Control and Optimization in Applied Mathematics
Volume:4 Issue: 1, Spring-Summer 2019

Location theory is an interstice field of optimization and operations research‎. ‎In the classic location models‎, ‎the goal is finding the location of one or more facilities such that some criteria such as transportation cost‎, ‎the sum of distances passed by clients‎, ‎total service time, and cost of servicing are minimized‎. ‎The goal Weber location problem is a special case of location models that have been considered recently by some researchers‎. ‎In this problem, the ideal is locating the facility in the distance $r_i$‎, ‎from the $i$-th client‎. ‎However‎, ‎in most instances‎, ‎the solution to this problem doesn't exist‎. ‎Therefore‎, ‎the minimizing sum of errors is considered‎. ‎In the previous versions of the goal location problem, the penalty functions have been considered by some symmetric functions such as square and absolute errors of distances between clients and ideal point‎. ‎In this paper‎, ‎we consider the asymmetric linex function as the error function‎. ‎We consider the case that the distances are measured by $L_p$ norm‎. ‎Some iterative methods are used to solve the problem and the results are compared with some previously examined methods.

In this paper‎, ‎we model and solve the problem of optimal shaping and placing to put sensors for a 3-D wave equation with constant damping in a bounded open connected subset‎ ‎of 3-dimensional space‎. ‎The place of sensor is modeled by a subdomain‎ ‎of this region of a given measure‎. ‎By using an approach based on the embedding process‎, ‎first‎, ‎the system is formulated in variational form; then‎, ‎by defining‎ ‎two positive Radon measures‎, ‎the problem is represented in a‎ ‎space of measures‎. ‎ In this way, the shape design problem is turned into an infinite linear problem whose solution is guaranteed. In this step, the optimal solution (optimal control, optimal region, and optimal energy) is identified by a 2-phase optimization search technique applying two subsequent approximation steps. Moreover, some numerical simulations are given to compare this new method with other methods.

This paper presents a numerical solution of a class of fractional optimal control problems (FOCPs) in a bounded domain having a noise function by the spectral Ritz method‎. ‎The Bernstein polynomials with the fractional operational matrix are applied to approximate the unknown functions‎. ‎By substituting these estimated functions into the cost functional‎, ‎an unconstrained nonlinear optimization problem is achieved‎. In order to solve this optimization problem‎, ‎the Matlab software and its optimization toolbox are used‎. ‎In the considered FOCP‎, ‎the performance index is expressed as a function of both state and control functions‎. ‎The method is robust enough because of its computational consistency in the presence of the noise function‎. ‎Moreover‎, ‎the proposed scheme has a good pliability satisfying the given initial and boundary conditions‎. ‎At last‎, ‎some test problems are investigated to confirm the efficiency and applicability of the new method.

Let $S= {e_1,,e_2‎, ‎ldots,,e_m}$ be an ordered subset of edges of a connected graph $G$‎. ‎The edge $S$-representation of an edge set $Msubseteq E(G)$ with respect to $S$ is the‎ ‎vector $r_e(M|S) = (d_1,,d_2,ldots,,d_m)$‎, ‎where $d_i=1$ if $e_iin M$ and $d_i=0$‎ ‎otherwise‎, ‎for each $iin{1,ldots‎ , ‎k}$‎. ‎We say $S$ is a global forcing set for maximal matchings of $G$‎ ‎if $r_e(M_1|S)neq r_e(M_2|S)$ for any two maximal matchings $M_1$ and $M_2$ of $G$‎. ‎A global forcing set for maximal matchings of $G$ with minimum cardinality is called‎ ‎a minimum global forcing set for maximal matchings‎, ‎and its cardinality‎, ‎denoted by $varphi_{gm}$‎, ‎is the‎ ‎global forcing number (GFN for short) for maximal matchings‎. ‎Similarly‎, ‎for an ordered subset $F = {v_1,,v_2‎, ‎ldots,,v_k}$ of $V(G)$‎, ‎the $F$-representation of a vertex set $Isubseteq V(G)$ with respect to $F$ is the‎ ‎vector $r(I|F) = (d_1,,d_2,ldots,,d_k)$‎, ‎where $d_i=1$ if $v_iin I$ and‎ ‎$d_i=0$ otherwise‎, ‎for each $iin{1,ldots‎ , ‎k}$‎. ‎We say $F$ is a global forcing set for independent dominatings of $G$‎ ‎if $r(D_1|F)neq r(D_2|F)$ for any two maximal independent dominating sets $D_1$ and $D_2$ of $G$‎. ‎A global forcing set for independent dominatings of $G$ with minimum cardinality is called‎ ‎a minimum global forcing set for independent dominatings‎, ‎and its cardinality‎, ‎denoted by $varphi_{gi}$‎, ‎is the‎ ‎GFN for independent dominatings‎. ‎In this paper, we study the GFN for maximal matchings‎ ‎under several types of graph products‎. ‎Also‎, ‎we present some upper bounds for this invariant‎. ‎Moreover‎, ‎we present some bounds for $varphi_{gm}$ of some well-known graphs.

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In this paper‎, ‎a synthesis method based on robust model predictive control is developed for compensation of uncertain time-delays in networked control systems with bounded disturbance‎. ‎The proposed method uses linear matrix inequalities and uncertainty polytope to model uncertain time-delays and system disturbances‎. ‎The continuous system with time-delay is discretized using uncertainty polytope‎. ‎Then‎, ‎the discretized model together with model disturbance is compensated‎. ‎Uncertain parameters and additive disturbances are included in the controller design explicitly and robust stability is guaranteed in this method‎. ‎The proposed method is applied to a level process‎. ‎It is simulated by applying conventional RMPC as well‎. ‎The simulation results show the effectiveness of the proposed method compared with the conventional algorithm of the RMPCfootnote{Robust Model Predictive Control} method.

‎Rough extreme learning machines (RELMs) are rough-neural networks with one hidden layer where the parameters between the inputs and hidden neurons are arbitrarily chosen and never updated‎. ‎In this paper‎, ‎we propose RELMs with a stable online learning algorithm for the identification of continuous-time nonlinear systems in the presence of noises and uncertainties‎, ‎and we prove the global asymptotically convergence of the proposed learning algorithm using the Lyapunov stability theory‎. ‎Then‎, ‎we use the proposed methodology to identify the chaotic systems of Duffing's oscillator and Lorentz system‎. ‎Simulation results show the efficiency of the proposed model.