To make a decision to select a suitable approximation for the solution of a functional inequality, we need reliable information. Two useful information ideas are quality and certainty, and the measure of quality and certainty approximation of the solution of a functional inequality helps us to find the optimum approximation. To measure quality and certainty, we used the idea of the Z-number (Z-N) and we introduced the generalized Z-N (GZ-N) as a diagonal matrix of the form $diag(X,Y,X\ast Y)$, where $X$ is a fuzzy set time-stamped, $Y$ is the probability distribution function and the third part is the fuzzy-random trace of the first and the second subjects. This kind of diagonal matrix allowed us to define a new model of control functions to stabilize our problem. Using stability analysis, we obtained the most suitable approximation for functional inequalities.

The main issue we are studying in this paper is that of aggregation maps, which refers to the process of combining various input values into a single output. We apply aggregated special maps on Mittag-Leffler-type functions in one parameter to get diverse approximation errors for fractional-order systems in Hilfer sense using an optimal method. Indeed, making use of various well-known special functions that are initially chosen, we establish a new class of matrix-valued fuzzy controllers to evaluate maximal stability and minimal error. An example is given to illustrate the numerical results by charts and tables.