Originally published in 2000, this is the second volume of a comprehensive two-volume treatment of quadratic optimal control theory for partial differential equations over a finite or infinite time horizon, and related differential (integral) and algebraic Riccati equations. Both continuous theory and numerical approximation theory are included. The authors use an abstract space, operator theoretic approach, which is based on semigroups methods, and which unifies across a few basic classes of evolution. The various abstract frameworks are motivated by, and ultimately directed to, partial differential equations with boundary/point control. Volume 2 is focused on the optimal control problem over a finite time interval for hyperbolic dynamical systems. A few abstract models are considered, each motivated by a particular canonical hyperbolic dynamics. It presents numerous fascinating results. These volumes will appeal to graduate students and researchers in pure and applied mathematics and theoretical engineering with an interest in optimal control problems.

Control Theory for Partial Differential Equations: Continuous and Approximation Theories

In part I of this paper [S. Chen, X. Li, and X. Zhou, SIAM J. Control Optim., 36 (1998), pp. 1685--1702], an optimization model of stochastic linear quadratic regulators (LQRs) with indefinite control cost weighting matrices is proposed and studied. In this sequel, the problem of solving LQR models with system diffusions dependent on both state and control variables, which is left open in part I, is tackled. First, the solvability of the associated stochastic Riccati equations (SREs) is studied in the normal case (namely, all the state and control weighting matrices and the terminal matrix in the cost functional are nonnegative definite, with at least one positive definite), which in turn leads to an optimal state feedback control of the LQR problem. In the general indefinite case, the problem is decomposed into two optimal LQR problems, one with a forward dynamics and the other with a backward dynamics. The well-posedness and solvability of the original LQR problem are then obtained by solving these two subproblems, and an optimal control is explicitly constructed. Examples are presented to illustrate the results.

Stochastic Linear Quadratic Regulators with Indefinite Control Weight Costs. II

1. Introduction


1.1. Searching the Mechanism of Evaluations of Risky Assets


1.2. Axiomatic Assumptions for Evaluations of Derivatives


1.3. Organization of the Lecture






2. Brownian Filtration Consistent Evaluations and Expectations


2.1. Main Notations and Definitions


2.2. Open image in new window -Consistent Nonlinear Expectations


2.3. Open image in new window -Consistent Nonlinear Evaluations






3. Backward Stochastic Differential Equations: g-Evaluations and g-Expectations


3.1. BSDE: Existence, Uniqueness and Basic Estimates


3.2. 1-Dimensional BSDE


3.3. A Monotonic Limit Theorem of BSDE


3.4. g-Martingales and (Nonlinear) g-Supermartingale Decomposition Theorem






4. Finding the Mechanism: Is an Open image in new window -Expectation a g-Expectation?


4.1. Open image in new window -Dominated Open image in new window -Expectations


4.2. Open image in new window -Consistent Martingales


4.3. BSDE under Open image in new window -Consistent Nonlinear Expectations


4.4. Decomposition Theorem for Open image in new window -Supermartingales


4.5. Representation Theorem of an Open image in new window -Expectation by a g-Expectation


4.6. How to Test and Find g?


4.7. A General Situation: Open image in new window -Evaluation Representation Theorem






5. Dynamic Risk Measures


6. Numerical Solution of BSDEs: Euler’s Approximation


7. Appendix


7.1. Martingale Representation Theorem


7.2. A Monotonic Limit Theorem of Ito’s Processes


7.3. Optional Stopping Theorem for Open image in new window -Supermartingale






References


References on BSDE and Nonlinear Expectations

/pdf/nonlinear-expectations-nonlinear-evaluations-and-risk-1df84s74m8.pdf

Nonlinear Expectations, Nonlinear Evaluations and Risk Measures

Inverse/Observability Estimates for Second-Order Hyperbolic Equations with Variable Coefficients

