Semiconcavity for the minimum time problem in presence of time delay effects

Elisa Continelli; Cristina Pignotti

doi:10.3934/mcrf.2024061

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2025, Volume 15, Issue 3: 1020-1048. Doi: 10.3934/mcrf.2024061

This issue Previous Article Sampled-data based adaptive mean field games for leader-follower stochastic multi-agent systems Next Article Optimal control of semilinear elliptic partial differential equations with non-Lipschitzian nonlinearities

Semiconcavity for the minimum time problem in presence of time delay effects

Elisa Continelli and
Cristina Pignotti^,

Dipartimento di Ingegneria e Scienze dell'Informazione e Matematica, Università degli Studi dell'Aquila, Via Vetoio, Loc. Coppito, 67100 L'Aquila, Italy

^*Corresponding author: Cristina Pignotti
^*Corresponding author: Cristina Pignotti

Received: April 2024

Revised: October 2024

Early access: November 28, 2024

Published online: November 28, 2024

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Abstract

In this paper, we deal with a minimum time problem in presence of a time delay $ \tau. $ The value function of the optimal control problem is no longer defined in a subset of $ \mathbb{R}^{n} $, as it happens in the undelayed case, but its domain is a subset of the Banach space $ C([-\tau, 0];\mathbb{R}^{n}) $. For the undelayed minimum time problem, it is known that the value function associated with it is semiconcave in a subset of the reachable set and is a viscosity solution of a suitable Hamilton-Jacobi-Bellman equation. The Hamilton-Jacobi theory for optimal control problems involving time delays has been developed by several authors. Here, we are rather interested in investigating the regularity properties of the minimum time functional. Extending classical arguments, we are able to prove that the minimum time functional is semiconcave in a suitable subset of the reachable set.

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Mathematics Subject Classification: Primary: 34K05, 49K15, 49L20.

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