Control and Optimization in Applied Mathematics
Volume:5 Issue: 2, Summer-Autumn 2020

In view of the tremendous importance of patients’ stability in medical sciences‎, ‎this paper addresses the application of a sliding mode control in medical devices‎. ‎In doing so‎, ‎we consider a nonlinear dynamic system that shows the mathematical model of the human immunodeficiency virus‎. ‎This nonlinear model has three variable states‎: ‎healthy cells‎, ‎infected cells‎, ‎and free viruses‎. ‎The proposed controller displays the effect of medication on preventing the production of the virus and blocking the new infection‎. ‎This controller ensures the stability of this dynamic system provided for HIV in the event of a bounded disturbance‎. ‎The stability and convergence of this process are proved by the Lyapunov theorem‎. ‎Finally‎, ‎a numerical example is given to demonstrate the efficiency of the proposed method.

A shearlet frame approach is used to solve $n$-dimensional wave equations numerically‎. ‎By the presented procedure‎, ‎the shearlet coefficients are obtained via separate time-independent partial differential equations‎. ‎The proposed method has the advantage of separation of spatial and temporal parameters‎. ‎The issues of convergence and best approximation are also discussed.

Evaluation of advertising marketing campaigns is a very important and complex task‎, ‎so far no comprehensive model has been presented in this regard‎. ‎The present study aims to provide a decision framework for evaluating marketing campaigns‎. ‎This article collects real-world data from an Iranian bank deposit marketing campaign‎. ‎For this purpose‎, ‎250 cases were considered to extract the rules and 60 cases were considered as test data‎. ‎Information is provided on 15 important parameters of marketing education‎, ‎defaults‎, ‎age‎, ‎occupation‎, ‎marriage‎, ‎day‎, ‎contact‎, ‎balance‎, ‎housing‎, ‎loans‎, ‎previous contact‎, ‎previous outcome‎, ‎month‎, ‎call duration, and campaigns‎. ‎A fuzzy expert system was designed with 12 rules after reviewing the rules and removing similar and contradictory rules by using their degree calculation‎. ‎In this system‎, ‎by integrating some factors‎, ‎finally, 6 input variables and one output variable were considered that were used by the product inference engine‎, ‎singleton fuzzifier, and center average defuzzifier‎. ‎It was observed that the designed fuzzy expert system provides very good results.

Iterative feedback tuning (IFT) is an algorithm for adjusting the coefficients of the integer-order type proportional-integral-derivative (PID) controllers without needing a system model‎. ‎The IFT algorithm is performed iteratively with the aim of optimizing the control coefficients at each stage via an objective function‎. ‎In this research‎, ‎for the first time‎, ‎the IFT algorithm is used to adjust all the coefficients of the fractional order PID controllers‎, ‎i.e.‎, ‎PI^α D^β controllers to have optimal performance‎. ‎For this purpose‎, ‎fractional order calculations and the integer-order version of the IFT algorithm are firstly presented‎, ‎and the novel IFT algorithm is then used to adjust coefficients of the PI^α D^β controller‎. ‎Finally‎, ‎the performance of the proposed method is illustrated and verified through some examples.

In this paper‎, ‎the benefits of 1/G'-expansion technique are utilized to create a direct scheme for extracting approximate solutions for a class of optimal control problems‎. ‎In the given approach‎, ‎first state and control functions have been parameterized as a power series‎, ‎which is constructed according to the solutions of a Bernoulli differential equation‎, ‎where the number of terms in produced power series is determined by the balance method‎. ‎A proportionate replacement and solving the created optimization problem lead to suitable solutions close to the analytical ones for the main problem‎. ‎Numerical experiments are given to evaluate the quality of the proposed method.

In this paper‎, ‎we study nonsmooth optimization problems with quasiconvex functions using topological subdifferential‎. ‎We present some necessary and sufficient optimality conditions and characterize topological pseudoconvex functions‎. ‎Finally‎, ‎the Mond-Weir type weak and strong duality results are stated for the problems.