Additional properties of an extended maximum principle†

This paper establishes conditions of optimality for linear systems whoso control functions are quantized either in magnitude or in switching instants, or both. Optimization of such systems hitherto has been limited primarily to dynamic programming techniques and direct enumeration. The results are derived from the total variations of the cost functional. The main theorem provides sufficient conditions of optimality for linear systems with quadratic and positive semi-definite performance index. The results are significant for digital control systems, pulse-code-modulated control systems and some types of pulse-frequency-modulated systems which are subject to quantized variations only. For the special case where the control functions are not quantized, the optimality conditions provided by the extended maximum principle are shown to be necessary and sufficient.