A simplification of the Agrachev–Gamkrelidze second-order variation for bang–bang controls

We consider an expression for the second-order variation (SOV) of bang–bang controls derived by Agrachev and Gamkrelidze. The SOV plays an important role in both necessary and sufficient second-order optimality conditions for bang–bang controls. These conditions are stronger than the one provided by the first-order Pontryagin maximum principle (PMP). For a bang–bang control with kk switching points, the SOV contains k(k+1)/2k(k+1)/2 Lie-algebraic terms. We derive a simplification of the SOV by relating kk of these terms to the derivative of the switching function, defined in the PMP, evaluated at the switching points. We prove that this simplification can be used to reduce the computational burden associated with applying the SOV to analyze optimal controls. We demonstrate this by using the simplified expression for the SOV to show that the chattering control in Fuller’s problem satisfies a second-order sufficient condition for optimality.