# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2020371

## Stabilizability in optimization problems with unbounded data

 1 Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza Università di Roma, Via Scarpa, 16, Roma 00181, Italy 2 Dipartimento di Matematica "Tullio Levi-Civita", Università di Padova, Via Trieste, 63, Padova 35121, Italy

* Corresponding author

Received  April 2020 Revised  September 2020 Published  November 2020

Fund Project: This research is partially supported by the Padua University grant SID 2018 "Controllability, stabilizability and infimun gaps for control systems", prot. BIRD 187147, and by the INdAM-GNAMPA Project 2020 "Extended control problems: gap, higher order conditions and Lyapunov functions"

In this paper we extend the notions of sample and Euler stabilizability to a set of a control system to a wide class of systems with unbounded controls, which includes nonlinear control-polynomial systems. In particular, we allow discontinuous stabilizing feedbacks, which are unbounded approaching the target. As a consequence, sampling trajectories may present a chattering behaviour and Euler solutions have in general an impulsive character. We also associate to the control system a cost and provide sufficient conditions, based on the existence of a special Lyapunov function, which allow for the existence of a stabilizing feedback that keeps the cost of all sampling and Euler solutions starting from the same point below the same value, in a uniform way.

Citation: Anna Chiara Lai, Monica Motta. Stabilizability in optimization problems with unbounded data. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020371
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