Jiongmin Yong - Böcker

The maximum principle and dynamic programming are the two most commonly used approaches in solving optimal control problems. These approaches have been developed independently. The theme of this book is to unify these two approaches, and to demonstrate that the viscosity solution theory provides the framework to unify them.

Infinite dimensional systems can be used to describe many physical phenomena in the real world. Well-known examples are heat conduction, vibration of elastic material, diffusion-reaction processes, population systems and others. Thus, the optimal control theory for infinite dimensional systems has a wide range of applications in engineering, economics and some other fields. On the other hand, this theory has its own mathematical interests since it is regarded as a generalization for the classical calculus of variations and it generates many interesting mathematical questions. The Pontryagin maximum principle, the Bellman dynamic programming method and the Kalman optimal linear quadratic regulator theory are regarded as the three milestones of modern (finite dimensional) control theory. Since the 1960s, the corresponding theory for infinite dimensional systems has also been developed. The essential difficulties for the infinite dimensional theory come from two aspects: the unboundedness of the differential operator or the generator of the strongly continuous semigroup and the lack of the local compactness of the underlying spaces.The purpose of this book is to introduce optimal control theory for infinite dimensional systems. The authors present the existence theory for optimal control problems. Some applications are also included in this volume.

As is well known, Pontryagin's maximum principle and Bellman's dynamic programming are the two principal and most commonly used approaches in solving stochastic optimal control problems. * An interesting phenomenon one can observe from the literature is that these two approaches have been developed separately and independently. Since both methods are used to investigate the same problems, a natural question one will ask is the fol­ lowing: (Q) What is the relationship betwccn the maximum principlc and dy­ namic programming in stochastic optimal controls? There did exist some researches (prior to the 1980s) on the relationship between these two. Nevertheless, the results usually werestated in heuristic terms and proved under rather restrictive assumptions, which were not satisfied in most cases. In the statement of a Pontryagin-type maximum principle there is an adjoint equation, which is an ordinary differential equation (ODE) in the (finite-dimensional) deterministic case and a stochastic differential equation (SDE) in the stochastic case. The system consisting of the adjoint equa­ tion, the original state equation, and the maximum condition is referred to as an (extended) Hamiltonian system. On the other hand, in Bellman's dynamic programming, there is a partial differential equation (PDE), of first order in the (finite-dimensional) deterministic case and of second or­ der in the stochastic case. This is known as a Hamilton-Jacobi-Bellman (HJB) equation.

Infinite dimensional systems can be used to describe many phenomena in the real world. As is well known, heat conduction, properties of elastic­ plastic material, fluid dynamics, diffusion-reaction processes, etc., all lie within this area. The object that we are studying (temperature, displace­ ment, concentration, velocity, etc.) is usually referred to as the state. We are interested in the case where the state satisfies proper differential equa­ tions that are derived from certain physical laws, such as Newton's law, Fourier's law etc. The space in which the state exists is called the state space, and the equation that the state satisfies is called the state equation. By an infinite dimensional system we mean one whose corresponding state space is infinite dimensional. In particular, we are interested in the case where the state equation is one of the following types: partial differential equation, functional differential equation, integro-differential equation, or abstract evolution equation. The case in which the state equation is being a stochastic differential equation is also an infinite dimensional problem, but we will not discuss such a case in this book.

This book gathers the most essential results, including recent ones, on linear-quadratic optimal control problems, which represent an important aspect of stochastic control.

This book gathers the most essential results, including recent ones, on linear-quadratic optimal control problems, which represent an important aspect of stochastic control.

Backward Stochastic Volterra Integral Equations (BSVIEs) have evolved into one of the most powerful and flexible mathematical frameworks for modeling systems with memory, time‑inconsistency, nonlinear dynamics, and path‑dependent uncertainty. Spanning foundational theory through cutting‑edge research, this comprehensive monograph offers the first unified and rigorous treatment of BSVIEs in their full generality.This landmark volume develops the analytic core of the subject—from classical stochastic calculus and Malliavin techniques to the modern theory of M‑solutions, adapted solutions, comparison principles, and representation PDEs. Building systematically from BSDEs and forward Volterra equations, the book presents the most complete framework to date for well‑posedness, stability, regularity, and qualitative analysis of BSVIEs, including equations with non‑uniform, quadratic, and superquadratic generators.Beyond theory, the manuscript showcases the profound role of BSVIEs across contemporary applied mathematics. Readers will find deep connections to optimal control with memory, dynamic risk measures, recursive utilities, rough volatility models, mean‑field interactions, stochastic games, and nonlinear pricing. The book also elaborates maximum principles, duality structures, and variational methods that place BSVIEs at the center of modern stochastic control and mathematical finance.Key features include:A complete and rigorous development of Type I, Type II, and anticipated BSVIEsDetailed well‑posedness theory under Lipschitz, Osgood, quadratic, and superquadratic growthModern tools including Malliavin calculus, BMO martingales, nonlocal PDE representations, and comparison principlesFull treatment of mean‑field BSVIEs and McKean–Vlasov interactionsOptimal control of systems with memory: adjoint equations, variational inequalities, and maximum principlesApplications to finance, recursive utilities, risk measures, equilibrium pricing, and rough volatilityOver 200 references connecting classical Volterra theory to the most recent advances (up to 2025)Comprehensive, rigorous, and forward‑looking, this monograph is an essential reference for graduate students, researchers, and practitioners working in stochastic analysis, optimal control, mathematical finance, engineering, and applied probability. It not only consolidates the existing theory of BSVIEs but also lays the groundwork for their next decade of development.

