Solving control problems - a numerical perspective

There is a continuing and growing need in the control community for good algorithms and robust numerical software for increasingly challenging applications. Consequently, during the past several decades, the control field has been a rich source of computational problems for applied mathematicians and numerical analysts alike. This has led to the development of several control design software packages, both as commercial and free software. In view of this positive situation, the question arises: Is numerical awareness in control an issue of pressing importance? The proposed talk will address, from both user and algorithm developer perspectives, the following ideas: 1. General strategies to solve control problems (role of coordinate transformations, using orthogonal canonical forms, computational building blocks based approaches, checking via alternative methods, etc.) 2. Principles for algorithm development (exploiting/preserving problem structure, avoiding unstable computations, favouring blocking, etc.) 3. "Never do" issues 4. Choosing the right system representation (e.g., no polynomials) 5. Good algorithms (classes of problems, perspectives, challenges); most algorithms are bad! 6. Well formulated problems (solution does not lies on a manifold, robustification issues, genericity) 7. Role of problem sensitivity (e.g., how scaling can help, but also can destroy any hope to solve a problem) 8. Roles of tolerances (types, caveat in software, relation to scaling, epsilon-canonical forms, robustification of structural algorithms using adaptive tolerances, etc.) 9. Implementing algorithms as robust numerical software (not only using good algorithms is an issue but also handling of bad data, employing safe computations, handling of trivial solutions, etc.) The presentation will be complemented by a number of simple examples to illustrate all these aspects (e.g., showing also some spectacular failures). The level of presentation will be as non-technical as possible, but - till with enough details to show some real challenges.