The paper deals with an optimal control problem in one dimension, having affine dynamics, and a running cost which is discontinuous in the control variable. More precisely, besides terms which are quadratic in the state and control variables, a mean field dependent fixed cost is paid as long as the control is activated (not null). This leads to a problem that, although simple, does not seem to fall into a known class. Moreover, the outcome is completely different according to the magnitude of the fixed cost in comparison to other parameters, such as the (constant) disturbance appearing into the state equation. By means of some tools of Bellman’s Dynamic Programming and viscosity solutions, we are able to provide an explicit formula for the value function in all different cases, as well as a feedback formula for the optimal control, leading in some subcases to chattering optimal controls. Finally a mean field game is introduced where a set of homogeneous players minimize each the cost functional of the single player control problem, sharing the fixed cost. For such a game the existence of equilibrium points is proved and their characterization is displayed.

A linear quadratic control problem with mean field dependent fixed costs

FAGGIAN, Silvia;PESENTI, Raffaele
2014-01-01

Abstract

The paper deals with an optimal control problem in one dimension, having affine dynamics, and a running cost which is discontinuous in the control variable. More precisely, besides terms which are quadratic in the state and control variables, a mean field dependent fixed cost is paid as long as the control is activated (not null). This leads to a problem that, although simple, does not seem to fall into a known class. Moreover, the outcome is completely different according to the magnitude of the fixed cost in comparison to other parameters, such as the (constant) disturbance appearing into the state equation. By means of some tools of Bellman’s Dynamic Programming and viscosity solutions, we are able to provide an explicit formula for the value function in all different cases, as well as a feedback formula for the optimal control, leading in some subcases to chattering optimal controls. Finally a mean field game is introduced where a set of homogeneous players minimize each the cost functional of the single player control problem, sharing the fixed cost. For such a game the existence of equilibrium points is proved and their characterization is displayed.
2014
Proceedings of the 52nd IEEE Conference on Decision and Control
File in questo prodotto:
File Dimensione Formato  
1761.pdf

non disponibili

Tipologia: Documento in Post-print
Licenza: Accesso chiuso-personale
Dimensione 138.15 kB
Formato Adobe PDF
138.15 kB Adobe PDF   Visualizza/Apri

I documenti in ARCA sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10278/38562
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 0
  • ???jsp.display-item.citation.isi??? 0
social impact