# American Institute of Mathematical Sciences

ISSN:
2155-3289

eISSN:
2155-3297

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## Numerical Algebra, Control and Optimization

September 2021 , Volume 11 , Issue 3

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2021, 11(3): 333-351 doi: 10.3934/naco.2020029 +[Abstract](1845) +[HTML](627) +[PDF](1545.16KB)
Abstract:

Control algorithms can affect the performance and cost-effectiveness of the control system of a structure. This study presents an active neuro-fuzzy optimized control algorithm based on a new optimization method taken from Tug of War competition, which is highly efficient for civil structures. The performance of the proposed control method has been evaluated on the finite element model of a nonlinear highway benchmark bridge; which is consisted of nonlinear structural elements and isolation bearings and equipped with hydraulic actuators. The nonlinear control rules are approximated with a five-layer optimized neural network which transmits instructions to the actuators installed between the deck and abutments. The stability of control laws are obtained based on Lyapunov theory. The performance of the proposed algorithm in controlling bridge structural responses is investigated in six different earthquakes. The results are presented in terms of a well-defined set of performance indices that are comparable to previous methods. The results show that despite the simple description of nonlinearities and non-detailed structural information, the proposed control method can effectively reduce the performance indices of the structure. The application of artificial neural networks is a privilege, which in so far as which, despite their simplicity, they have significant effects even on complex structures such as nonlinear highway bridges.

2021, 11(3): 353-362 doi: 10.3934/naco.2020030 +[Abstract](2163) +[HTML](633) +[PDF](359.91KB)
Abstract:

Many kinds of practical problems can be formulated as nonlinear complementarity problems. In this paper, an inexact alternating direction method of multipliers for the solution of a kind of nonlinear complementarity problems is proposed. The convergence analysis of the method is given. Numerical results confirm the theoretical analysis, and show that the proposed method can be more efficient and faster than the modulus-based Jacobi, Gauss-Seidel and Successive Overrelaxation method when the dimension of the problem being solved is large.

2021, 11(3): 363-375 doi: 10.3934/naco.2020031 +[Abstract](1664) +[HTML](632) +[PDF](432.61KB)
Abstract:

This paper deals with an optimal control problem for a viral infection model with cytotoxic T-lymphocytes (CTL) immune response. The model under consideration describes the interaction between the uninfected cells, the infected cells, the free viruses and the CTL cells. The two treatments represent the efficiency of drug treatment in inhibiting viral production and preventing new infections. Existence of the optimal control pair is established and the Pontryagin's maximum principle is used to characterize these two optimal controls. The optimality system is derived and solved numerically using the forward and backward difference approximation. Finally, numerical simulations are performed in order to show the role of optimal therapy in controlling the infection severity.

2021, 11(3): 377-389 doi: 10.3934/naco.2020032 +[Abstract](2073) +[HTML](765) +[PDF](456.25KB)
Abstract:

To achieve a conjugate gradient method which is strong in theory and efficient in practice for solving unconstrained optimization problem, we propose a hybridization of the Hager and Zhang (HZ) and Polak-Ribière and Polyak (PRP) conjugate gradient methods which possesses an important property of the well known PRP method: the tendency to turn towards the steepest descent direction if a small step is generated away from the solution, averting a sequence of tiny steps from happening, the new scalar \begin{document}$\beta_k$\end{document} is obtained by convex combination of PRP and HZ under the wolfe line search we prove the sufficient descent and the global convergence. Numerical results are reported to show the effectiveness of our procedure.

2021, 11(3): 391-405 doi: 10.3934/naco.2020033 +[Abstract](1900) +[HTML](565) +[PDF](585.61KB)
Abstract:

In this paper, a predator-prey interaction model among juvenile prey, adult prey and predator has been developed where stage structure is considered on prey species. The functional responses has been considered as ratio dependent. It is assumed that that the adult prey is strong enough such that it has an anti-predator characteristic. Global dynamics of the co-existing equilibrium point has been discussed with the help of the geometric approach. Furthermore, it is established that the proposed system undergoes through a Hopf bifurcation with respect to some important parameters. Finally, some numerical simulations have been done to test our theoretical results.

2021, 11(3): 407-429 doi: 10.3934/naco.2020034 +[Abstract](1936) +[HTML](532) +[PDF](1234.21KB)
Abstract:

Differential Riccati equations are at the heart of many applications in control theory. They are time-dependent, matrix-valued, and in particular nonlinear equations that require special methods for their solution. Low-rank methods have been used heavily for computing a low-rank solution at every step of a time-discretization. We propose the use of an all-at-once space-time solution leading to a large nonlinear space-time problem for which we propose the use of a Newton–Kleinman iteration. Approximating the space-time problem in a higher-dimensional low-rank tensor form requires fewer degrees of freedom in the solution and in the operator, and gives a faster numerical method. Numerical experiments demonstrate a storage reduction of up to a factor of 100.

2021, 11(3): 431-441 doi: 10.3934/naco.2020035 +[Abstract](1305) +[HTML](514) +[PDF](324.09KB)
Abstract:

In this paper, we define and introduce some new concepts of the higher order strongly preinvex functions and higher order strongly monotone operators involving an arbitrary bifunction. Some new relationships among various concepts of higher order strongly preinvex functions have been established. We have shown that the optimality conditions for the preinvex functions can be characterized by class of higher order strongly variational-like inequalities, which appears to be new ones. As a novel applications of the higher order strongly preinvex functions, we have obtained the parallelogram-like laws for the uniformly Banach spaces. As special cases, one can obtain various new and known results from our results. Results obtained in this paper can be viewed as refinement and improvement of previously known results.

2021, 11(3): 443-448 doi: 10.3934/naco.2020036 +[Abstract](1771) +[HTML](565) +[PDF](268.2KB)
Abstract:

The paper deals with an application of survival theory in mineral processing industry. We consider the problem of maximizing copper recovery and determine the best operating conditions based on survival theory. The survival of the system reduces to a problem of maximizing a radius of a sphere inscribed into a polyhedral set defined by the linear regression equations for a flotation process. To demonstrate the effectiveness of the proposed approach, we present a case study for the rougher flotation process of copper-molybdenum ores performed at the Erdenet Mining Corporation(Mongolia).

2021, 11(3): 449-460 doi: 10.3934/naco.2020037 +[Abstract](1761) +[HTML](499) +[PDF](384.68KB)
Abstract:

In this paper, by considering that the objective function of the least squares NP-hard absolute value equations (AVE) \begin{document}$Ax-\vert x\vert = b$\end{document}, is non-convex and non-smooth, two types of proximal algorithms are proposed to solve it. One of them is the proximal difference-of-convex algorithm with extrapolation and another is the proximal subgradient method. The convergence results of the proposed methods are proved under certain assumptions. Moreover, a numerical comparison is presented to demonstrate the effectiveness of the suggested methods.

2021, 11(3): 461-485 doi: 10.3934/naco.2020038 +[Abstract](1941) +[HTML](473) +[PDF](504.66KB)
Abstract:

In this paper, we consider a class of non-autonomous nonlinear evolution equations in separable reflexive Banach spaces. First, we consider a linear problem and establish the approximate controllability results by finding a feedback control with the help of an optimal control problem. We then establish the approximate controllability results for a semilinear differential equation in Banach spaces using the theory of linear evolution systems, properties of resolvent operator and Schauder's fixed point theorem. Finally, we provide an example of a non-autonomous, nonlinear diffusion equation in Banach spaces to validate the results we obtained.

2021 CiteScore: 1.9