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Numerical Algebra, Control & Optimization

June 2017 , Volume 7 , Issue 2

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Computing minimum norm solution of linear systems of equations by the generalized Newton method
Saeed Ketabchi, Hossein Moosaei, M. Parandegan and Hamidreza Navidi
2017, 7(2): 113-119 doi: 10.3934/naco.2017008 +[Abstract](1002) +[HTML](10) +[PDF](354.6KB)
Abstract:

The aim of this paper is to find the minimum norm solution of a linear system of equations. The proposed method is based on presenting a view of solution on the dual exterior penalty problem of primal quadratic programming. To solve the unconstrained minimization problem, the generalized Newton method was employed and to guarantee its finite global convergence, the Armijo step size regulation was adopted. This method was tested on all systems selected in NETLIB 1. Numerical results were compared with the MOSEK Optimization Software 2 on linear systems in NETLIB (Table 1) and on linear systems generated by the Linear systems generator (Table 2).

1www.netlib.org

2 www.mosek.com

SISO H-Optimal synthesis with initially specified structure of control law
Evgeny I. Veremey and Vladimir V. Eremeev
2017, 7(2): 121-138 doi: 10.3934/naco.2017009 +[Abstract](744) +[HTML](20) +[PDF](437.7KB)
Abstract:

The paper is devoted to particular cases of H-optimization problems for LTI systems with scalar control and external disturbance. The essence of these problems is to find an output feedback optimal controller having initially given structure to attenuate disturbances action with respect to controlled variable and control. An admissible set of controllers can be additionally restricted by the requirement to assign given poles spectrum of the closed-loop system. Specific features of the posed problems are considered and three simple numerical methods of synthesis are proposed to design correspondent H-optimal controllers. To show the simplicity and effectiveness of the proposed approach and the benefits of developed methods, illustrative examples are enclosed to the paper.

Computational networks and systems-homogenization of self-adjoint differential operators in variational form on periodic networks and micro-architectured systems
Erik Kropat, Silja Meyer-Nieberg and Gerhard-Wilhelm Weber
2017, 7(2): 139-169 doi: 10.3934/naco.2017010 +[Abstract](747) +[HTML](22) +[PDF](8272.8KB)
Abstract:

Micro-architectured systems and periodic network structures play an import role in multi-scale physics and material sciences. Mathematical modeling leads to challenging problems on the analytical and the numerical side. Previous studies focused on averaging techniques that can be used to reveal the corresponding macroscopic model describing the effective behavior. This study aims at a mathematical rigorous proof within the framework of homogenization theory. As a model example, the variational form of a self-adjoint operator on a large periodic network is considered. A notion of two-scale convergence for network functions based on a so-called two-scale transform is applied. It is shown that the sequence of solutions of the variational microscopic model on varying networked domains converges towards the solution of the macroscopic model. A similar result is achieved for the corresponding sequence of tangential gradients. The resulting homogenized variational model can be easily solved with standard PDE-solvers. In addition, the homogenized coefficients provide a characterization of the physical system on a global scale. In this way, a mathematically rigorous concept for the homogenization of self-adjoint operators on periodic manifolds is achieved. Numerical results illustrate the effectiveness of the presented approach.

An infeasible full NT-step interior point method for circular optimization
Behrouz Kheirfam and Guoqiang Wang
2017, 7(2): 171-184 doi: 10.3934/naco.2017011 +[Abstract](824) +[HTML](18) +[PDF](389.8KB)
Abstract:

In this paper, we design a primal-dual infeasible interior-point method for circular optimization that uses only full Nesterov-Todd steps. Each main iteration of the algorithm consisted of one so-called feasibility step. Furthermore, giving a complexity analysis of the algorithm, we derive the currently best-known iteration bound for infeasible interior-point methods.

The optimal stabilization of orbital motion in a neighborhood of collinear libration point
Alexander Shmyrov and Vasily Shmyrov
2017, 7(2): 185-189 doi: 10.3934/naco.2017012 +[Abstract](578) +[HTML](17) +[PDF](354.1KB)
Abstract:

In this paper we consider the special problem of stabilization of controllable orbital motion in a neighborhood of collinear libration point \begin{document}$L_2$\end{document} of Sun-Earth system. The modification of circular three-body problem -nonlinear Hill's equations, which describe orbital motion in a neighborhood of libration point is used as a mathematical model. Also, we used the linearized equations of motion. We investigate the problem of spacecraft arrival on the unstable invariant manifold. When a spacecraft reaches this manifold, it does not leave the neighborhood of \begin{document}$L_2$\end{document} by long time. The distance to the unstable invariant manifold is described by a special function of phase variables, so-called ''hazard function". The control action directed along Sun-Earth line.

Sufficient optimality conditions for extremal controls based on functional increment formulas
Vladimir Srochko, Vladimir Antonik and Elena Aksenyushkina
2017, 7(2): 191-199 doi: 10.3934/naco.2017013 +[Abstract](502) +[HTML](14) +[PDF](273.2KB)
Abstract:

Optimal control problem without phase and terminal constraints is considered. Conceptions of strongly extremal controls are introduced on the basis of nonstandard functional increment formulas. Such controls are optimal in linear and quadratic problems. In general case optimality property is guaranteed by concavity condition of the Pontryagin function with respect to phase variables.

Feedback necessary optimality conditions for a class of terminally constrained state-linear variational problems inspired by impulsive control
Stepan Sorokin and Maxim Staritsyn
2017, 7(2): 201-210 doi: 10.3934/naco.2017014 +[Abstract](621) +[HTML](19) +[PDF](331.5KB)
Abstract:

We consider a class of rightpoint-constrained state-linear (but non convex) optimal control problems, which takes its origin in the impulsive control framework. The main issue is a strengthening of the Pontryagin Maximum Principle for the addressed problem. Towards this goal, we adapt the approach, based on feedback control variations due to V.A. Dykhta [4,5,6,7]. Our necessary optimality condition, named the feedback maximum principle, is expressed completely in terms of the classical Maximum Principle, but is shown to discard non-optimal extrema. As a connected result, we derive a certain form of duality for the considered problem, and propose the dual version of the proved necessary optimality condition.

Global optimization reduction of generalized Malfatti's problem
Rentsen Enkhbat, Evgeniya A. Finkelstein, Anton S. Anikin and Alexandr Yu. Gornov
2017, 7(2): 211-221 doi: 10.3934/naco.2017015 +[Abstract](1049) +[HTML](27) +[PDF](839.1KB)
Abstract:

In this paper, we generalize Malfatti's problem as a continuation of works [6,7]. The problem has been formulated as a global optimization problem. To solve Malfatti's problem numerically, we propose the co-called ''Hill method'' which is based on a heuristic approach. Some computational results for two and three-dimensional test problems are provided.

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