Stabilizability in optimization problems with unbounded data

Anna Chiara Lai; Monica Motta

doi:10.3934/dcds.2020371

Article Contents

2021, Volume 41, Issue 5: 2447-2474. Doi: 10.3934/dcds.2020371

This issue Previous Article Strong blow-up instability for standing wave solutions to the system of the quadratic nonlinear Klein-Gordon equations Next Article Orbital and asymptotic stability of a train of peakons for the Novikov equation

Stabilizability in optimization problems with unbounded data

Anna Chiara Lai^1, and
Monica Motta^2, ,

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
^* Corresponding author

Received: March 31, 2020

Revised: August 31, 2020

Early access: November 2020

Published: May 2021

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"

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Abstract

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.

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Mathematics Subject Classification: Primary: 93D05, 93D20; Secondary: 49J15, 49N25.

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