Aris and Amundson [1–3] originally analyzed the design of a continuous stirredtank reactor (CSTR) with a firstorder, irreversible exothermic reaction. The reactor is controlled by varying the flow of a cooling fluid through a coil inside the reactor. The state equations for the CSTR are given by , , where and are the deviations from steadystate concentration and temperature, respectively, and is the control action. This Demonstration finds the optimal control, which minimizes the performance index given by , where the normalized control action is and is a weighting parameter. Assume a final time of the control [4, 5]. You can set the initial conditions (i.e. both and ) and choose to apply upper and lower bounds on . We plot the component of the state vector ( and shown in red and blue, respectively), the normalized control , and the costates. We also compute the value of the performance index . For the case where is unbounded, if you select the initial conditions and , then , in close agreement with the result found in [4] and [5]. The costates and are plotted in magenta and cyan. The costates verify and (yellow dot).
 Contributed by: Housam Binous and Ahmed Bellagi
 (King Fahd University of Petroleum & Minerals, KSA; University of Monastir, Tunisia)
[1] R. Aris and N. R. Amundson, "An Analysis of Chemical Reactor Stability and Control—I: The Possibility of Local Control, with Perfect or Imperfect Control Mechanisms," Chemical Engineering Science, 7(3), 1958 pp. 121–131. doi:10.1016/00092509(58)800196. [2] R. Aris and N. R. Amundson, "An Analysis of Chemical Reactor Stability and Control—II: The Evolution of Proportional Control", Chemical Engineering Science, 7(3), 1958 pp. 132–147. doi:10.1016/00092509(58)800202. [3] R. Aris and N. R. Amundson, "An Analysis of Chemical Reactor Stability and Control—III: The Principles of Programming Reactor Calculations. Some Extensions", Chemical Engineering Science, 7(3), 1958 pp. 148–155. doi:10.1016/00092509(58)800214. [4] R. Luus and L. Lapidus, "The Control of Nonlinear Systems. Part II: Convergence by Combined First and Second Variations," AIChE Journal, 13(1), 1967 pp. 108–113. doi:10.1002/aic.690130120. [5] D. E. Kirk, Optimal Control Theory: An Introduction, Mineola, NY: Dover Publications, 2004.
