Control and Optimization in Applied Mathematics
Volume:7 Issue: 1, Winter-Spring 2022

Drones are among the most valuable and versatile technologies in the world‎, ‎with applications in a vast number of ‎‎‎fields such as traffic control‎, ‎agriculture‎, ‎firefighting and‎ ‎rescue‎, ‎and filmmaking‎, ‎to name a few‎. ‎As the development of unmanned aerial vehicles (UAVs) accelerates‎, ‎the‎ ‎safety of UAVs becomes increasingly important‎. ‎In this paper‎, ‎a robust adaptive controller is designed to improve the safety of a hexa-rotor UAV‎, ‎and a robust adaptive controller is developed to control our system‎. ‎In doing so‎, ‎the wind parameters from the aerodynamic forces and moments acting on the hexa-rotor are estimated using an observer with the adaptive algorithm‎. ‎This proposed controller guarantees stability and reliable function in the midst of parametric and non-parametric uncertainties‎. ‎The process's global stability and tracking convergence are investigated using the Lyapunov theorem‎. ‎The performance and effectiveness of the proposed controller are tested through two simulation studies‎, ‎which take into account external disturbances that are a function of time.

Data envelopment analysis models are able to rank decision-making units (DMUs) based on their efficiency scores‎. ‎In spite of the fact that there exists a unique ranking of inefficient DMUs‎, ‎ranking efficient DMUs is problematic‎. ‎However‎, ‎rather than ranking methods‎, ‎another way to choose one of the efficient units is to determine the most efficient DMU‎. ‎Up to the present‎, ‎many models have been proposed to rank DMUs and determine the most efficient one‎. ‎These models require solving nonlinear or integer programs‎, ‎which are NP-hard and time-consuming‎. ‎Considering efficient DMU's characteristics‎, ‎this paper proposes a procedure to find the most efficient DMU through some simple operations‎. ‎The validity of the proposed approach is verified and tested via some numerical examples.

Fixed point theorems can be used to prove the solvability of optimization problems‎, ‎differential equations and equilibrium problems, and‎ ‎the intrinsic flexibility of probabilistic metric spaces makes it possible to extend the idea of contraction mapping in several inequivalent ways‎. ‎In this paper‎, ‎we extend very recent fixed point theorems in the setting of Menger probabilistic metric spaces‎. ‎We present some fixed point theorems for self-mappings satisfying a generalized (ϕ , ψ ) - contractive condition in Menger probabilistic metric spaces which are contractions used extensively in global optimization problems‎. ‎On the other hand‎, ‎we consider a more general class of auxiliary functions in the contractivity condition and prove the existence of fixed points of non-expansive mappings on Menger probabilistic metric spaces.

‎‎‎In this paper‎, ‎a stable multi-objective model of location‎, ‎inventory‎, ‎and supply chain routing is presented under conditions of uncertainty and using a passive defense approach‎. ‎Parameters such as demand‎, ‎cost of setting up the facility and cost of maintaining inventory are considered uncertain and in the form of triangular fuzzy numbers‎.‎ ‎Also‎, ‎in order to increase supply chain resilience‎, ‎the characteristics and capabilities of passive defense in the supply chain‎, ‎such as ``ready flow rate''‎, ‎``security of backup routes''‎, ‎``possibility of deployment of resources and equipment''‎, ‎and ``the principle of dispersion for location'' are considered‎. ‎Multipurpose‎, ‎multipartite algorithms‎, ‎based on the Pareto archive and genetic algorithm‎, ‎are used to solve the model‎. ‎‎The results of validation show that the proposed model is valid and feasible‎, ‎and the proposed algorithm is also valid and converges to the optimal solution. ‎Sample problems‎, ‎in three groups of small‎, ‎medium and large‎, ‎are solved by two algorithms‎, ‎and the results are compared based on quality‎, ‎dispersion‎, ‎uniformity and execution time‎.‎ ‎The results of this section show that in all cases‎, ‎the multi-objective particle mass algorithm has a higher ability than the GA to produce solutions of higher quality and to explore and extract the scalable area of‎ ​‎​the solution. ‎Also‎, ‎the comparison of the execution times of the algorithms indicates that the multi-objective particle mass algorithm has a higher solution time.

The purpose of this paper is to develop nonsmooth optimization problems (P) in which all emerging functions are assumed to be real-valued quasiconvex functions that are defined on a finite-dimensional Euclidean space‎. First‎, ‎we introduce two linear optimization problems with the same optimal value of the considered problem‎. ‎Then‎, ‎we introduce a real-valued non-negative gap function for (P)‎, ‎and we provide some conditions which ensure that its null points are the same as the optimal solution of problem (P)‎. ‎The results are based on incident subdifferential‎, ‎which is an important concept in the analysis of quasiconvex functions.

In this paper‎, ‎we prove that every orthogonally higher ring derivation is a higher ring derivation‎. ‎Also we find the general solution of the pexider orthogonally higher ring derivations‎‎begin{align*}‎‎left{‎‎begin{array}{lr}‎‎f_n(x+y)=g_n(x)+h_n(y)‎, ‎;leftlangle x,y rightrangle =0,\‎‎f_n(xy) = sum_{i+j=n} g_i(x)h_j(y)‎.‎end{array}‎‎right‎.‎end{align*}‎‎Then we prove that for any approximate pexider orthogonally higher ring derivation under some control functions $ varphi(x,y) $ and $ psi(x,y) $‎, ‎there exists a unique higher ring derivation $ D={d_n}_{n=0}^infty $‎, ‎near $ {f_n}_{n=0}^infty $‎, ‎$ {g_n}_{n=0}^infty $ and $ {h_n}_{n=0}^infty $ estimated by $ varphi $ and $ psi $.