This winter school is aimed to introduce and encourage researchers to the enriched field of control of partial differential equations in stochastic and deterministic case. The special emphasis will be given to fluid control problems with underlying governing equations being Navier-Stokes.
Ph.D. students, post-doctoral fellows, junior faculty and scientists working under R&D sector of Government laboratories are encouraged to apply. Applications from talented and motivated M.Sc. students will also be considered. It is expected that the selected participants would attend the entire program.
It is required that the applicant has sound knowledge of techniques used in partial differential equations. A good source of prerequisite material is the book by S. Kesavan titled Topics in functional analysis and applications. The applicant should also have knowledge on first course on differential geometry, measure theory and probabilty.
No prior knowledge on control theory and Navier-Stokes equations is essential.
Boarding, lodging and partial/full travel support are available to all selected participants.
Application window opens: 1st July, 2012
Application window closes: 30th September, 2012
Initimation to the selected applicants: 15th October, 2012
Detailed description of the topics to be covered in the winter school
Below we describe briefly the content of material which is proposed to be covered in the winter school by various speakers. The description is self explanatory and would help aspirant applicants to have clear idea about what they would gain from the school.
- Topic – I: Review of Certain Topics in Functional Analysis
Duration: First week - 4 lecture hours
Speaker: Prof. Adimurthi
Description: Certain topics in function analysis, e.g., basics of Sobolev spaces, embedding theorem, Lax Milgram theorem, weak solutions will be reviewed.
1. S. Kesavan, Topics in Functional Analysis and Applications, Wiley, 1989.
2. L. C. Evans, Partial Differential Equations, 2nd Edn., AMS, 2010.
Topic – II: An Introduction to determinstic optimal control and controllability
Duration: First week - 6 lecture hours
Speaker: Prof. A. K. Nandkumaran
Description: Beginning with the old concept of optimization and calculus of variation as motivation, optimal control theory will be introduced with suitable examples. The notions like Lagrange multipliers, Pontryagin maximum principle, dynamic programming principle, Hamilton-Jacobi-Bellman equations (HJB equations) will be explained in details. In the second part of the series of lectures the concept of exact controllability will be introduced first for finite dimensional systems and then for partial differential equations.
1. L. C. Evans, An Introduction to Mathematical Optimal Control Theory, Berkeley Lecture Notes and the references therein.
2. J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, 1971.
3. J. L. Lions, Exact controllability, Stabilization and Perturbations for Distributed Systems, SIAM Review, 30 (1988), no.1, 1-68.
4. S. Micu and E. Zuazua, An introduction to the Controllability of Partial Differential Equations, Lecture notes.
5. A. K. Nandakumaran, Exact controllability of Linear Wave Equations and Hilbert Uniqueness Method, Proceedings, Computational Mathematics, Narosa (2005).
Topic – III: Introduction to Stochastic Integration and Stochastic Differential Equations
Duration: First week - 7 lecture hours
Speaker: Prof. Barbara Ruediger
Description: Hilbert space valued Wiener processes will be introduced and corresponding stochastic integration theory, Ito formula, stochastic Fubini theorem will be discussed in details. Existence and uniqueness of solutions (in the sense of strong, weak, mild and martingale) results of stochastic differential equations with additive and multiplicative will be covered.
1. G. DaPrato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, 1992.
2. M. Me’tivier, Stochastic Partial Differential Equations in Infinite-Dimensional Spaces, Scuola Normale Superiore di Pisa, Quaderni, Pisa, 1988.
Topic – IV: Solution Techniques for Partial Differential Equations Arising in Deterministic and Stochastic Control
Duration: First week - 9 lecture hours
Speakers: Prof. M. K. Ghosh, Dr. A. J. Shaiju and Dr. Sursh Kumar (3 lecture hours each)
Description: Controlled Diffusion Processes - M. K. Ghosh: Introduction to controlled diffusion processes with the essence of corresponding strong and weak solutions will be brought out. Optimal control of diffusion processes which gives rise to Hamilton-Jacobi-Bellman (HJB) equations will be discussed in details. Some results related to strong solutions to HJB equation will be covered.