Consider the minimization of the following quadratic cost functional: $$J(u):=E\langle Mx_T,x_T\rangle +E\int_0^T(\langle Q_sx_s,x_s\rangle +\langle N_su_s,u_s\rangle )\, ds,$$ where x is the solution of the following linear stochastic control system: $$ \eq{dx_t=&(A_tx_t+B_tu_t)\, dt +\sum_{i=1}^d(C_t^ix_t+D_t^iu_t)\, dW_t^i,\cr x_0=h \cr} $$ u is a square integrable adapted process. The problem is conventionally called the stochastic LQ (the abbreviation of "linear quadratic") problem. We are concerned with the following general case: the coefficients A,B,Ci,Di, Q, N, and M are allowed to be adapted processes or random matrices. We prove the existence and uniqueness result for the associated Riccati equation, which in our general case is a backward stochastic differential equation with the generator (the drift term) being highly nonlinear in the two unknown variables. This solves Bismut and Peng's long-standing open problem (for the case of a Brownian filtration), which was initially proposed by the French mathematician J. M. Bismut [in Seminaire de Probabilites XII, Lecture Notes in Math. 649, C. Dellacherie, P. A. Meyer, and M. Weil, eds., Springer-Verlag, Berlin, 1978, pp. 180--264]. We also provide a rigorous derivation of the Riccati equation from the stochastic Hamilton system. This completes the interrelationship between the Riccati equation and the stochastic Hamilton system as two different but equivalent tools for the stochastic LQ problem.
There are two key points in our arguments. The first one is to connect the existence of the solution of the Riccati equation to the homomorphism of the stochastic flows derived from the optimally controlled system. Actually, we establish their equivalence. As a consequence, we can construct solutions to a sequence of suitably modified Riccati equations in terms of the associated stochastic Hamilton systems (and the optimal controls). The second key point is to establish a new type of a priori estimate for solutions of Riccati equations, with which we show that the sequence of constructed solutions has a limit which is a solution to the original Riccati equation.

General Linear Quadratic Optimal Stochastic Control Problems with Random Coefficients: Linear Stochastic Hamilton Systems and Backward Stochastic Riccati Equations

This paper is concerned with the stochastic linear quadratic optimal control problem (LQ problem, for short) for which the coefficients are allowed to be random and the cost functional is allowed to have a negative weight on the square of the control variable. Some intrinsic relations among the LQ problem, the stochastic maximum principle, and the (linear) forward—backward stochastic differential equations are established. Some results involving Riccati equation are discussed as well.

Stochastic Linear Quadratic Optimal Control Problems

Controllability properties of a partial differential equation (PDE) model describing a thermoelastic plate are studied. The PDE is comprised of a Kirchoff plate equation coupled to a heat equation on a bounded domain, with the coupling taking place on the interior and boundary of the domain. The coupling in this PDE is parameterized by a > 0. Control is exerted through the (two) free boundary conditions of the pla:te equation, and through the Robin boundary condition of the temperature. These controls have the physical interpretation, respectively, of inserted forces and moments, and prescribed temperature, all of which act on the edges of the plate. The main result here is that under such boundary control, and with initial data in the basic space of wellposedness, one can simultaneously control the displacement of the plate exactly, and the temperature approximately. Moreover, the thermal control may be taken to be arbitrarily smooth in time and space, and the thermal control region may be any nonempty subset of the boundary. This controllability holds for arbitrary values of the coupling parameter a.

Control of Distributed Parameter and Stochastic Systems

Necessary and sufficient conditions are established in this paper for the existence of positive- and/or negative-definite solutions to the algebraic Riccati equation with indefinite coefficient. An iterative procedure is also given for computing such a solution.

Necessary and sufficient conditions for the existence of positive solutions to algebraic Riccati equations with indefinite quadratic term

A linear quadratic optimal control problem with coexisting initial and persistent disturbances is studied. Upper and lower values and relevant algebraic Riccati equations (ARE for short) are introduced. Various relations among these values are presented. The solvability of the resulting AREs is shown to be closely related to the solvability of the original optimal control problem. A formula is obtained for the solution of one of the AREs, which is of nonstandard form. Several unexpected features of the original problem are revealed from the standpoint of differential games. Some known results on the so-calledH∞ optimal control problem are recovered.

Shuping Chen

Papers

Stochastic Linear Quadratic Optimal Control Problems

Control of Distributed Parameter and Stochastic Systems

Necessary and sufficient conditions for the existence of positive solutions to algebraic Riccati equations with indefinite quadratic term

A linear quadratic optimal control problem with disturbances—An algebraic Riccati equation and differential games approach