Linear-quadratic optimal control problems are considered for mean-field stochastic differential equations with deterministic coefficients. By a variational method, the optimality system is derived, which is a linear mean-field forward-backward stochastic differential equation. Using a decoupling technique, two Riccati differential equations are obtained which are uniquely solvable under certain conditions. Then a feedback representation is obtained for the optimal control.

/pdf/linear-quadratic-optimal-control-problems-for-mean-field-2c28v8u3ev.pdf

Linear-Quadratic Optimal Control Problems for Mean-Field Stochastic Differential Equations

A variable-order adaptive pseudospectral method is presented for solving optimal control problems. The method developed in this paper adjusts both themesh spacing and the degree of the polynomial on eachmesh interval until a specified error tolerance is satisfied. In regions of relatively high curvature, convergence is achieved by refining the mesh, while in regions of relatively low curvature, convergence is achieved by increasing the degree of the polynomial. An efficient iterativemethod is then described for accurately solving a general nonlinear optimal control problem. Using four examples, the adaptive pseudospectral method described in this paper is shown to be more efficient than either a global pseudospectral method or a fixed-order method.

/pdf/direct-trajectory-optimization-using-a-variable-low-order-1fuwne1xf0.pdf

Direct Trajectory Optimization Using a Variable Low-Order Adaptive Pseudospectral Method

Numerical analysis presents different faces to the world. For mathematicians it is a bona fide mathematical theory with an applicable flavour. For scientists and engineers it is a practical, applied subject, part of the standard repertoire of modelling techniques. For computer scientists it is a theory on the interplay of computer architecture and algorithms for realnumber calculations. The tension between these standpoints is the driving force of this book, which presents a rigorous account of the fundamentals of numerical analysis both of ordinary and partial differential equations. The point of departure is mathematical, but the exposition strives to maintain a balance among theoretical, algorithmic and applied aspects of the subject. This new edition has been extensively updated, and includes new chapters on developing subject areas: geometric numerical integration, an emerging paradigm for numerical computation that exhibits exact conservation of important geometric and structural features of the underlying differential equation; spectral methods, which have come to be seen in the last two decades as a serious competitor to finite differences and finite elements; and conjugate gradients, one of the most powerful contemporary tools in the solution of sparse linear algebraic systems. Other topics covered include numerical solution of ordinary differential equations by multistep and Runge–Kutta methods; finite difference and finite elements techniques for the Poisson equation; a variety of algorithms to solve large, sparse algebraic systems; methods for parabolic and hyperbolic differential equations and techniques for their analysis. The book is accompanied by an appendix that presents brief back-up in a number of mathematical topics.

A First Course In The Numerical Analysis Of Differential Equations [Book News & Reviews]

This paper first presents necessary and sufficient conditions for the solvability of discrete time, mean-field, stochastic linear-quadratic optimal control problems. Then, by introducing several sequences of bounded linear operators, the problem becomes an operator stochastic LQ problem, in which the optimal control is a linear state feedback. Furthermore, from the form of the optimal control, the problem changes to a matrix dynamic optimization problem. Solving this optimization problem, we obtain the optimal feedback gain and thus the optimal control. Finally, by completing the square, the optimality of the above control is validated.

Discrete Time Mean-Field Stochastic Linear-Quadratic Optimal Control Problems

Uncertain differential game investigates interactive decision making of players over time, and the system dynamics is described by an uncertain differential equation. This paper goes further to study the two-player zero-sum uncertain differential game. In order to guarantee the saddle-point Nash equilibrium, a Max–Min theorem is provided. Furthermore, when the system dynamics is described by a linear uncertain differential equation and the performance index function is quadratic, the existence of saddle-point Nash equilibrium is obtained via the solvability of a corresponding Riccati equation. Finally, a resource extraction problem is analyzed by using the theory proposed in this paper.

Linear–Quadratic Uncertain Differential Game With Application to Resource Extraction Problem

The Forward-Backward Sweep Method is a numerical technique for solving optimal control problems. The technique is one of the indirect methods in which the differential equations from the Maximum Principle are numerically solved. After the method is briefly reviewed, two convergence theorems are proved for a basic type of optimal control problem. The first shows that recursively solving the system of differential equations will produce a sequence of iterates converging to the solution of the system. The second theorem shows that a discretized implementation of the continuous system also converges as the iteration and number of subintervals increases. The hypotheses of the theorem are a combination of basic Lipschitz conditions and the length of the interval of integration. An example illustrates the performance of the method.

Convergence of the forward-backward sweep method in optimal control

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.

http://hilltop.bradley.edu/~mou/7b2OpenLoopLQGame.pdf

Two-person zero-sum linear quadratic stochastic differential games by a Hilbert space method

For a controlled stochastic differential equation with a Bolza type performance functional, a variational formula for the functional in a given control process direction is derived, by means of backward stochastic differential equations. As applications, some Pontryagin type maximum principles are established for optimal controls of control problems, for saddle points of open-loop two-person zero-sum differential games, and for Nash equilibria of N -person nonzero-sum differential games. The results presented in this paper generalizes/simplifies the relevant ones found in [12] [17]. In addition, a sufficient existence condition of Nash equilibria is proved for nonzero-sum games.

https://www.intlpress.com/site/pub/files/_fulltext/journals/pamq/2007/0003/0002/PAMQ-2007-0003-0002-a007.pdf

A Variational Formula for Stochastic Controls and Some Applications

http://bradley.bradley.edu/~mou/Riccati.pdf

Generalized Riccati equations arising in stochastic games

Here we obtain everywhere regularity of weak solutions of some nonlinear elliptic systems with borderline growth, includingn-harmonic maps between manifolds or map with constant volumes. Other results in this paper include regularity up to the boundary and a removability theorem for isolated singularities.

Libin Mou

Papers

Convergence of the forward-backward sweep method in optimal control

Two-person zero-sum linear quadratic stochastic differential games by a Hilbert space method

A Variational Formula for Stochastic Controls and Some Applications

Generalized Riccati equations arising in stochastic games

Regularity forn-harmonic maps