The paper deals with a control system which has a plant consisting of two integrators with a saturable control input. With a command signal (which may be a step plus a ramp) the optimum switching curve and the optimum controller for two different performance criteria (‘Minimum integral square of error’ and ‘Minimum integral modulus of error’) are found explicitly by purely algebraic and analytic methods which start from Pontryagin's Maximum Principle and involves Taylor's series expansion of the Hamiltonian system of equations between two arbitrary successive switch points on the forward path of an arbitrary optimum trajectory. The equations of the switching curves in the phase plane obtained by this method are verified with the known results already determined by Brennan and Roberts (1962), Eggleston (1963), Fuller (1960 a) and Wonham (1963) by other methods.