Computational stochastic control: basic foundations, complexity and techniques

Much research and education in control systems is purely mathematical, but computational advances in stochastic control problem solving can be used beyond the limits of theoretical mathematics. Theoretical and computational mathematics are complementary. Computation is important where the problem is mathematically intractable, highly dimensional as in stochastic dynamic programming or urgently in need of answers as in competitive financial predictions. Many advances in solving large scale control problems have been gained through technical improvements in computing hardware, but as many advances have been made in the development of better algorithms. Both analysis and computation are important in solving problems. Both rely on mathematics, but rely on them in different ways. An important part of educational training is general preparation for problem solving since postgraduate jobs are uncertain in the current world. In this expository control education paper, some basic computational considerations, high performance computers and useful algorithms are surveyed. Much of the knowledge gained in research has been transferred to classes in computation and control, so that student instruction is at the leading edge, a buffer against obsolescence. An important general lesson in computational education is that the computation forms the other half of mathematics, beyond regular mathematics courses. Computation has its own algebra, oriented to finite precision arithmetic, and its own numerically oriented analysis. The view is that of an applied analyst and computational control scientist explaining the field to those with a regular mathematics background.