# American Institute of Mathematical Sciences

ISSN:
2155-3289

eISSN:
2155-3297

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## Numerical Algebra, Control and Optimization

June 2022 , Volume 12 , Issue 2

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2022, 12(2): 213-253 doi: 10.3934/naco.2022004 +[Abstract](605) +[HTML](253) +[PDF](1388.24KB)
Abstract:

This paper presents a survey on some of the recent progress on numerical solutions for controlled switching diffusions. We begin by recalling the basics of switching diffusions and controlled switching diffusions. We then present regular controls and singular controls. The main objective of this paper is to provide a survey on some recent advances on Markov chain approximation methods for solving stochastic control problems numerically. A number of applications in insurance, mathematical biology, epidemiology, and economics are presented. Several numerical examples are provided for demonstration.

2022, 12(2): 255-278 doi: 10.3934/naco.2021004 +[Abstract](1627) +[HTML](976) +[PDF](529.46KB)
Abstract:

In this paper, we present a new modified self-adaptive inertial subgradient extragradient algorithm in which the two projections are made onto some half spaces. Moreover, under mild conditions, we obtain a strong convergence of the sequence generated by our proposed algorithm for approximating a common solution of variational inequality problem and common fixed point of a finite family of demicontractive mappings in a real Hilbert space. The main advantages of our algorithm are: strong convergence result obtained without prior knowledge of the Lipschitz constant of the related monotone operator, the two projections made onto some half-spaces and the inertial technique which speeds up rate of convergence. Finally, we present an application and a numerical example to illustrate the usefulness and applicability of our algorithm.

2022, 12(2): 279-291 doi: 10.3934/naco.2021005 +[Abstract](1440) +[HTML](532) +[PDF](425.03KB)
Abstract:

The aim of this work is to apply the results of R. Gabasov et al. [4,14] to an extended class of optimal control problems in the Bolza form, with intermediate phase constraints and multivariate control. In this paper, the developed iterative numerical method avoids the discretization of the dynamical system. Indeed, by using a piecewise constant control, the problem is reduced for each iteration to a linear programming problem, this auxiliary task allows to improve the value of the quality criterion. The process is repeated until the optimal or the suboptimal control is obtained. As an application, we use this method to solve an extension of the deterministic optimal cash management model of S.P. Sethi [31,32]. In this extension, we assume that the bank overdrafts and short selling of stock are allowed, but within the authorized time limit. The results of the numerical example show that the optimal decision for the firm depends closely on the intermediate moment, the optimal decision for the firm is to purchase until a certain date the stocks at their authorized maximum value in order to take advantage of the returns derived from stock. After that, it sales the stocks at their authorized maximum value in order to satisfy the constraint at the intermediate moment.

2022, 12(2): 293-308 doi: 10.3934/naco.2021006 +[Abstract](1473) +[HTML](601) +[PDF](399.6KB)
Abstract:

A real interval vector/matrix is an array whose entries are real intervals. In this paper, we consider the real linear interval equations \begin{document}$\bf{Ax} = \bf{b}$\end{document} with \begin{document}${{\bf{A}} }$\end{document}, \begin{document}$\bf{b}$\end{document} respectively, denote an interval matrix and an interval vector. The aim of the paper is to study the numerical solution of the linear interval equations for various classes of coefficient interval matrices. In particular, we study the convergence of interval AOR method when the coefficient interval matrix is either interval strictly diagonally dominant matrices, interval \begin{document}$L$\end{document}-matrices, interval \begin{document}$M$\end{document}-matrices, or interval \begin{document}$H$\end{document}-matrices.

2022, 12(2): 309-320 doi: 10.3934/naco.2021007 +[Abstract](1113) +[HTML](461) +[PDF](672.72KB)
Abstract:

The approximation of the fractional-order controller (FOC) has already been recognized as a distinguished field of research in the literature of system and control. In this paper, a two-step design approach is presented to realize a fractional-order proportional integral controller (FOPI) for a class of fractional-order plant model. The design goals are based on some frequency domain specifications. The first stage of the work is focused on developing the pure continuous-time FOC, while the second stage actually realizes the FOPI controller in discrete-time representation. The presented approach is fundamentally dissimilar with respect to the conventional approaches of z -domain. In the process of realizing the FOC, the delta operator has been involved as a generating function due to its exclusive competency to unify the discrete-time system and its continuous-time counterpart at low sampling time limit. The well-known continued fraction expansion (CFE) method has been employed to approximate the FOPI controller in delta-domain. Simulation outcomes exhibit that the discrete-time FOPI controller merges to its continuous-time counterpart at the low sampling time limit. The robustness of the overall system is also investigated in delta-domain.

2022, 12(2): 321-337 doi: 10.3934/naco.2021008 +[Abstract](1581) +[HTML](540) +[PDF](3370.69KB)
Abstract:

Ordinary differential equations are converted into a constrained optimization problems to find their approximate solutions. In this work, an algorithm is proposed by applying particle swarm optimization (PSO) to find an approximate solution of ODEs based on an expansion approximation. Since many cases of linear and nonlinear ODEs have singularity point, Padé approximant which is fractional expansion is employed for more accurate results compare to Fourier and Taylor expansions. The fitness function is obtained by adding the discrete least square weighted function to a penalty function. The proposed algorithm is applied to 13 famous ODEs such as Lane Emden, Emden-Fowler, Riccati, Ivey, Abel, Thomas Fermi, Bernoulli, Bratu, Van der pol, the Troesch problem and other cases. The proposed algorithm offer fast and accurate results compare to the other methods presented in this paper. The results demonstrate the ability of proposed approach to solve linear and nonlinear ODEs with initial or boundary conditions.

