A multistage irreversible Carnot heat engine system operating between a finite thermal capacity high-temperature fluid reservoir and an infinite thermal capacity low-temperature environment with generalized convective heat transfer law [q proportional to (Delta T)(m)] and the irreversibility of heat resistance and internal dissipation is investigated in this paper. Optimal control theory is applied to derive the continuous Hamilton-Jacobi-Bellman (HJB) equations, which determine optimal fluid temperature configurations for maximum power output under the conditions of fixed duration and fixed initial temperature of the driving fluid. Based on general optimization results, the analytical solution for the case with Newtonian heat transfer law (m = 1) is further obtained. Since there are no analytical solutions for the other heat transfer laws (m not equal 1), the continuous HJB equations are discretized and dynamic programming (DP) algorithm is adopted to obtain complete numerical solutions of the optimization problem, and the relationships among the maximum power output of the system, the process period and the fluid temperature are discussed in detail. (C) 2010 Elsevier Ltd. All rights reserved.