# American Institute of Mathematical Sciences

ISSN:
2155-3289

eISSN:
2155-3297

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## Numerical Algebra, Control & Optimization

September 2017 , Volume 7 , Issue 3

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2017, 7(3): 223-250 doi: 10.3934/naco.2017016 +[Abstract](2901) +[HTML](186) +[PDF](751.4KB)
Abstract:

In modern material sciences and multi-scale physics homogenization approaches provide a global characterization of physical systems that depend on the topology of the underlying microgeometry. Purely formal approaches such as averaging techniques can be applied for an identification of the averaged system. For models in variational form, two-scale convergence for network functions can be used to derive the homogenized model. The sequence of solutions of the variational microcsopic models and the corresponding sequence of tangential gradients converge toward limit functions that are characterized by the solution of the variational macroscopic model. Here, a further extension of this result is proved. The variational macroscopic model can be equivalently represented by a homogenized model on the superior domain and a certain number of reference cell problems. In this way, the results obtained by averaging strategies are supported by notions of convergence for network functions on varying domains.

2017, 7(3): 251-271 doi: 10.3934/naco.2017017 +[Abstract](2832) +[HTML](156) +[PDF](575.5KB)
Abstract:

In the present paper, the vibration supression of a smart plate with the use of ANFIS (Adaptive Neuro-Fuzzy Inference System) is investigated. The whole system consists of a nonlinear mechanical model, which is an extension of the von Kármán plate model with control. The structure is subjected to external disturbances and generalized control forces. Initial and boundary conditions are set up. The initial boundary value problem is spatially-discretized by a time spectral method. The obtained discretized model is a system of nonlinear ordinary differential equations (ODEs) with respect to time. A neuro-fuzzy inference system is built and tested in order to create a nonlinear controller for the vibration supression of the plate. More specifically, a Sugeno-type fuzzy inference system is employed and trained through ANFIS. The inputs of the controller are the displacement and the velocity and the output is the control force. An effective optimization procedure is proposed and numerical results are presented.

2017, 7(3): 273-287 doi: 10.3934/naco.2017018 +[Abstract](3049) +[HTML](194) +[PDF](391.1KB)
Abstract:

In this paper we propose an efficient and easy-to-implement numerical method for an $α$-th order Ordinary Differential Equation (ODE) when $α∈ (0, 1)$, based on a one-point quadrature rule. The quadrature point in each sub-interval of a given partition with mesh size $h$ is chosen judiciously so that the degree of accuracy of the quadrature rule is 2 in the presence of the singular integral kernel. The resulting time-stepping method can be regarded as the counterpart for fractional ODEs of the well-known mid-point method for 1st-order ODEs. We show that the global error in a numerical solution generated by this method is of the order $\mathcal{O}(h^{2})$, independently of $α$. Numerical results are presented to demonstrate that the computed rates of convergence match the theoretical one very well and that our method is much more accurate than a well-known one-step method when $α$ is small.

2017, 7(3): 289-299 doi: 10.3934/naco.2017019 +[Abstract](2552) +[HTML](153) +[PDF](621.9KB)
Abstract:

Pinning control on complex dynamical networks has emerged as a very important topic in recent trends of control theory due to the extensive study of collective coupled behaviors and their role in physics, engineering and biology. In practice, real-world networks consist of a large number of vertices and one may only be able to perform a control on a fraction of them only. Controllability of such systems has been addressed in [17], where it was reformulated as a global asymptotic stability problem. The goal of this short note is to refine the analysis proposed in [17] using recent results in singular value perturbation theory.

2017, 7(3): 301-323 doi: 10.3934/naco.2017020 +[Abstract](4196) +[HTML](212) +[PDF](600.9KB)
Abstract:

In this paper, we propose a new hybrid grey wolf optimizer (GWO) algorithm with simplex Nelder-Mead method in order to solve integer programming and minimax problems. We call the proposed algorithm a Simplex Grey Wolf Optimizer (SGWO) algorithm. In the the proposed SGWO algorithm, we combine the GWO algorithm with the Nelder-Mead method in order to refine the best obtained solution from the standard GWO algorithm. We test it on 7 integer programming problems and 10 minimax problems in order to investigate the general performance of the proposed SGWO algorithm. Also, we compare SGWO with 10 algorithms for solving integer programming problems and 9 algorithms for solving minimax problems. The experiments results show the efficiency of the proposed algorithm and its ability to solve integer and minimax optimization problems in reasonable time.

2017, 7(3): 325-344 doi: 10.3934/naco.2017021 +[Abstract](2315) +[HTML](140) +[PDF](454.2KB)
Abstract:

This paper is concerned with optimal boundary control of a three dimensional reaction-diffusion system in a more general form than what has been presented in the literature. The state equations are analyzed and the optimal control problem is investigated. Necessary and sufficient optimality conditions are derived. The model is widely applicable due to its generality. Some examples in applications are discussed.

2017, 7(3): 345-357 doi: 10.3934/naco.2017022 +[Abstract](2458) +[HTML](152) +[PDF](596.0KB)
Abstract:

A type of new consensus protocol for a two-dimension multi-agent system (MAS) is proposed. By introducing the conventional MAS and protocol, a dynamic equation of the first-order two-dimension MAS is proposed. then a new protocol with its Laplacian matrix is adopted. According to two types possible roots of character equations, two lemmas are proposed to show consensus asymptotical conditions. Furthermore, the convergence conditions of parameters are analyzed. Several simulated examples illustrate that consensus is achieved if the convergence conditions are satisfied.

2017, 7(3): 359-377 doi: 10.3934/naco.2017023 +[Abstract](3362) +[HTML](198) +[PDF](445.7KB)
Abstract:

This paper presents a multistage stochastic programming model to deal with multi-period, cardinality constrained portfolio optimization. The presented model aims to minimize investor's expected regret, while ensuring achievement of a minimum expected return. To generate scenarios of market index returns, a random walk model based on the empirical distribution of market-representative index returns is proposed. Then, a single index model is used to estimate stock returns based on market index returns. Afterward, historical returns of a number of stocks, selected from Frankfurt Stock Exchange (FSE), are used to implement the presented scenario generation method, and solve the stochastic programming model. In addition, the impact of cardinality constraints, transaction costs, minimum expected return and predetermined investor's target wealth are investigated. Results show that the inclusion of cardinality constraints and transaction costs significantly influences the investors risk-return tradeoffs. This is also the case for investors target wealth.