2022, 12(2): 339-351 doi: 10.3934/naco.2021009 +[Abstract](1271) +[HTML](453) +[PDF](364.11KB)
Abstract:

This paper discusses exact (approximate) controllability and exact (approximate) observability of stochastic implicit systems in Banach spaces. Firstly, we introduce the stochastic GE-evolution operator in Banach space and discuss existence and uniqueness of the mild solution to stochastic implicit systems by stochastic GE-evolution operator in Banach space. Secondly, we discuss conditions for exact (approximate) controllability and exact (approximate) observability of the systems considered in terms of stochastic GE-evolution operator and the dual principle. Finally, an illustrative example is given.

2022, 12(2): 353-371 doi: 10.3934/naco.2021010 +[Abstract](1341) +[HTML](494) +[PDF](416.06KB)
Abstract:

The paper deals with the controllability and observability of second order discrete linear time varying and linear time-invariant continuous systems in matrix form. To this case, we generalize the classical conditions for linear systems of the first order, without reducing them to systems of the first order. Within the framework of Kalman-type criteria, we investigate these concepts for second-order linear systems with discrete / continuous time; we define the initial values and input functions uniquely if and only if the observability and controllability matrices have full rank, respectively. Also a conceptual partner of controllability, that is, reachability of second order discrete time-varying systems is formulated and a necessary and sufficient condition for complete reachability is derived. Also the transfer function of the second order continuous-time linear state-space system is constructed. We have given numerical examples to illustrate the feasibility and effectiveness of the theoretical results obtained.

2022, 12(2): 373-393 doi: 10.3934/naco.2021011 +[Abstract](1278) +[HTML](460) +[PDF](609.99KB)
Abstract:

In this paper, we introduce and study a modified extragradient algorithm for approximating solutions of a certain class of split pseudo-monotone variational inequality problem in real Hilbert spaces. Using our proposed algorithm, we established a strong convergent result for approximating solutions of the aforementioned problem. Our strong convergent result is obtained without prior knowledge of the Lipschitz constant of the pseudo-monotone operator used in this paper, and with minimized number of projections per iteration compared to other results on split variational inequality problem in the literature. Furthermore, numerical examples are given to show the performance and advantage of our method as well as comparing it with related methods in the literature.

2022, 12(2): 395-426 doi: 10.3934/naco.2021013 +[Abstract](1305) +[HTML](841) +[PDF](842.13KB)
Abstract:

Caputo derivative operational matrices of the arbitrary scaled Legendre and Chebyshev wavelets are introduced by deriving directly from these wavelets. The Caputo derivative operational matrices are used in quadratic optimization of systems having fractional or integer orders differential equations. Using these operational matrices, a new quadratic programming wavelet-based method without doing any integration operation for finding solutions of quadratic optimal control of traditional linear/nonlinear fractional time-delay constrained/unconstrained systems is introduced. General strategies for handling different types of the optimal control problems are proposed.

2022, 12(2): 427-443 doi: 10.3934/naco.2021014 +[Abstract](1692) +[HTML](584) +[PDF](415.57KB)
Abstract:

In this paper, a new concept of generalized convexity is introduced for not necessarily differentiable vector optimization problems with \begin{document}$E$\end{document}-differentiable functions. Namely, for an \begin{document}$E$\end{document}-differentiable vector-valued function, the concept of \begin{document}$V$\end{document}-\begin{document}$E$\end{document}-invexity is defined as a generalization of the \begin{document}$E$\end{document}-differentiable \begin{document}$E$\end{document}-invexity notion and the concept of \begin{document}$V$\end{document}-invexity. Further, the sufficiency of the so-called \begin{document}$E$\end{document}-Karush-Kuhn-Tucker optimality conditions are established for the considered \begin{document}$E$\end{document}-differentiable vector optimization problems with both inequality and equality constraints under \begin{document}$V$\end{document}-\begin{document}$E$\end{document}-invexity hypotheses. Furthermore, the so-called vector \begin{document}$E$\end{document}-dual problem in the sense of Mond-Weir is defined for the considered \begin{document}$E$\end{document}-differentiable multiobjective programming problem and several \begin{document}$E$\end{document}-duality theorems are derived also under appropriate \begin{document}$V$\end{document}-\begin{document}$E$\end{document}-invexity assumptions.

2022, 12(2): 445-467 doi: 10.3934/naco.2021017 +[Abstract](1161) +[HTML](428) +[PDF](477.29KB)
Abstract:

We consider some important computational aspects of the long-step path-following algorithm developed in our previous work and show that a broad class of complicated optimization problems arising in quantum information theory can be solved using this approach. In particular, we consider one difficult optimization problem involving the quantum relative entropy in quantum key distribution and show that our method can solve problems of this type much faster in comparison with (very few) available options.

2021 CiteScore: 1.9