Deterministic and Stochastic Viscosity Solutions - A. J. Shaiju & Suresh Kumar: The value function associated with the given control problem satisfies a dynamic programming principle. The value function if it is smooth can be shown to satisfy HJB equation. But in general value functions are not necessarily smooth and hence one has to look for weak solutions. The very effective technique of viscosity solutions, a kind of a weak solution for these equations, will be discussed in details in the case of deterministic as well as stochastic control problems. Some sufficient conditions, using the above analysis, for the optimality of control inputs will be derived and the use of this method in the particular case of linear quadratic control problems will be illustrated, the associated Riccati equation and formulae for the value function and optimal control will be derived. Associated results for the stochastic setting will also be discussed in details.
1. A. Arapostathis, V. S. Borkar and M. K. Ghosh, Ergodic Control of Diffusion Processes, Cambridge University Press, 2011.
2. M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhauser, 2008.
3. G. Barles, Discontinuous viscosity solutions of first-order Hamilton-Jacobi equations: a guided visit. Nonlinear Anal. 20 (1993), no. 9, 1123--1134.
4. M. G. Crandall and P. L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983), no. 1, 1--42.
5. W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, second edition, Springer, 2006.
Topic – V: Deterministic and Stochastic Navier-Stokes Equations - Solvability, Control and Large Deviations
Duration: 10 lecture hours (Spreaded over 3 weeks)
Speaker: Prof. Sivaguru S. Sritharan
Description: The first few lectures in the first week will be spent on the derivation of the incompressible Euler and Navier-Stokes equations, the abstract formulation of the model in a Hilbert space, various properties of the linear and nonlinear operators, known existence (weak and mild) and uniqueness results and open problems. A general overview of the status of solvability theory for compressible flow will also be elaborated.
The remaining lectures will be focused mostly on the solvability and control of stochastic Navier-Stokes equations. An introduction to Navier-Stokes equations with deterministic and stochastic controls will be done. The typical control problems for viscous flow past obstacles, flow through channels and pipes will be discussed. Existence theorems for ordinary and chattering (relaxed) controls for the above viscous flow problems explaining the typical machinery such as lower semicontinuity of cost functional, Young measures, etc will be discussed. The necessary conditions for optimal controls in the form of adjoint system using (in the most general constrained case) Ekeland variational principle will be derived which is close to Pontryagin maximum principle. Feedback synthesis in terms of Hamilton-Jacobi-Bellman as well as Hamilton-Jacobi-Isaac (in the case of differential games) will be derived for the control of Navier-Stokes equations and solvability in terms of viscosity solutions and Nisio semigroups will be discussed. The variational and quasi-variational inequalities associated with the optimal stopping and impulse control of stochastic Navier-Stokes equations will be derived and their solvability, optimal strategies for control will be discussed. The nonlinear filtering as well as the problem of control with partial observation will be explained and the corresponding Hamilton-Jacobi-Mortensen equation for measures will be derived. It will follow discussion on the solvability in terms of Nisio semigroups. The large deviation theory of stochastic Navier-Stokes equation addressing both Wentzel-Friedlin as well as Donsker-Varadhan theory will be introduced in the last few lectures.
1. O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969.
2. F. Gozzi, S. S. Sritharan and A. Swiech, Viscosity solutions of dynamic programming equations for optimal control of Navier-Stokes equations, Archive for Rational Mechanics and Analysis, 163 (2002), no. 4, 295--327.
3. F. Gozzi, S. S. Sritharan and A. Swiech, Bellman equation for the optimal feedback control of stochastic Navier-Stokes equations, Communications on Pure and Applied Mathematics, 58 (2005), no. 5, 671--700.
4. J. L. Menaldi and S. S. Sritharan, Impulse Control of Stochastic Navier-Stokes Equation, Nonlinear Analysis, Theory, Methods and Applications, 52 (2003), no. 2, 357--381.
