Optimality conditions for controls of semilinear evolution systems with mixed constraints
Jiongmin Yong
In this paper, we study an optimal control problem for semilinear evolution equations with a mixed constraint of the state and the control. One of the motivations is the problem with a state dependent control domain. Our main result is the Pontryagin type necessary conditions for the optimal controls. The main tools we use are the Ekeland variational principle and the spike perturbation technique.
keywords: mixed constraint maximum principle. distributed parameter system
Remarks on some short rate term structure models
Jiongmin Yong
Some widely used short interest rate term structure models are discussed. By establishing certain estimates on the solutions of stochastic differential inequalities, we found that the interest rate processes obtained from these models do not have enough integrability which leads to some defects in several applications.
keywords: Term structure interest rate models stochastic differential inequalities.
Goong Chen Peng-Fei Yao Jiongmin Yong Bingyu Zhang Xu Zhang
Professor David L. Russell just celebrated his 70th birthday in 2009. His numerous important and fundamental contributions to the control theory of partial differential equations have won him recognitions as a true pioneer and giant in this field. And his pleasant personality and ready helpfulness have won our hearts as his admirers, students, and friends.
   Prof. Russell's lifetime love and interests are control theory and partial differential equations. We have asked Goong Chen, a former PhD student of Prof. Russell, to write a survey of his career and mathematical work. Moreover, as a way to pay our tribute to him, it is only proper here that we put together a collection of papers by researchers who work on these and closely related subjects of Prof. Russell's. Indeed, the topics collected are extensive, including the theory and applications of analysis, computation, control, optimization, etc., for various systems arising from conservation laws and quasilinear hyperbolic systems, fluids, mathematical finance, neural networks, reaction-diffusion, structural dynamics, stochastic PDEs, thermoelasticity, viscoelasticity, etc. They are covered in great depth by the authors. There are a total of 35 papers finally accepted for publication after going through strict manuscript reviews and revision. Due to the large number of papers on hand, we divide them into Groups I (20 papers) and II (15 papers). Papers in Group I are published in Discrete and Continuous Dynamical Systems-Series B (DCDS-B) Volume 14, No. 4 (2010), while those in Group II are published in Journal of Systems Science and Complexity (JSSC) Volume 23, No. 3 (2010). This division is up to the preference of the authors. We thank all the authors and reviewers for their critical contributions and valuable assistance that make the publication of these two journal issues possible.

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A biographical note and tribute to xunjing li on his 80th birthday
Jin Ma Shige Peng Jiongmin Yong Xunyu Zhou
Professor Xunjing Li was born in Qingdao, Shandong Province, China, on the 13th June 1935. Shandong is a province with a rich culture that has nurtured a great number of influential intellectuals during its long history, including Confucius. Immediately after his graduation from the Department of Mathematics at Shandong University in 1956, Professor Li was enrolled into the master program at Fudan University specializing in function approximation theory, supervised by Professor Jiangong Chen, one of the most prominent Chinese mathematicians in modern history. He stayed at Fudan as an Assistant Lecturer upon graduation in 1959, and was promoted to Lecturer, Associate Professor and Professor in 1962, 1980, and 1984, respectively. He became a Chair Professor in 1997, before retiring in 2001. He died of cancer in February 2003 at the age of 68.

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Two-person zero-sum linear quadratic stochastic differential games by a Hilbert space method
Libin Mou Jiongmin Yong
An open-loop two-person zero-sum linear quadratic (LQ for short) stochastic differential game is considered. The controls for both players are allowed to appear in both the drift and diffusion of the state equation, the weighting matrices in the payoff/cost functional are not assumed to be definite/non-singular, and the cross-terms between two controls are allowed to appear. A forward-backward stochastic differential equation (FBSDE, for short) and a generalized differential Riccati equation are introduced, whose solvability leads to the existence of the open-loop saddle points for the corresponding two-person zero-sum LQ stochastic differential game, under some additional mild conditions. The main idea is a thorough study of general two-person zero-sum LQ games in Hilbert spaces.
keywords: Hilbert method. open-loop controls Stochastic games linear-quadratic saddle points
A deterministic linear quadratic time-inconsistent optimal control problem
Jiongmin Yong
A time-inconsistent optimal control problem is formulated and studied for a controlled linear ordinary differential equation with a quadratic cost functional. A notion of time-consistent equilibrium strategy is introduced for the original time-inconsistent problem. Under certain conditions, we construct an equilibrium strategy which can be represented via a Riccati--Volterra integral equation system. Our approach is based on a study of multi-person hierarchical differential games.
keywords: linear-quadratic optimal control problem Time-inconsistency multi-person hierarchical differential games time-consistent equilibrium strategy Riccati--Volterra integral equation system.
Time-inconsistent optimal control problems and the equilibrium HJB equation
Jiongmin Yong
A general time-inconsistent optimal control problem is considered for stochastic differential equations with deterministic coefficients. Under suitable conditions, a Hamilton-Jacobi-Bellman type equation is derived for the equilibrium value function of the problem. Well-posedness such an equation is studied, and time-consistent equilibrium strategies are constructed. As special cases, the linear-quadratic problem and a generalized Merton's portfolio problem are investigated.
keywords: Time-inconsistent optimal control problem forward-backward stochastic differential equation. equilibrium value function equilibrium Hamilton-Jacobi-Bellman equation
Second-order necessary conditions for optimal control of semilinear elliptic equations with leading term containing controls
Hongwei Lou Jiongmin Yong

An optimal control problem for a semilinear elliptic equation of divergenceform is considered. Both the leading term and the semilinear term of the state equationcontain the control. The well-known Pontryagin type maximum principle for the optimal controls is the first-order necessary condition. When such a first-order necessary condition is singular in some sense, certain type of the second-order necessary condition will come in naturally. The aim of this paper is to explore such kind of conditions for our optimal control problem.

keywords: Optimal control semilinear elliptic equation control in leading term second-order necessary conditions
Exact controllability of linear stochastic differential equations and related problems
Yanqing Wang Donghui Yang Jiongmin Yong Zhiyong Yu

A notion of $L^p$-exact controllability is introduced for linear controlled (forward) stochastic differential equations with random coefficients. Several sufficient conditions are established for such kind of exact controllability. Further, it is proved that the $L^p$-exact controllability, the validity of an observability inequality for the adjoint equation, the solvability of an optimization problem, and the solvability of an $L^p$-type norm optimal control problem are all equivalent.

keywords: Controlled stochastic differential equation $L^p$-exact controllability observability inequality norm optimal control problem
A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon
Jianhui Huang Xun Li Jiongmin Yong
A linear-quadratic (LQ, for short) optimal control problem is considered for mean-field stochastic differential equations with constant coefficients in an infinite horizon. The stabilizability of the control system is studied followed by the discussion of the well-posedness of the LQ problem. The optimal control can be expressed as a linear state feedback involving the state and its mean, through the solutions of two algebraic Riccati equations. The solvability of such kind of Riccati equations is investigated by means of semi-definite programming method.
keywords: MF-stabilizability Riccati equation. Mean-field stochastic differential equation linear-quadratic optimal control

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