Numerical Methods for Optimal Control Problems

This work presents recent mathematical methods in the area of optimal control with a particular emphasis on the computational aspects and applications. Optimal control theory concerns the determination of control strategies for complex dynamical systems, in order to optimize some measure of their performance. Started in the 60's under the pressure of the "space race" between the US and the former USSR, the field now has a far wider scope, and embraces a variety of areas ranging from process control to traffic flow optimization, renewable resources exploitation and management of financial markets. These emerging applications require more and more efficient numerical methods for their solution, a very difficult task due the huge number of variables. The chapters of this volume give an up-to-date presentation of several recent methods in this area including fast dynamic programming algorithms, model predictive control and max-plus techniques. This book is addressed to researchers, graduate students and applied scientists working in the area of control problems, differential games  and their applications.

1 M. Assellaou and A. Picarelli, A Hamilton-Jacobi-Bellman approach for the numerical computation of probabilistic state constrained reachable sets.- 2. A. Britzelmeier,  A. De Marchi, and M. Gerdts, An iterative solution approach for a bi-level optimization problem for congestion avoidance on road networks.- 3 S. Cacace, R. Ferretti, and Z. Rafiei, Computation of Optimal Trajectories for Delay Systems: an Optimize-Then-Discretize Strategy for General-Purpose NLP Solvers.-  4 L. Mechelli and S. Volkwein, POD-Based Economic Optimal Control of Heat-Convection Phenomena.- 5 A. Alla and V. Simoncini, Order reduction approaches for the algebraic Riccati equation and the LQR problem.- 6 F. Durastante and S. Cipolla, Fractional PDE constrained optimization: box and sparse constrained problems.- 7 M. C. Delfour, Control, Shape, and Topological Derivatives via Minimax Differentiability of Lagrangians.- 8 A. J. Krener, Minimum Energy Estimation Applied to the Lorenz Attractor.- 9 M. Akian and E. Fodjo, Probabilistic max-plus schemes for solving Hamilton-Jacobi-Bellman equations.- 10 P. M. Dower, An adaptive max-plus eigenvector method for continuous time optimal control problems.- 11 W. Mc Eneaney and R. Zhao, Diffusion Process Representations for a Scalar-Field Schr¨odinger Equation Solution in Rotating Coordinates.

Maurizio Falcone is Professor of Numerical Analysis at the University of Rome "La Sapienza" since 2001.   He held visiting positions at several institutions including ENSTA (Paris), the IMA (Minneapolis), Paris 6 and 7, the Russian Academy of Sciences (Moscow and Ekaterinburg) and UCLA. He serves as associate editor for the journal "Dynamic Games and Applications" and  has authored a monograph and about 80 papers in international journals.
His research interests include numerical analysis, control theory and differential games.

Roberto Ferretti is Associate Professor of Numerical Analysis at Roma Tre University since 2001. He has been an invited professor in UCLA (USA), Universitet Goroda Pereslavlya (Russia), ENSTA-Paristech and IRMA (France), TU Munich (Germany) and UP Madrid (Spain). He has authored a monograph and more than 40 papers on international journals/volumes, in topics ranging from semi-Lagrangian schemes to optimal control, level set methods, image processing and computational fluid Dynamics.

Lars Grüne is Professor for Applied Mathematics at the University of Bayreuth, Germany. He obtained his Ph.D. from the University of Augsburg in 1996 and his habilitation from Goethe University in Frankfurt/M in 2001. He held visiting positions at the Sapienza in Rome (Italy) and at the University of Newcastle (Australia) and is Editor-in-Chief of the journal Mathematics of Control, Signals and Systems. His research interests lie in the areas of mathematical systems theory and optimal control.

William M. McEneaney received B.S. and M.S. degrees in Mathematics from Rensselaer Polytechnic Inst., followed by M.S. and Ph.D. degrees in Applied Mathematics from Brown Univ. He has held academic positions at Carnegie Mellon Univ. and North Carolina State Univ., prior to his current appointment at Univ. of California, San Diego. His non-academic positions have included Jet Propulsion Laboratory and Air Force Office of Scientific Research. His interests include Stochastic Control and Games, Max-Plus Algebraic Numerical Methods, and the Principle of Stationary Action.