We consider nonlinear transport equations with non-local velocity describing the time-evolution of a measure. Such equations often appear when considering the mean-field limit of finite-dimensional systems modeling collective dynamics. We address the problem of controlling these equations by means of a time-varying bounded control action localized on a time-varying control subset of small Lebesgue measure. We first define dissipativity for nonlinear transport equations in terms of Lie derivatives of a Lyapunov function depending on the measure. Then, assuming that the uncontrolled system is dissipative, we provide an explicit construction of a control law steering the system to an invariant sublevel of the Lyapunov function. The control function and the control domain are designed in terms of the Lie derivatives of the Lyapunov function. In this sense the construction can be seen as an infinite-dimensional analogue of the well-known Jurdjevic-Quinn procedure. Moreover, the control law presents sparsity properties in the sense that the support of the control is small. Finally, we show that our result applies to a large class of kinetic equations modeling multi-agent dynamics.

Caponigro, M., Piccoli, B., Rossi, F., Tr('e)lat, E. (2017). Mean-field sparse Jurdjevic--Quinn control. MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES, 27(7), 1223-1253 [10.1142/S0218202517400140].

Mean-field sparse Jurdjevic--Quinn control

Caponigro, Marco
;
2017-01-01

Abstract

We consider nonlinear transport equations with non-local velocity describing the time-evolution of a measure. Such equations often appear when considering the mean-field limit of finite-dimensional systems modeling collective dynamics. We address the problem of controlling these equations by means of a time-varying bounded control action localized on a time-varying control subset of small Lebesgue measure. We first define dissipativity for nonlinear transport equations in terms of Lie derivatives of a Lyapunov function depending on the measure. Then, assuming that the uncontrolled system is dissipative, we provide an explicit construction of a control law steering the system to an invariant sublevel of the Lyapunov function. The control function and the control domain are designed in terms of the Lie derivatives of the Lyapunov function. In this sense the construction can be seen as an infinite-dimensional analogue of the well-known Jurdjevic-Quinn procedure. Moreover, the control law presents sparsity properties in the sense that the support of the control is small. Finally, we show that our result applies to a large class of kinetic equations modeling multi-agent dynamics.
2017
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/05
English
Con Impact Factor ISI
Multi-agent systems
crowd control
control systems
networked control
distributed parameter systems
control of partial differential equations
Lyapunov methods
Caponigro, M., Piccoli, B., Rossi, F., Tr('e)lat, E. (2017). Mean-field sparse Jurdjevic--Quinn control. MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES, 27(7), 1223-1253 [10.1142/S0218202517400140].
Caponigro, M; Piccoli, B; Rossi, F; Tr('e)lat, E
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/349144
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