AbstractA new and simple computational algorithm for obtaining suboptimal singular and/or bang‐bang solutions of typical lumped and distributed parameter control problems is proposed. The algorithm is based on the piecewise maximization of the Hamiltonian and a limiting process utilizing a penalty function of the control variables. Theoretical developments and computational applications of the algorithm to several linear lumped parameter control problems are presented. Extensions and applications of the algorithm to nonlinear and distributed parameter systems are given in Part II.