Robust optimality conditions for semi-infinite equilibrium problems involving data uncertainty

In this paper, we have first formulated semi-infinite equilibrium problems involving data uncertainty. For this class of problems, we have proposed the Linear Independence Constraint Qualification (LICQ) in terms of tangential subdifferentials and obtained robust Karush–Kuhn–Tucker (KKT) necessary optimality conditions. Also, the sufficiency theorem has been derived using pseudo and quasi convexity assumptions. Finally, we have shown the application of our problem in control systems for PID controllers. For that, we have formulated a Model to find the best tuning parameters for the PID controller and solved it using MATLAB. For verification of our results, the Nyquist plot of the loop transfer function is shown, which has come out to be as per Nyquist stability criteria. Moreover, the results for two more PID controller problems with combined sensitivity have been obtained, which are not solved using the concept of H-infinity error anywhere in the literature. Nontrivial examples are shown in appropriate places to justify the theorems derived in this paper.