Part I. Foundations: 1. Random variables 2. Probability measures 3. Stochastic processes 4. The stochastic integral Part II. Existence and Uniqueness: 5. Linear equations with additive noise 6. Linear equations with multiplicative noise 7. Existence and uniqueness for nonlinear equations 8. Martingale solutions Part III. Properties of Solutions: 9. Markov properties and Kolmogorov equations 10. Absolute continuity and Girsanov's theorem 11. Large time behaviour of solutions 12. Small noise asymptotic.

Stochastic Equations in Infinite Dimensions

Control theory provides a model of self-regulati on that we believe is useful in the analysis of human behavior. As an illustration of the breadth of its applicability, we present the basic construct of control theory—the discrepancy-reducing feedback loop—and discuss certain of its implications for theory in three separate areas of human psychology. In personality-s ocial, clinical, and health psychology, the construct proves to fit well with known phenomena and with the theories most recently developed to account for the phenomena. Moreover, in each case control theory appears to make a distinct and unique contribution to the state of the area. We conclude by noting the integrative potential that is suggested by these illustrations and by noting some issues that should receive attention in future work. Cybernetic or control theory is a general approach to the understanding of self-regulating systems. Its central ideas have been around for a long time (see, for example, Cannon's 1929, 1932, discussion of homeostatic physiological mechanisms), but its birth as a distinct body of thought is usually traced to the publication of Wiener's (1948) book, Cybernetics: Control and communication in the animal and the machine. Since then, control theory (in various forms) has had a major impact on areas of work as diverse as engineering (e.g., Dransfield, 1968; Ogata, 1970), applied mathematics (e.g.,

Control theory: A useful conceptual framework for personality–social, clinical, and health psychology.

/pdf/lectu-re-notes-in-economics-and-mathematical-systems-4gmrols5ux.pdf

Lectu re Notes in Economics and Mathematical Systems

For nonlinear programming problems with equality constraints, Hestenes and Powell have independently proposed a dual method of solution in which squares of the constraint functions are added as penalties to the Lagrangian, and a certain simple rule is used for updating the Lagrange multipliers after each cycle. Powell has essentially shown that the rate of convergence is linear if one starts with a sufficiently high penalty factor and sufficiently near to a local solution satisfying the usual second-order sufficient conditions for optimality. This paper furnishes the corresponding method for inequality-constrained problems. Global convergence to an optimal solution is established in the convex case for an arbitrary penalty factor and without the requirement that an exact minimum be calculated at each cycle. Furthermore, the Lagrange multipliers are shown to converge, even though the optimal multipliers may not be unique.

The multiplier method of Hestenes and Powell applied to convex programming

One of the fundamental problems in sensor networks is to estimate and track the state of targets (or dynamic processes) of interest that evolve in the sensing field. Kalman filtering has been an effective algorithm for tracking dynamic processes for over four decades. Distributed Kalman Filtering (DKF) involves design of the information processing algorithm of a network of estimator agents with a two-fold objective: 1) estimate the state of the target of interest and 2) reach a consensus with neighboring estimator agents on the state estimate. We refer to this DKF algorithm as Kalman-Consensus Filter (KCF). The main contributions of this paper are as follows: i) finding the optimal decentralized Kalman-Consensus filter and showing that its computational and communication costs are not scalable in n and ii) introducing a scalable suboptimal Kalman-Consensus Filter and providing a formal stability and performance analysis of this distributed and cooperative filtering algorithm. Kalman-Consensus Filtering algorithm is applicable to sensor networks with variable topology including mobile sensor networks and networks with packet-loss.

Kalman-Consensus Filter : Optimality, stability, and performance

Stochastic differential systems

Kalman filtering theory

Computing methods in optimization problems

Introduction to Optimization Theory in a Hilbert Space

Review. Random Processes: Basic Concepts, Properties. Stationary Random Processes: Covariance and Spectrum. Response of Linear Systems to Random Inputs: Discrete--Time Models. Response of Linear Systems to Random Inputs: Continuous--Time Models. Time Averages and the Ergodic Principle. Sampling Principle and Interpolation. Simulation of Random Processes. Random Fields. Linear Filtering Theory. Problems. Notes and Comments. References. Index.

A. V. Balakrishnan

Papers

Stochastic differential systems

Kalman filtering theory

Computing methods in optimization problems

Introduction to Optimization Theory in a Hilbert Space

Introduction to Random Processes in Engineering