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This book provides an introduction to nonsmooth constrained optimal control theory over infinite horizon, tackling the mathematical challenges that arise when classical finite-horizon methods prove inadequate. The work focuses on recent advances in handling nonsmooth time-dependent data and state constraints scenarios that commonly arise in fields such as engineering, machine learning, and artificial intelligence.
At its core, the book establishes foundational contributions, including viability results, extensions of Pontryagin's maximum principle to infinite horizons, regularity analysis of value functions, and necessary optimality conditions, with particular attention to transversality conditions and sensitivity relations. The analysis culminates in studying Hamilton-Jacobi-Bellman equations through weak solution notions suited to the nonsmooth framework.
Written as a brief course, the book aims to provide graduate students and researchers with the mathematical tools needed to analyze optimal control problems over infinite time horizons, where standard approaches may not apply.
List of contents
Preface.- Introduction.- Overview.- First Order Necessary Conditions for Infinite Horizon Optimal Control Problems Under State Constraints.- Lipschitz Continuity of the Value Functions.- Hamilton-Jacobi-Bellman Equations with Time-Measurable Data and infinite Horizons.- Representation of Fiber-Convex Hamiltonian with Time-Measureable Data.- Appendix.
About the author
Vincenzo Basco earned his M.S. degree in Pure and Applied Mathematics from the University of Rome Tor Vergata in 2015, and his Ph.D. in Mathematics from Sorbonne University, Paris, in 2019. From 2019 to 2020, he was a research fellow in optimal control theory at the Department of Electrical and Electronic Engineering, University of Melbourne, Australia. Between 2020 and 2024, he worked for a space systems company. He is currently with a multi-service company, where he leads the development of AI solutions. His primary research interests include optimal control theory, Hamilton–Jacobi–Bellman equations, nonsmooth analysis, and artificial intelligence.
