Minimax, Liapunov and “bang-bang”

Commentary on Norman Levinson's paper: Minimax, Liapunov and "Bang-Bang," J. Differential Equations (1966), 218-241.

In this paper, his only venture into control theory, Levinson solves the following basic problem of fundamental importance in the theory of linear control systems. Use IR m to denote real m-dimensional Euclidean space, thought of as a space of column vectors. Let P be a compact convex polyhedron in IR m , and let the set of vertices of P be denoted by V : Let a; b] be a given compact subinterva...

Strong Optimality for a Bang-Bang Trajectory

In this paper we give sufficient conditions for a bang-bang regular extremal to be a strong local optimum for a control problem in the Mayer form; strong means that we consider the C0 topology in the state space. The controls appear linearly and take values in a polyhedron, and the state space and the end point constraints are finite-dimensional smooth manifolds. In the case of bang-bang extrem...

Bang-Bang Operations from a Geometric Perspective

Strong, fast pulses, called ‘‘bang–bang’’ controls can be used to eliminate the effects of system-environment interactions. This method for preventing errors in quantum information processors is treated here in a geometric setting which leads to an intuitive perspective. Using this geometric description, we clarify the notion of group symmetrization as an averaging technique, provide a geometri...

The Bang-Bang Funnel Controller (long version)

A bang-bang controller is proposed which is able to ensure reference signal tracking with prespecified time-varying error bounds (the funnel) for nonlinear systems with relative degree one or two. For the design of the controller only the knowledge of the relative degree is needed. The controller is guaranteed to work when certain feasibility assumptions are fulfilled, which are explicitly give...

The Big Bang is bang on.