In a recent study [ibid. 39, 431-449 (1983; Zbl 0487.65039)], the effects of large penalty constants on Ritz penalty methods based on finite element approximations used in the solution of the control of a system governed by diffusion equation were established. The problem involves the selection of the input u(x,t) so as the miminize the cost \[ J(u)=\int^{1}_{0}\int^{1}_{0}\{u^ 2(x,t)+Z^ 2(x,t)\}dx dt \] subject to the constraint \(\partial z/\partial t=\partial^ 2z/\partial x^ 2+u(x,t)\), \(0\leq x\), \(t\leq 1\), with boundary conditions \(z(0,t)=z(1,t)=0\), \(0\leq t\leq 1\), and the initial state \(z(x,0)=z_ 0(x)\), \(0\leq x\leq 1.\) Our results verify that the Ritz penalty method exhibits good convergence properties, although the estimates for the convergence rate are cumbersome. In this paper, a conceptually simple procedure based on the conventional penalty method is presented. Some significant advantages of the method are the following. It allows easy estimation of its convergence rate. Furthermore, the multiplier method can be used to accelerate the rate of convergence of the method without essentially allowing the penalty constants to tend to infinity; thus, in this way, it is possible to retain the good convergence properties, an important feature which is often glossed over. The paper provides a clear mathematical analysis of how these advantages can be exploited and is illustrated with numerical examples.