Preliminary Announcement

Course 1

F. Bonnans (INRIA, Rocquencourt)
Numerical Methods for the Optimal Control of Ordinary Differential Equations

Overview
The course will give an overview of various approaches for solving optimal control problems of ordinary differential equations, namely (i) shooting algorithms, (ii) reduction to a nonlinear programming problem via a time discretization, (iii) solving the Hamilton-Jacobi-Bellman (HJB) equation, and (iv) the extension of the latter to stochastic control problems. For each of them, the basic steps will be carefully presented, and also some challenging problems related to the need for accurate solutions.

Contents
Each part will include two lectures.

• Shooting

  1. The basic unconstrained optimal control problem. Simple and multiple shooting.

  2. Constrained problems : control constraints, singular arcs, state constraints.

• Direct Optimal Control

  1. Time discretization of an optimal control problem : stability, error estimates.

  2. Numerical algorithms : sequential quadratic programming, augmented Lagrangians, interior point methods.

• Dynamic Programming

  1. The Hamilton-Jacobi-Bellman (HJB) equation for finite and infinite horizons. Extension to impulse control.

  2. Numerical schemes : finite differences and finite elements.

• Stochastic Control

  1. The stochastic control problem and associated HJB equation.

  2. Finite differences numerical schemes : the problem of consistency.

Basic knowledge
The course will assume nothing more than general knowledge on ordinary differential equations, the use of Lagrange multipliers for solving constrained optimization problems and Newton's method for solving a system of nonlinear equations.

References

1. M. Bardi and I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations, Systems and Control: Foundations and Applications, Birkhäuser, Boston, 1997

2. D. Bertsekas, Dynamic programming and optimal control (2 volumes), Athena Scientific, Belmont, Massachusetts,1995

3. H. J. Kushner and P. G. Dupuis, Numerical methods for stochastic control problems in continuous time, Applications of mathematics, vol. 24, Springer, New York, 1992

Course 2

K. Kunisch (University of Graz)
Numerical Methods for Optimal Control of Partial Differential Equations with Emphasis on Fluids

Overview
Due to the increasing computing power, optimal control of partial differential equations is rapidly becoming an accepted tool by an increasing number of scientists. In these lectures, we shall present some of the basic concepts of formulating and analyzing optimal control problems governed by partial differential equations. Up-to-date numerical techniques are discussed primarily by means of their application to optimal control of equations in fluid dynamics.

Contents

• Theoretical Concepts

  1. Optimality systems

  2. Optimal control of fluids

  3. Singular optimal control

  4. Hessian calculations

  5. Control and state constrained optimal control.

  
  

• Numerical Techniques

  1. Conjugate Gradients

  2. Newton and quasi-Newton methods

  3. SQP-methods

  4. Primal-dual active set strategy and semi-smooth Newton methods.

  
  

• Sub-optimal Techniques

  1. Instantaneous - and receding horizon control

  2. POD-based optimal control

Basic knowledge
Solid background in applied functional analysis and numerical analysis of partial differential equations, Sobolev spaces, basic familiarity with Navier-Stokes equations, basic concepts of optimization theory.

References

1. J.L Lions, Optimal control of systems governed by partial differential equations, Springer-Verlag, New York-Berlin, 1971.

2. P. Neittaanmaki and D. Tiba, Optimal Control of Nonlinear Parabolic Systems: Theory, Algorithms and Applications, M. Dekker, New York, 1994

3. K. Ito and K. Kunisch, Augmented Lagrangian-SQP-Methods for Nonlinear Optimal Control Problems of Tracking Type, SIAM J. Control and Optimization, Vol. 34, 3 (1996), 874-891.

4. H. Choi, M. Hinze and K. Kunisch, Instantaneous Control of Backward Facing Step, Applied Numerical Mathematics, 31(1999), 133-158.

5. K. Kunisch and St. Volkwein, Control of Burgers' Equation by Reduced Order Approach using Proper Orthogonal Decomposition, Journal of Optimization Theory and Applications 102(1999), 345-371.

Course 3

F.A. Potra (University of Maryland, Baltimore County)
Semidefinite Programming and Applications

Overview
Semidefinite programming (SDP) consists in minimizing a linear functional of a matrix subject to linear equality and inequality constraints, the inequalities being understood in the sense of the cone of positive semidefinite matrices. It contains as particular cases linear programming, (linearly constrained) quadratic programming, quadratically constrained quadratic programming and other optimization problems (see [2] and [3]). Semidefinite programming has numerous applications in such diverse areas as optimal control, combinatorial optimization, structural optimization, pattern recognition, trace factor analysis in statistics, matrix completions, etc. (see the excellent survey paper by Vandenberghe and Boyd [3] and the recent handbook edited by Wolkowicz et al. [4]). However it was only after the advent of interior point methods that SDP problems could be efficiently solved.

In a survey paper [1] published in October 1996, Freund and Mizuno state that ``Interior point methods in mathematical programming have been the largest and most dramatic area of research in optimization since the development of the simplex method... Interior point methods have permanently changed the landscape of mathematical programming theory, practice and computation...''. While most of the research in this area was devoted to linear programming, in the opinion of the authors, ``semidefinite programming is the most exciting development in mathematical programming in 1990s.''

Our lectures will focus on interior point methods for semidefinite programming with applications to several areas of optimization and control.

Contents

1. Linear Matrix Inequalities and Semidefinite Programming;

2. The Geometry of SDP;

3. Interior Point Methods for SDP (theoretical results);

4. Interior Point Methods for SDP (software developments);

5. SDP in Robust Optimization;

6. SDP in Structural Design;

7. SDP in Experimental Design;

8. SDP in Control System Analysis and Design.

Basic knowledge
We will try to make the lectures self-contained. However it will be very useful for the audience to have some familiarity with the papers in the following list of references.

References

1. R. M. Freund and S. Mizuno, Interior point methods: current status and future directions, Optima, 51 (1996), 1-9.

2. Y.E. Nesterov and A.S. Nemirovsky, Interior point polynomial methods in convex programming: theory and applications, SIAM Publications, SIAM, Philadelphia, USA, 1994.

3. L. Vandenberghe and S. Boyd, Semidefinite Programming, SIAM Review, 38 (1) (1996), 49-95.

4. H. Wolkowicz, R. Saigal and L. Vandenberghe, Handbook of Semidefinite Programming, Kluwer, Norwell, Massachussetts, 2000.

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