An Optimization Problem in Circuits

Let be a linear, time-invariant, lumped two-port driven at its port 1 by a voltage source e. and loaded at its port 2 by a variable capacitor . The values of C are restricted by , where and , are given positive constants. Given this inequality constraint on , any initial state of , a time interval , and a performance criterion it is shown that the law of variation of C as a function of time which maximizes the value taken by at the state at time is bang-bang, i.e., is piecewise constant and takes only the values and . A subsidiary result as well as some interpretations are also given.