On Switching Times Optimization and Second‐Order Sufficient Conditions for Optimal Bang‐Bang Boundary Control of the Heat Equation

ABSTRACT We consider optimal control of the heat equation where the control variable appears linearly as a boundary control. We establish switching times optimization for bang–bang controls with multiple switching times. Particularly, representations of variational derivatives of the state and the cost w.r.t. switching times are given. Necessary and sufficient optimality conditions in the induced optimization problem are discussed. The results are compared with switching times optimization for ODE optimal control problems. This leads to new second‐order sufficient optimality conditions in the control problem. The results will be illustrated with two numerical examples.