One-Dimensional Forward–Forward Mean-Field Games
| Autor: | Diogo A. Gomes, Marc Sedjro, Levon Nurbekyan |
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| Rok vydání: | 2016 |
| Předmět: |
Class (set theory)
Conservation law Control and Optimization Relation (database) Logarithm Applied Mathematics 010102 general mathematics 01 natural sciences Conserved quantity 010101 applied mathematics Mean field theory Convergence (routing) Applied mathematics Fokker–Planck equation 0101 mathematics Mathematics |
| Zdroj: | Applied Mathematics & Optimization. 74:619-642 |
| ISSN: | 1432-0606 0095-4616 |
| DOI: | 10.1007/s00245-016-9384-y |
| Popis: | While the general theory for the terminal-initial value problem for mean-field games (MFGs) has achieved a substantial progress, the corresponding forward–forward problem is still poorly understood—even in the one-dimensional setting. Here, we consider one-dimensional forward–forward MFGs, study the existence of solutions and their long-time convergence. First, we discuss the relation between these models and systems of conservation laws. In particular, we identify new conserved quantities and study some qualitative properties of these systems. Next, we introduce a class of wave-like equations that are equivalent to forward–forward MFGs, and we derive a novel formulation as a system of conservation laws. For first-order logarithmic forward–forward MFG, we establish the existence of a global solution. Then, we consider a class of explicit solutions and show the existence of shocks. Finally, we examine parabolic forward–forward MFGs and establish the long-time convergence of the solutions. |
| Databáze: | OpenAIRE |
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