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We consider the problem of determining the optimal profile of doping concentration that minimizes the base transit time in homojunction bipolar transistors. This is a well-studied problem in the electronics literature, but typically only numerical optimization is used to find solutions. Also, we include a fixed base resistance, a constraint often ignored in numerical approaches. In this paper we give an explicit analytic solution to the problem using the Pontryagin maximum principle with state-space constraints and outline how its optimality can be proven using synthesis type arguments.