we establish new sufficient optimality conditions for second order continuous-time evolution inclusions. This approach and results make a bridge between optimal control problem with higher order differential inclusion (PFC) and constrained mathematical programming problems in finite-dimensional spaces. Formulation of the transversality and complementary slackness conditions for second order differential inclusions play a substantial role in the next investigations without which it is hardly ever possible to get any optimality conditions; consequently, these results are generalized to the problem with an arbitrary higher order differential inclusion. Furthermore, application of these results is demonstrated by solving some semilinear problem with second and third order differential inclusions.

Analysis and design of thermo-mechanical interfaces

2012 ◽  
Vol 60 (2) ◽  
pp. 205-213
Author(s):  
K. Dems ◽  
Z. Mróz
Keyword(s):  
Optimality Conditions ◽  
Mechanical Loading ◽  
Material Properties ◽  
Structural Response ◽  
External Boundary ◽  
Mechanical Loads ◽  
Internal Interface ◽  
Analysis And Design ◽  
First Order ◽  
Mechanical Nature

Abstract. An elastic structure subjected to thermal and mechanical loading with prescribed external boundary and varying internal interface is considered. The different thermal and mechanical nature of this interface is discussed, since the interface form and its properties affect strongly the structural response. The first-order sensitivities of an arbitrary thermal and mechanical behavioral functional with respect to shape and material properties of the interface are derived using the direct or adjoint approaches. Next the relevant optimality conditions are formulated. Some examples illustrate the applicability of proposed approach to control the structural response due to applied thermal and mechanical loads.

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