Minimum time problem with impulsive and ordinary controls

Monica Motta

doi:10.3934/dcds.2018252

Article Contents

2018, Volume 38, Issue 11: 5781-5809. Doi: 10.3934/dcds.2018252

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$L^p$

mapping properties for nonlocal Schrödinger operators with certain potentials

Minimum time problem with impulsive and ordinary controls

Monica Motta

Università di Padova, Via Trieste, 63, Padova 35121, Italy

Received: December 31, 2017

Revised: March 31, 2018

Published: August 2018

This research is partially supported by the INdAM-GNAMPA Project 2017 "Optimal impulsive control: higher order necessary conditions and gap phenomena"; and by the Padova University grant PRAT 2015 "Control of dynamics with reactive constraints"

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Abstract

Given a nonlinear control system depending on two controls $u$ and $v$, with dynamics affine in the (unbounded) derivative of $u$ and a closed target set $\mathcal{S}$ depending both on the state and on the control $u$, we study the minimum time problem with a bound on the total variation of $u$ and $u$ constrained in a closed, convex set $U$, possibly with empty interior. We revisit several concepts of generalized control and solution considered in the literature and show that they all lead to the same minimum time function $T$. Then we obtain sufficient conditions for the existence of an optimal generalized trajectory-control pair and study the possibility of Lavrentiev-type gap between the minimum time in the spaces of regular (that is, absolutely continuous) and generalized controls. Finally, under a convexity assumption on the dynamics, we characterize $T$ as the unique lower semicontinuous solution of a regular HJ equation with degenerate state constraints.

Keywords:

Impulsive optimal control problems,
existence of optimal controls,
Lavrentiev phenomenon,
Hamilton Jacobi equations,
lower semicontinuous viscosity solutions.

Mathematics Subject Classification: Primary: 49N25, 49L25; Secondary: 34H5, 49J15.

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