On optimal control of norm-invariant systems

Optimal control of norm-invariant systems is discussed in relation to two constraints : (i) control constrained to a hypersphere ; and (ii) control constrained to a hyper-rectangle. In the case of the first (smooth) constraint set, it is well known that, for certain optimal control problems, the solutions have a particularly simple structure ; a generalization of the time-optimal regulator is given. In the case of the second (non-smooth) constraint set, it is shown that, for a particular optimal control problem, the solution is again of simple form and may be synthesized via switching hyperplanes, as illustrated in two examples of (a) controlled Euler equations and (b) a controlled harmonic oscillator.