Investigation of worst-case errors when inputs and their rate of change are bounded

The worst-case error analysis is extended to include the problem of bounded input <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r(t)</tex> and its rate of change <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dr/dt</tex> for a a dynamical system described by a set of differential equations with separable forcing function. The problem is reformulated as a bounded-input, bounded-state variable problem, and Pontryagin's Maximum Principle is applied to maximize a given error function. For a wide class of systems, the time derivative of the worst forcing function is shown to be "bang-bang" for the open region defined by the constraint of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r(t)</tex> and zero on its boundary. A computational algorithm is developed to solve the resulting two-point boundary value problem.