In chemical processes, there are constraints imposed on the inputs and outputs. For constraints on the outputs, which are not hard, i.e., the constraints can be violated to some extent, we can use fuzzy membership functions to describe the violation. However, in general, the fuzzy model predictive control problem is hard to be solved. Another method dealing with the violation of the output constraints is soft constraint method, i.e., allow the constraints on the outputs to be violated and then penalize the violation. For one kind of fuzzy model predictive control problems with specific membership functions and aggregation operators, we can transform it to a soft constraint optimization problem, which is basically a quadratic programming. Hence the computational burden is largely reduced. The simulation results verify the equivalence and the savings in the computational burden.

Using classical Lie symmetry theory，boundary control for a class of parabolic distributed parameter systems has been studied，
and the control laws on open loop and closed loop have been designed to control the system reaching the fixed stationary state． By
means of infinitesimal generators and invariance condition，the analytic control conditions on open loop and closed loop are presented after
having the classical Lie symmetry of the system，which is expressed by infinitesimal generators． The control purpose can be satisfied
under the two types of control laws by setting system parameters，initial condition and control purpose ahead． Comparatively，from the
simulation，the convergence of output error under open loop control is slower than that under closed loop control，and however there are
overshoot phenomena nearby the intake under closed loop control． The result proposed in the paper can be applied for controlling a class
of temperature or concentration models with conduction and convention characteristics．