5. S. S. Sritharan (Ed.), Optimal Control of Viscous Flow, SIAM, 1998.
6. S. S. Sritharan and P. Sundar, Large Deviations for Two-dimensional Navier-Stokes Equations with Multiplicative Noise, Stochastic Processes and their Applications, 116 (2006), no. 11, 1636--1659.
7. R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland, Amsterdam, 1984.
Topic – VI: Stabilization of Navier-Stokes Equations - Deterministic and Stochastic
Duration: Second week - 13 lecture hours
Speakers: Prof. V. Barbu and Prof. J-P Raymond
Description: Prof. V. Barbu (7 lecture hours): The stabilization of stationary fluid flows, which for large Reynolds numbers are unstable, is of crucial importance in fluid dynamics. This course is dedicated to recent results and methods for stabilization of steady state and non-stationary regimes in fluid dynamics governed by Navier-Stokes equations. A brief presentation of Fursikov and Imanuvilov results on exact controllability of Navier-Stokes equations via observability inequality will be done. The spectral-Ricatti method in stabilization of stationary solution will be discussed. The method is the following: one designs first a stabilizable feedback controller for the associated Oseen-Stokes equation by controlling the unstable modes, put this feedback controller into Navier-Stokes equation and show that it is stabilized. An explicit design of the stabilizing feedback controller as a linear combination of unstable eigenfunctions will be shown. Further the stabilization of steady state solution via a linear stochastic feedback controller will be discussed. Finally the stabilization of non-stationary Navier-Stokes flows, which requires a different approach, will be covered.
Prof. J-P Raymond (6 lecture hours): Introduction to the Feedback stabilization of unstable nonlinear dynamical systems will be given with specific examples in fluid mechanics. Controllability and Stabilization of linearized models will be discussed. Stabilization via the solution to the Algebraic Bernoulli Equation (A.B.E) will be done. The linearized Navier-Stokes equations with boundary control will be represented as a control system. The optimal control theory can be used as a tool for determining feedback control law. The Riccati equation for infinite horizon optimal control problems helps in this. This technique will be demonstrated. The control of finite dimension for the boundary control of the Linearized Navier-Stokes equations will be determined and stabilization of the system will be discussed. The estimation problem for the linearized Navier-Stokes equations and the Navier-Stokes equations will be defined. Further detectability of an observed linear system, observers of finite dimension, Best location for actuators and sensors for the control and the observation of the linearized Navier-Stokes equations will be discussed. Coupling between controllers and observers of finite dimension in the case of the Linearized Navier-Stokes equations will be detailed.
1. V. Barbu, Stabilization of Navier-Stokes Flows, Springer-Verlag, New-York, 2011.
2. V. Barbu, The internal stabilization by noise of the linearized Navier-Stokes equation, ESAIM Control Optim. Calc. Var. 17 (2011), no. 1, 117--130.
3. J-P Raymond, Feedback boundary stabilization of the two-dimensional Navier-Stokes equations, SIAM J. Control Optim. 45 (2006), no. 3, 790--828.
Topic – VII: Invariant Measures and Ergodicity for Stochastic Navier-Stokes Equations
Duration: Second and Third week - 6 lecture hours
Speaker: Prof. P. Sundar
Description: The lectures will begin with a discussion of stationary measures and ergodic behavior of a large class of Markov processes. Several examples will be given to illustrate the importance and usefulness of this research area. Next, invariant measures and ergodicity will be studied in the infinite-dimensional context for stochastic evolution equations, and in particular, for stochastic Navier-Stokes equations. Several criteria that ensure uniqueness of invariant measures will be discussed. Some applications and research problems will be briefly mentioned.