The IFIP-TC7, WG 7.2 Conference on Control Theory of Distributed Parameter Systems and Applications was held at Fudan University, Shanghai, China, May 6-9, 1990. The papers presented cover a wide variety of topics, e.g. the theory of identification, optimal control, stabilization, controllability, stochastic control as well as appplications in heat exchangers, elastic structures, nuclear reactor, meteorology etc.

This book is a survey/monograph on the recently developed theory of forward-backward stochastic differential equations (FBSDEs). Basic techniques such as the method of optimal control, the "Four Step Scheme", and the method of continuation are presented in full. Related topics such as backward stochastic PDEs and many applications of FBSDEs are also discussed in detail. The book is suitable for readers with basic knowledge of stochastic differential equations, and some exposure to the stochastic control theory and PDEs. The book can be used for researchers and/or senior graduate students in the areas of probability, control theory, mathematical finance, and other related fields.

The symposium discusses and explores the current and future development of some aspects of the theory of nonlinear control systems, adaptive control and filtering, robust control and H∞ optimization, stochastic systems and white noise analysis, etc.

The book deals with topics such as the pricing of various contingent claims within different frameworks, risk-sensitive problems, optimal investment, defaultable term structure, etc. It also reflects on some recent developments in certain important aspects of mathematical finance.

Mathematical analysis serves as a common foundation for many research areas of pure and applied mathematics. It is also an important and powerful tool used in many other fields of science, including physics, chemistry, biology, engineering, finance, and economics. In this book, some basic theories of analysis are presented, including metric spaces and their properties, limit of sequences, continuous function, differentiation, Riemann integral, uniform convergence, and series.After going through a sequence of courses on basic calculus and linear algebra, it is desirable for one to spend a reasonable length of time (ideally, say, one semester) to build an advanced base of analysis sufficient for getting into various research fields other than analysis itself, and/or stepping into more advanced levels of analysis courses (such as real analysis, complex analysis, differential equations, functional analysis, stochastic analysis, amongst others). This book is written to meet such a demand. Readers will find the treatment of the material is as concise as possible, but still maintaining all the necessary details.

Xunjing Li (1935-2003) was a pioneer in control theory in China. He was known in the Chinese community of applied mathematics, and in the global community of optimal control theory of distributed parameter systems. He has made important contributions to the optimal control theory of distributed parameter systems, in particular regarding the first-order necessary conditions (Pontryagin-type maximum principle) for optimal control of nonlinear infinite-dimensional systems. He directed the Seminar of Control Theory at Fudan towards stochastic control theory in 1980s, and mathematical finance in 1990s, which has led to several important subsequent developments in both closely interactive fields. These remarkable efforts in scientific research and education, among others, gave birth to the so-called “Fudan School”.This proceedings volume includes a collection of original research papers or reviews authored or co-authored by Xunjing Li's former students, postdoctoral fellows, and mentored scholars in the areas of control theory, dynamic systems, mathematical finance, and stochastic analysis, among others.

Mathematically, most of the interesting optimization problems can be formulated to optimize some objective function, subject to some equality and/or inequality constraints. This book introduces some classical and basic results of optimization theory, including nonlinear programming with Lagrange multiplier method, the Karush-Kuhn-Tucker method, Fritz John's method, problems with convex or quasi-convex constraints, and linear programming with geometric method and simplex method.A slim book such as this which touches on major aspects of optimization theory will be very much needed for most readers. We present nonlinear programming, convex programming, and linear programming in a self-contained manner. This book is for a one-semester course for upper level undergraduate students or first/second year graduate students. It should also be useful for researchers working on many interdisciplinary areas other than optimization.

This book uses a small volume to present the most basic results for deterministic two-person differential games. The presentation begins with optimization of a single function, followed by a basic theory for two-person games. For dynamic situations, the author first recalls control theory which is treated as single-person differential games. Then a systematic theory of two-person differential games is concisely presented, including evasion and pursuit problems, zero-sum problems and LQ differential games.The book is intended to be self-contained, assuming that the readers have basic knowledge of calculus, linear algebra, and elementary ordinary differential equations. The readership of the book could be junior/senior undergraduate and graduate students with majors related to applied mathematics, who are interested in differential games. Researchers in some other related areas, such as engineering, social science, etc. will also find the book useful.