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King Abdullah University of Science and Technology

1. Evangelista, David. Stationary Mean-Field Games with Congestion.

Degree: Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division, 2019, King Abdullah University of Science and Technology

URL: http://hdl.handle.net/10754/655679

Mean-field games (MFG) are models of large populations of rational agents who seek to optimize an objective function that takes into account their state variables and the distribution of the state variable of the remaining agents. MFG with congestion model problems where the agentsâ€™ motion is hampered in high-density regions.
First, we study radial solutions for first- and second-order stationary MFG with congestion on Rd. The radial case, which is one of the simplest non one-dimensional MFG, is relatively tractable. As we observe, the Fokker-Planck equation is integrable with respect to one of the unknowns. Consequently, we obtain a single equation substituting this solution into the Hamilton-Jacobi equation. For the first-order case, we derive explicit formulas; for the elliptic case, we study a variational formulation of the resulting equation. For the first case, we use our approach to compute numerical approximations to the solutions of the corresponding MFG systems.
Next, we consider second-order stationary MFG with congestion and prove the existence of stationary solutions. Because moving in congested areas is difficult, agents prefer to move in non-congested areas. As a consequence, the model becomes singular near the zero density. The existence of stationary solutions was previously obtained for MFG with quadratic Hamiltonians thanks to a very particular identity. Here, we develop robust estimates that give the existence of a solution for general subquadratic Hamiltonians.
Additionally, we study first-order stationary MFG with congestion with quadratic or power-like Hamiltonians. Using explicit examples, we illustrate two key difficulties: the lack of classical solutions and the existence of areas with vanishing densities. Our
main contribution is a new variational formulation for MFG with congestion. With this formulation, we prove the existence and uniqueness of solutions. Finally, we devise a discretization that is combined with optimization algorithms to numerically solve various MFG with congestion.
*Advisors/Committee Members: Gomes, Diogo A. (advisor), Tempone, Raul (committee member), Santamarina, Carlos (committee member), Fusco, Nicola (committee member).*

Subjects/Keywords: mean-field games; congestion problems; stationary problems; calculus f variations

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Evangelista, D. (2019). Stationary Mean-Field Games with Congestion. (Thesis). King Abdullah University of Science and Technology. Retrieved from http://hdl.handle.net/10754/655679

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Evangelista, David. “Stationary Mean-Field Games with Congestion.” 2019. Thesis, King Abdullah University of Science and Technology. Accessed April 14, 2021. http://hdl.handle.net/10754/655679.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Evangelista, David. “Stationary Mean-Field Games with Congestion.” 2019. Web. 14 Apr 2021.

Vancouver:

Evangelista D. Stationary Mean-Field Games with Congestion. [Internet] [Thesis]. King Abdullah University of Science and Technology; 2019. [cited 2021 Apr 14]. Available from: http://hdl.handle.net/10754/655679.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Evangelista D. Stationary Mean-Field Games with Congestion. [Thesis]. King Abdullah University of Science and Technology; 2019. Available from: http://hdl.handle.net/10754/655679

Not specified: Masters Thesis or Doctoral Dissertation

2. Bhan, Sankalp Kishan. Sufficient Conditions for Optimal Control Problems with Terminal Constraints and Free Terminal Times with Applications to Aerospace.

Degree: PhD, Electrical & Systems Engineering, 2019, Washington University in St. Louis

URL: https://openscholarship.wustl.edu/eng_etds/440

Motivated by the flight control problem of designing control laws for a Ground Collision Avoidance System (GCAS), this thesis formulates sufficient conditions for a strong local minimum for a terminally constrained optimal control problem with a free-terminal time. The conditions develop within the framework of a construction of a field of extremals by means of the method of characteristics, a procedure for the solution of first-order linear partial differential equations, but modified to apply to the Hamilton-Jacobi-Bellman equation of optimal control. Additionally, the thesis constructs these sufficient conditions for optimality with a mathematically rigorous development. The proof uses an approach which generalizes and differs significantly from procedures outlined in the classical literature on control engineering, where similar formulas are derived, but only in a cursory, formal and sometimes incomplete way. Additionally, the thesis gives new arrangements of the relevant expressions arising in the formulation of sufficient conditions for optimality that lead to more concise formulas for the resulting perturbation feedback control schemes. These results are applied to an emergency perturbation-feedback guidance scheme which recovers an aircraft from a dangerous flight-path angle to a safe one. Discussion of required background material contrasts nonlinear and linear optimal control theory are contrasted in the context of aerospace applications. A simplified version of the classical model for an F-16 fighter aircraft is used in numerical computation to very, by example, that the sufficient conditions for optimality developed in this thesis can be used off-line to detect possible failures in perturbation feedback control schemes, which arise if such methods are applied along extremal controlled trajectories and which only satisfy the necessary conditions for optimality without being locally optimal. The sufficient conditions for optimality developed in this thesis, on the other hand, guarantee the local validity of such perturbation feedback control schemes. This thesis presents various graphs that compare the neighboring extremals which were derived from the perturbation feedback control scheme with optimal ones that start from the same initial condition. Future directions for this work include extending the perturbation feedback control schemes to optimization problems that are further constrained, possibly through control constraints, state-space constraints or mixed state-control constraints.
*Advisors/Committee Members: Heinz Schaettler, ShiNung Ching, Jr-Shin Li, Ron Cytron, Hiro Mukai.*

Subjects/Keywords: Calculus of Variations; F-16; Flight Control; GCAS; Optimal Control; Sufficient Conditions; Aerospace Engineering; Applied Mathematics; Systems Engineering

…equations. The
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*calculus* of *variations* and optimal control theory solve this type of… …*calculus* of
*variations* for unconstrained problems with fixed terminal times, they complicate when… …study and
pushing me to find new applications of control theory on the *F*-15X. I thank my… …many teammates on the Dominator team, *F*-15SA team, and the MQ-25
teams, thank you for… …version of the classical model for an *F*-16
fighter aircraft is used in numerical computation to…

Record Details Similar Records

❌

APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Bhan, S. K. (2019). Sufficient Conditions for Optimal Control Problems with Terminal Constraints and Free Terminal Times with Applications to Aerospace. (Doctoral Dissertation). Washington University in St. Louis. Retrieved from https://openscholarship.wustl.edu/eng_etds/440

Chicago Manual of Style (16^{th} Edition):

Bhan, Sankalp Kishan. “Sufficient Conditions for Optimal Control Problems with Terminal Constraints and Free Terminal Times with Applications to Aerospace.” 2019. Doctoral Dissertation, Washington University in St. Louis. Accessed April 14, 2021. https://openscholarship.wustl.edu/eng_etds/440.

MLA Handbook (7^{th} Edition):

Bhan, Sankalp Kishan. “Sufficient Conditions for Optimal Control Problems with Terminal Constraints and Free Terminal Times with Applications to Aerospace.” 2019. Web. 14 Apr 2021.

Vancouver:

Bhan SK. Sufficient Conditions for Optimal Control Problems with Terminal Constraints and Free Terminal Times with Applications to Aerospace. [Internet] [Doctoral dissertation]. Washington University in St. Louis; 2019. [cited 2021 Apr 14]. Available from: https://openscholarship.wustl.edu/eng_etds/440.

Council of Science Editors:

Bhan SK. Sufficient Conditions for Optimal Control Problems with Terminal Constraints and Free Terminal Times with Applications to Aerospace. [Doctoral Dissertation]. Washington University in St. Louis; 2019. Available from: https://openscholarship.wustl.edu/eng_etds/440