This book presents a rigorous and modern treatment of variational and quasi–variational inequalities, with applications to impulse control, nonlinear partial differential equations, and mathematical finance. It develops the theory of state-dependent constrained systems, where the admissible set depends on the unknown solution itself, leading to challenging nonlinear analytical structures.

The book covers existence and stability theory, fixed-point methods, impulse control problems, asymptotic analysis, penalty techniques, and numerical approximation methods. Special attention is devoted to applications in optimal control and American option pricing.

Combining functional analysis, nonlinear PDEs, optimization, and computational mathematics, this book offers a unified framework that bridges theoretical developments with practical modeling applications. Detailed numerical examples and computational approaches complement the analytical results.

Designed for researchers, graduate students, and specialists in applied mathematics, control theory, numerical analysis, and financial mathematics, the book provides both advanced theoretical insight and practical tools for studying complex systems governed by state-dependent constraints.

Professor Salah Boulaaras is a Professor of Mathematics at the Department of Mathematics, College of Science, Qassim University, Saudi Arabia. His research interests include numerical analysis, partial differential equations, variational and quasi-variational inequalities, and mathematical modeling. He obtained his Ph.D. in numerical and functional analysis with distinction and was awarded the HDR (Habilitation à Diriger des Recherches) in numerical analysis from Annaba University, Algeria. He has more than 15 years of experience in teaching and research and has held academic positions at several universities in Algeria and Saudi Arabia. Professor Boulaaras is ranked among the top 2% of scientists worldwide over the past five years, according to internationally recognized bibliometric rankings. He has authored and coauthored more than 597 scientific publications, including journal articles and books, with many of his works published in internationally indexed journals. His research is characterized by a strong combination of theoretical analysis and numerical methods, with applications to nonlinear partial differential equations, variational inequalities, and complex physical models.

Professor Mohamed Haiour is a Professor of Mathematics at the Department of Mathematics, Badji Mokhtar University, Annaba, Algeria. He holds a Ph.D. in applied mathematics and his research focuses on variational inequalities, free boundary problems, impulse control, and numerical methods for constrained partial differential equations. His work covers both theoretical and computational aspects of elliptic and parabolic variational and quasi-variational inequalities, including regularity analysis, subsolution and supersolution techniques, finite element methods, and asymptotic behavior. He has published extensively in these areas and has supervised several graduate students. Professor Haiour has contributed to the development of reliable numerical approaches for applications in control theory, reliability engineering, and mathematical finance, particularly in the study of American options and impulse control models.