Fictitious Control Technique in PDE-Constrained Optimization

Erscheinungsjahr 2026
ISBN: 978-3-032-27319-2
Verlag: Springer

Buch, Englisch, 190 Seiten, Format (B × H): 168 mm x 240 mm

Reihe: Synthesis Lectures on Mathematics & Statistics

This book develops the concept of fictitious controls for studying optimal control problems associated with nonlinear elliptic and parabolic equations. The authors address several challenges currently faced by researchers, particularly those arising in optimization problems for degenerate boundary value problems (BVPs), boundary value problems associated with coupled systems of essentially nonlinear partial differential equations (PDEs), as well as in problems where uniqueness of solutions cannot be expected in standard functional spaces due to degeneracy and/or 1-data.  An emphasis is placed on optimization problems that are ill-posed or involve nonstandard nonlinearity such as quadratic terms, exponential terms, and/or degenerate elliptic operators. These problems are not just theoretical, but rather have practical applications in various fields, including mechanics, chemistry, and image processing. As such, the authors provide mathematical tools that can be used to solve complex problems in different areas of fundamental and applied science. The book further explores optimization problems for nonlinear degenerate partial differential equations with control and state constraints using the concept of fictitious controls and provides a comprehensive treatment of the wide class of optimal control problems for Perona-Malik equations, investigating the existence of optimal solutions and methods for their approximation. Additionally, the book includes innovative research in the regularity issues of the variational and duality solutions of boundary value problems for degenerate elliptic equations with mixed boundary conditions. The book is a valuable resource for researchers and graduate students in optimization theory, approximation methods, and partial differential equations.

Zielgruppe

Professional/practitioner

Autoren/Hrsg.

Kogut, Peter

Kohut, Yaroslav

Fachgebiete

• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Optimierung
• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen
• Technische Wissenschaften Elektronik | Nachrichtentechnik Elektronik
• Technische Wissenschaften Elektronik | Nachrichtentechnik Nachrichten- und Kommunikationstechnik Regelungstechnik
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Numerische Mathematik
• Mathematik | Informatik EDV | Informatik Informatik Bildsignalverarbeitung

Inhaltsverzeichnis

Introduction.- Functional Framework.- Fictitious Control Technique in Optimization Problems for Steady-State Perona-Malik Equations.- Fictitious Control Technique in Optimization Problems for Evolutionary Perona-Malik Equation.- Fictitious Control Technique in Optimization Problems for One Class of Quasi-Static Thermo–Mechanical Damage Models.- Fictitious Control Technique for One Class of Optimal Control Problems for Ill-Posed Nonlinear Elliptic Equations.- Regularity Issues in Variational and Duality Solutions for Linear Degenerate Elliptic Equations. References.

Über Autor(innen)

Peter Kogut, Ph.D., Habilitated Doctor in Mathematics, is a Professor in the Department of Mathematical Analysis and Optimization at Oles Honchar Dnipro National University, where his teaching focuses on variational methods in mathematical physics, optimal control theory for distributed systems, and nonlinear functional analysis. His research interests include the different areas of optimization theory, optimal control problems for PDEs, image processing, asymptotic analysis, optimization in partially ordered spaces, and homogenization theory. He has published over 200 journal articles in mathematical sciences and three books.

Yaroslav Kohut, Ph.D., is a Researcher in the Department of Mathematical Analysis and Optimization at Oles Honchar Dnipro National University, where his scientific activity focuses on variational methods in mathematical physics and optimal control theory for distributed systems. He received a Ph.D. in Mathematical Analysis from Oles Honchar Dnipro National University in 2025, and his research interests include optimization theory, optimal control problems for PDEs, and image processing.