1. G. DaPrato and J. Zabczyk, Ergodicity for Infinite-dimensional Systems, Cambridge University Press, 1996.
2. R. Khasminskii and G.N. Milstein, Stochastic Stability of Differential Equations, 2nd Edition, Springer, 2012.
Topic – VIII: Stochastic Landau-Lifshitz-Gilbert Equation
Duration: Second and Third week - 6 lecture hours
Speaker: Prof. Zdzislaw Brzezniak
Description: The stochastic Landau-Lifshitz-Gilbert equation (LLGE) is a highly nontrivial example of a fully nonlinear stochastic partial differential equation motivated by physics and at the same time possessing rich geometric structure. The LLGE is a fundamental equation in the theory of ferromagnetic materials. In fact, the stationary states of a ferromagnetic material minimize the energy functional and the LLGE defines an associated flow, which is expected to converge to the minimizer. For physical reasons solutions to the LLGE must be concentrated on a sphere. It was noticed very early that in the case of a thin magnetic domain the LLGE combines two other important partial differential equations, namely the nonlinear Schrodinger equation in two dimensions and the equation for the heat flow of harmonic maps from a domain into a sphere.
The stochastic LLGE, with its unquestioned significance for physics and technology and its structural similarity to a number of partial differential equations arising in geometry of manifolds, provides an excellent starting point for studying the theory of geometric stochastic partial differential equations. This is a new emerging area in the theory of stochastic partial differential equations with fascinating interactions between stochastic analysis, theory of nonlinear PDEs, geometry and physics. Plan is to have fair introduction and some physical motivations for solving LLGE. Existence of weak martingale solutions, partially regular solutions and singularities of stochastic LLGE will be discussed in details. Finally some results related to large deviation and limiting cases (stochastic version of the heat flow of harmonic maps and the stochastic Heisenberg model) will be put forth.
1. Z. Brzezniak, B. Goldys and T. Jegaraj, Large deviations for a stochastic Landau-Lifshitz equation, arXiv:1202.0370.
2. Z. Brzezniak, B. Goldys, and T. Jegaraj, Weak Solutions of the Stochastic Landau-Lifshitz-Gilbert Equation, arXiv:0901.0039.
3. R. V. Kohn, M. G. Reznikoff and E. Vanden-Eijnden, Magnetic elements at finite temperature and large deviation theory, J. Nonlinear Sci. 15 (2005), 223--253.
4. L. Landau and E. Lifshitz, On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, Phys. Z. Sowj. 8, 153 (1935); terHaar, D. (eds.) Reproduced in: Collected Papers of L. Landau, 101--114, New York, Pergamon Press 1965.
Topic – IX: Densities for the Navier-Stokes equations with Noise
Duration: Third week - 6 lecture hours
Speaker: Prof. Marco Romito
Description: One of the most important open problem for the Navier--Stokes equations concerns regularity of solutions. There is an extensive literature devoted to the problem, both in the non-random and randomly forced case. Existence of densities for the distribution of the solution is a probabilistic form of regularity and the course addresses some attempts at understanding, characterizing and proving existence of such mathematical objects.
While the topic is somewhat specific, it offers the opportunityto introduce the mathematical theory of the Navier-Stokes equations, together with some of the most recent results, as well as a good selection of tools in stochastic analysis and in the theory of (stochastic) partial differential equations.
In the first part of the course we give a quick introduction to the equations and discuss a few results in the literature related to densities and absolute continuity. We then present four different methods to prove existence of a density with respect to the Lebesgue measure for the law of finite dimensional functionals of the solutions of the Navier-Stokes equations forced by Gaussian noise.
Each of the four methods has some advantages, as well as disadvantages. The first two methods provide a qualitative result, while the other two provide quantitative estimates and ensure a bit of regularity.
1. F. Flandoli, An introduction to 3D stochastic fluid dynamics, SPDE in hydrodynamic: recent progress and prospects, Lecture Notes in Math., vol. 1942, Springer, Berlin, 2008, Lectures given at the C.I.M.E. Summer School held in Cetraro, August 29 - September 3, 2005, Edited by Giuseppe Da Prato and Michael Roeckner, pp. 51--150.
2. J. C. Mattingly and E. Pardoux, Malliavin calculus for the stochastic 2D Navier-Stokes equation, Comm. Pure Appl. Math. 59, no. 12 (2006), 1742--1790.
3. R. S. Liptser and A. N. Shiryaev, Statistics of random processes. I, expanded ed., Applications of Mathematics (New York), vol. 5, Springer-Verlag, Berlin, 2001, General theory, Translated from the 1974 Russian original by A. B. Aries, Stochastic Modelling and Applied Probability.
