Well-posedness of H ∞ optimal control problems

This paper considers the effect of perturbations of a nominal single-input/single-output linear plant, or a band of uncertainty, on the solution to certain frequency domain optimal control problems. Weighted $H^\infty $ sensitivity minimization, mixed, and robust sensitivity minimization will be considered along with problems of more general type. A brief discussion of $H^p $ optimal control problems will also be given. Typical examples will be presented where optimal sensitivity does not depend continuously on the plant (ill-posedness) and conditions for well-posedness will be given. It is demonstrated that similar discontinuities can occur for more general problems. Also suggested are ways of defining an optimization problem so that continuous dependence of the infimum is ensured.