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 subinterval of IR, and suppose we x a piecewise analytic nm matrix-valued function a; b] 3 t ! F(t) 2 IR nm. Write BB((a; b]; V) to denote the set of all maps a; b] 3 t ! w(t) 2 V having a nite number of discontinuities on a; b]. Since V is a nite set, the members of BB((a; b]; V) are exactly the piecewise constant V-valued maps on a; b]. In control theory, such maps w are called bang-bang controls. The basic questions are: (i) Which vectors 2 IR n can be represented in the form = w;a;b , where w;a;b := Z b a F(t)w(t)dt ; (1) and w 2 BB((a; b];V)? (ii) How does the minimum number of discontinuities of the piecewise constant control function w on the given interval a; b] depend on the vector ? (iii) For a given a, , is there, among all the pairs (b; w) such that = w;a;b , one with a minimum value of b? (In other words, is there a minimum-time bang-bang control?) THE CONTROL THEORY CASE: In control theory, one studies linear control systems in IR n of the form _ x = A(t)x + B(t)w ; (2) where the real n-vector x(t) represents the state of the system at time t, the real m-vector w(t) is the control, and A; B are integrable functions on an interval a; b], with values in the spaces IR nn , IR nm of n n and n m matrices, respectively. For a given initial condition x(a) = x, and a given control function t ! w(t), the solution w; x of (2) is given by w; x (t) = M(t; a) x + Z t a M(t; s)B(s)w(s)ds ; where a; b]a; b] 3 (t; s) ! M(t; s) 2 IR nn is the fundamental matrix solution of the linear system _ X(t) = A(t)X(t). The set Reach A;B; x;a;b of all possible vectors w; …