4. A. Debussche and M. Romito, Existence of densities for the 3D Navier-Stokes equations driven by Gaussian noise, preprint, arXiv:1203.0417.
Topic – X: Applications of Optimization and Control Methods in Fluid Mechanics
Duration: Second week - 3 lecture hours
Speaker: Prof. Bartosz Protas
The lectures will survey applications of optimization and control techniques to a number of classical and emerging problems in fluid mechanics. Our main focus will be on problems which can be formulated in terms of PDE-constrained optimization and solved with gradient-based techniques. We will review numerical approaches to the calculation of gradients (sensitivities) in problems with different structure based on suitably-defined adjoint systems. In addition, we will also discuss some problems involving linear feedback control applied to simplified fluid models. The specific topics which will be covered include
(i) optimal control of vortex flows,
(ii) probing fundamental bounds in hydrodynamics using variational optimization methods,
(iii) reconstruction of constitutive relations in multiphysics systems as an optimal control problem,
(iv) shape optimization.
The presentations will offer a tutorial-style introduction to several research problems. While the main focus will be on techniques of large-scale computations and related results, the lectures will also feature a number of rigorous developments.
- M. D. Gunzburger, Perspectives in Flow Control and Optimization", SIAM (2003).
- B. Protas, "Vortex Dynamics Models in Flow Control Problems", Nonlinearity 21, R203-R250 (review paper), 2008
Topic – XI: Numerical Methods for Navier-Stokes Equations: Model Reduction, Control and Random Inputs
Duration: Third week - 6 lecture hours
Speaker: Prof. S. S. Ravindran
This lecture will consists of three parts. In the first part of lecture, we will cover model reduction for nonlinear infinite dimensional PDEs using proper-orthogonal decomposition (POD) Galerkin method. Resulting reduced-order model can be useful for simulation and control of both deterministic and stochastic Navier-Stokes equations. After a short review of the POD and its derivation, we will consider error analysis and convergence of POD-Galerkin approximations of Navier-Stokes equations.
In the second of this lecture, we will consider nonstationary Navier-Stokes equations with Dirichlet control when the control space is L2(Γ). The use of L2-control space yields localized optimality conditions, whereas the use of more regular control spaces such as H1/2(Γ) or H1(Γ) yields optimality conditions involving non-local Laplace-Beltrami operators. On the other hand in the L2 control space setting, control is expressed in terms of normal derivative of adjoint state variable. To overcome these difficulties, a boundary penalty method will be presented. Various issues associated with penalized optimal boundary control including existence of optimal solutions and convergence of solutions of penalized control problem to Dirichlet boundary control problem will be discussed.
Third part of this lecture will cover numerical methods for Navier-Stokes equations with random input data, such as viscosity, forcing terms, initial and boundary conditions or geometry. The resulting stochastic Navier-Stokes systems will be investigated
from numerical point of view. We will begin with a short review of available methods.
In particular, the lectures will cover polynomial chaos representations, Karhunen-Loeve expansion, stochastic Galerkin finite element method (SGFEM). We will also discuss the use of reduced-order-modeling to mitigate the computational burden (curse of dimensionality) associated with SGFEM when the number of random variables is large.
1. S.S. Ravindran, Convergence of Extrapolated BDF2 Finite Element Schemes for Unsteady Penetrative Convection Model, Numerical Functional Analysis and Optimization, Volume 33(1) (2012), pp.48-79.
2. S.S. Ravindran, Error Analysis for Galerkin POD Approximation of the Non-stationary Boussinesq Equations, Numerical Methods for Partial Differential Equations, Volume 27 (6) (2011), pp. 1639-1665.
3. L.S. Hou and S.S. Ravindran, A Penalized Neumann Control Approach for Solving an Optimal Dirichlet Control Problem for the Navier-Stokes Equations, SIAM Journal on Control and Optimization, Volume 36(5)(1998), pp. 1795–1814.
4. R.G. Ghanem, RG and P.D. Spanos, Stochastic finite elements: A spectral approach. New York, Springer; 1991