Control and Optimization in Applied Mathematics
Volume:3 Issue: 1, Winter-Spring 2019

In this paper‎, ‎a modern method is presented to solve a class of fractional optimal control problems (FOCPs) indirectly‎. ‎First‎, ‎the necessary optimality conditions for the FOCP are obtained in the form of two fractional differential equations (FDEs)‎. ‎Then‎, ‎the unknown functions are approximated by the hybrid functions‎, ‎including Bernoulli polynomials and Block-pulse functions based on the spectral Ritz method‎. ‎Also‎, ‎two new methods are proposed for calculating the left Caputo fractional derivative and right Riemann-Liouville fractional derivative operators of the hybrid functions that are proportional to the Ritz method‎. ‎The FOCP is converted into a system of the algebraic equations by applying the fractional derivative operators and collocation method‎, ‎which determines the solution of the problem‎. ‎Error estimates for the hybrid function approximation‎, ‎fractional operators and‎, ‎the proposed method are provided‎. ‎Finally‎, ‎the efficiency of the proposed method and its accuracy in obtaining optimal solutions are shown by some test problems.

In this paper‎, ‎two extended three-term conjugate gradient methods based on the Liu-Storey ({tt LS})‎ ‎conjugate gradient method are presented to solve unconstrained optimization problems‎. ‎A remarkable property of the proposed methods is that the search direction always satisfies‎ ‎the sufficient descent condition independent of line search method‎, ‎based on eigenvalue analysis‎. ‎The global convergence of proposed algorithms is established under suitable conditions‎. ‎Preliminary numerical results show that the proposed methods are efficient and robust‎ ‎to solve the unconstrained optimization problems.

The existing modeling methods using Petri Nets‎, ‎have been successfully applied to model and analyze dynamic systems‎. ‎However‎, ‎these methods are not capable of modeling all dynamic systems such as systems with the current sample time signals‎, ‎systems including various subsystems and multi-mode systems‎. ‎This paper proposes Hybrid Time Delay Petri Nets (HTDPN) to solve the problem‎. ‎In this approach‎, ‎discrete and continuous Petri Nets are combined so that the continuous PNs part and the discrete PNs are responsible for past time samples and current sample time‎, ‎respectively‎. ‎To evaluate the performance of the proposed tool‎, ‎it is employed to model a legless piezoelectric capsubot robot as a multi modes system and a $PID$ controller‎, ‎in which the gains tuned by the Genetic Algorithm are designed for the resulting model by HTDPN‎. ‎Results show that the proposed method is faster in terms of mathematical calculations which can reduce the simulation time and complexity of complicated systems‎. ‎It would be observed that the proposed approach makes the $PID$ controller design simpler as well‎. ‎In addition‎, ‎a comparative study of capsubot has been performed‎. ‎Simulation results show that the presented method is encouraging compared to the predictive control‎, ‎which is used in the literature.

In this paper‎, ‎we introduce a new type of graph contraction using a special class of functions and give a best proximity point theorem for this contraction in complete metric spaces endowed with a graph under two different conditions‎. ‎We then support our main theorem by a non-trivial example and give some consequences of best proximity point of it for usual graphs.

In this paper, first the space of hyperbolic tangent functions is introduced and then the universal approximator property of this space is proved. In fact, by using this space, any nonlinear continuous function can be uniformly approximated with any degree of accuracy. Also, as an application, this space of functions is utilized to design feedback control for a nonlinear dynamical system.

o enhance the performances of rough-neural networks (R-NNs) in the system identification‎, ‎on the base of emotional learning‎, ‎a new stable learning algorithm is developed for them‎. ‎This algorithm facilitates the error convergence by increasing the memory depth of R-NNs‎. ‎To this end‎, ‎an emotional signal as a linear combination of identification error and its differences is used to achieve the learning laws‎. ‎In addition‎, ‎the error convergence and the boundedness of predictions and parameters of the model are proved‎. ‎To illustrate the efficiency of proposed algorithm‎, ‎some nonlinear systems including the cement rotary kiln are identified using this method and the results are compared with some other models.