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

IN two previous notes1, Prof. Max Born and I have shown that one can obtain a theory of superconductivity by taking account of the fact that the interaction of the electrons with the ionic lattice is appreciable only near the boundaries of Brillouin zones, and particularly strong near the corners of these. This leads to the criterion that the metal should be superconductive if a set of corners of a Brillouin zone is lying very near the Fermi surface, considered as a sphere, which limits the region in the momentum space completely filled with electrons.

On the theory of superconductivity

A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

On the Navier-Stokes equations

The Theory of Stochastic Processes. By D. R. Cox and H. D. Miller. Pp. x, 398. 70s. (Methuen)

The evolution of the state of many systems modeled by linear partial diﬁerentialequations (PDEs) or linear delay-diﬁerential equations can be described by operatorsemigroups. The state of such a system is an element in an inﬂnite-dimensionalnormed space, whence the name \inﬂnite-dimensional linear system".The study of operator semigroups is a mature area of functional analysis, which isstill very active. The study of observation and control operators for such semigroupsis relatively more recent. These operators are needed to model the interactionof a system with the surrounding world via outputs or inputs. The main topicsof interest about observation and control operators are admissibility, observability,controllability, stabilizability and detectability. Observation and control operatorsare an essential ingredient of well-posed linear systems (or more generally systemnodes). Inthisbookwedealonlywithadmissibility, observabilityandcontrollability.We deal only with operator semigroups acting on Hilbert spaces.This book is meant to be an elementary introduction into the topics mentionedabove. By \elementary" we mean that we assume no prior knowledge of ﬂnite-dimensional control theory, and no prior knowledge of operator semigroups or ofunbounded operators. We introduce everything needed from these areas. We doassume that the reader has a basic understanding of bounded operators on Hilbertspaces, diﬁerential equations, Fourier and Laplace transforms, distributions andSobolev spaces on

Observation and Control for Operator Semigroups

Controllability of Evolution equations

The existence of solutions to optimal control problems Optimality system for optimal control problems The solvability of boundary value problems for a dense set of data The problem of work minimization in accelerating still fluid to a prescribed velocity Optimal boundary control for nonstationary problems of fluid flow and nonhomogeneous boundary value problems for the Navier-Stokes equations The Cauchy problem for elliptic equations in a conditionally well-posed formulation The local exact controllability of the flow of incompressible viscous fluid Bibliography Index.

Optimal Control of Distributed Systems: Theory and Applications

Table Contents- 1: Functional-Analytic Expansions of Solution of Evolution Equations- 2: Elements of Measure Theory- 3: Moment Theory for Small Reynolds Numbers- 4: Space-Time Statistical Solutions of the Navier-Stokes Equations for Arbitrary Reynolds Numbers- 5: The Hopf Equation- 6: Moment Theory for Arbitrary Reynolds Numbers- 7: Homogeneous Space-Time Statistical Solutions of Navier-Stokes Equations- 8: Individual Solutions with Unbounded Energy for Navier-Stokes Equations and Other Problems- 9: Analytic First Integrals and Asymptotic Behaviour as t ? ? of Fourier Coefficients of Solutions of Two-Dimensional Navier Stokes Equations- 10: Navier-Stokes System With White Noise In A Bounded Domain- 11: The Direct and Inverse Kolmogorov Equations Corresponding to a Stochastic Navier-Stokes System- 12: Homogeneous In x Solutions of the Stochastic Navier-Stokes System With White Noise- Appendix 1: Unique Solvability "In Large" of the Three-Dimensional Navier-Stokes System and Moment Equations for a Dense Set of Data- Appendix 2: Periodic Approximations of Homogeneous Measures- Comments- References

Mathematical Problems of Statistical Hydromechanics

Optimal control of distributed systems

ContentsIntroduction § 1. Statement of the problem. Main results1.1. Exact boundary controllability of the Navier-Stokes equations1.2. Local exact distributed controllability of the Navier-Stokes equations1.3. Exact controllability of the Boussinesq system1.4. Exact local controllability and approximate controllability of the Boussinesq system1.5. Some applications § 2. Carleman estimates2.1. Preliminaries2.2. Carleman estimates for solutions of system (2.9), (2.10)2.3. Definitive estimates § 3. Solubility of the exact controllability problem for the linearized Boussinesq system3.1. Statement of the problem3.2. An auxiliary extremal problem3.3. Proof of the main result § 4. Local exact controllability of the Boussinesq system § 5. Approximate controllability of the Boussinesq system: reduction to a linear system of a special form5.1. The idea of the proof5.2. Approximate controllability of the Boussinesq system § 6. Exact controllability of a linear system6.1. Proof of Lemma 5.16.2. Proof of Theorem 5.1 Bibliography

/pdf/exact-controllability-of-the-navier-stokes-and-boussinesq-1zghm8oub9.pdf

Andrei V. Fursikov

Papers

Controllability of Evolution equations

Optimal Control of Distributed Systems: Theory and Applications

Mathematical Problems of Statistical Hydromechanics

Optimal control of distributed systems

Exact controllability of the Navier-Stokes and Boussinesq equations