Symbolic operations are used together with delta functions to derive the generalized adjoint method for physical processes that contain first-order discontinuities caused by parameterized on/off switches with zero-order discontinuities in the source term. Generalized adjoint solutions are obtained analytically for simple heuristic examples and verified by direct perturbation analyses. Errors due to the conventional treatment with the “classic” adjoint method (which ignores the variation of the switch point) are quantified and found to be significant. The classic adjoint method encounters more serious problems when the parameterized process causes on/off oscillations in a numerical integration of the equation. In the limit of a vanishing computational time step, the on/off oscillations approach a marginal state that can be well treated by the generalized adjoint method. It is found that the marginal state imposes a constraint on the perturbation.

Three basic issues are raised and addressed concerning whether and how discontinuous on/off switches may affect (i) the existence of adjoint and gradient, (ii) the nonlinearity and sensitivity, and (iii) the bifurcation properties. It is found that the gradient becomes discontinuous and has a regular (or singular) jump at a non-bifurcated (or bifurcated) branch point but still can be correctly computed by the generalized adjoint. Unless the switch is branched at a bifurcation point, its nonlinearity is local and lower by a half-order than the quadratic nonlinearity. The linear sensitivity of the solution to the initial state will be reduced (or enhanced) by a discontinuous switch if the perturbation is reduced (or amplified) by the switch.

Smoothing modifications of switches with their jumps fitted by continuous functions are examined for their effectiveness in making the switches suitable for the classic adjoint method. It is found that fitting the jump with a continuous function of time (control variable) cannot (can) make the switch suitable for the classic adjoint method.