ISSN:

2155-3289

eISSN:

2155-3297

## Numerical Algebra, Control & Optimization

2013 , Volume 3 , Issue 3

Special Issue dedicated to Professor George Leitmann on the
occasion of his 88th birthday (Part II)

Special Issue Papers: 389-489; Regular Papers: 491-599

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2013, 3(3): 389-406
doi: 10.3934/naco.2013.3.389

*+*[Abstract](477)*+*[PDF](767.0KB)**Abstract:**

The research is focused on an algorithm for constructing solutions of optimal control problems with infinite time horizon arising, for example, in economic growth models.

There are several significant difficulties which complicate solution of the problem, such as: (1) stiffness of Hamiltonian systems generated by the Pontryagin maximum principle; (2) non-stability of equilibrium points; (3) lack of initial conditions for adjoint variables.

The analysis of the Hamiltonian system implemented in this paper for optimal control problems with infinite horizon provides results effective for construction of optimal solutions, namely, (1) if a steady state exists and satisfies regularity conditions then there exists a nonlinear stabilizer for the Hamiltonian system; (2) a nonlinear stabilizer generates the system with excluding adjoint variables whose trajectories (according to qualitative analysis of the corresponding differential equations) approximate solutions of the original Hamiltonian system in a neighborhood of the steady state;(3) trajectories of the stabilized system serve as first approximations and localize search of optimal trajectories.

The results of numerical experiments are presented by modeling of an economic growth system with investment in capital and enhancement of the labor efficiency.

2013, 3(3): 407-424
doi: 10.3934/naco.2013.3.407

*+*[Abstract](507)*+*[PDF](463.0KB)**Abstract:**

A generalization of the cubic algorithm is presented for global optimization of nonconvex nonsmooth multiobjective optimization programs $\min f_{s}(x),\ s=1,\dots,k,$ with box constraints $x\in X=[a_{1},b_{1}]\times \dots\times\lbrack a_{n},b_{n}]$.

This monotonic set contraction algorithm converges onto the entire exact Pareto set, if nonempty, and yields its approximation with given precision in a finite number of iterations. Simultaneously, approximations for the ideal point and for the function values over Pareto set are obtained. The method is implemented by Maple code, and this code does not create ill-conditioned situations.

Results of numerical experiments are presented, with graphs, to illustrate the use of the code, and the solution set can be visualized in projections on coordinate planes. The code is ready for engineering and economic applications.

2013, 3(3): 425-443
doi: 10.3934/naco.2013.3.425

*+*[Abstract](503)*+*[PDF](443.3KB)**Abstract:**

This paper develops a new, simple, general, and explicit form of the equations of motion for general constrained mechanical systems that can have holonomic and/or nonholonomic constraints that may or may not be ideal, and that may contain either positive semi-definite or positive definite mass matrices. This is done through the replacement of the actual unconstrained mechanical system, which may have a positive semi-definite mass matrix, with an unconstrained auxiliary system whose mass matrix is positive definite and which is subjected to the same holonomic and/or nonholonomic constraints as those applied to the actual unconstrained mechanical system. A simple, unified fundamental equation that gives in closed-form both the acceleration of the constrained mechanical system and the constraint force is obtained. The results herein provide deeper insights into the behavior of constrained motion and open up new approaches to modeling complex, constrained mechanical systems, such as those encountered in multi-body dynamics.

2013, 3(3): 445-462
doi: 10.3934/naco.2013.3.445

*+*[Abstract](430)*+*[PDF](447.0KB)**Abstract:**

In this paper, we develop a Direct Model Reference Adaptive Tracking Controller for linear systems with unknown time varying input delays. This controller can also reject bounded disturbances of known waveform but unknown amplitude, e.g. steps or sinusoids. In this paper a robustness result is developed for DMRAC of linear systems with unknown small constant or time varying input delays using the concept of un-delayed ideal trajectories. We will show that the adaptively controlled system is globally stable, but the adaptive tracking error is no longer guaranteed to approach the origin. However, exponential convergence to a neighborhood can be achieved as a result of the control design. A simple example will be provided to illustrate this adaptive control method.

2013, 3(3): 463-469
doi: 10.3934/naco.2013.3.463

*+*[Abstract](391)*+*[PDF](244.1KB)**Abstract:**

We define and analyse partial Newton iterations for the solutions of a system of algebraic equations. Firstly we focus on a linear system of equations which does not require a line search. To apply a partial Newton method to a system of nonlinear equations we need a line search to ensure that the linearized equations are valid approximations of the nonlinear equations. We also focus on the use of one or two components of the displacement vector to generate a convergent sequence. This approach is inspired by the Simplex Algorithm in Linear Programming. As expected the partial Newton iterations are found not to have the fast convergence properties of the full Newton method. But the proposed partial Newton iteration makes it significantly simpler and faster to compute in each iteration for a system of equations with many variables. This is because it uses only one or two variables instead of all the search variables in each iteration.

2013, 3(3): 471-489
doi: 10.3934/naco.2013.3.471

*+*[Abstract](401)*+*[PDF](1444.3KB)**Abstract:**

On June 18, 2008 at the Plenary Meeting of the International Conference ``Differential Equations and Topology" dedicated to the 100-th anniversary of L.S. Pontryagin, the report [15] was submitted by Isaev V.K. and Leitmann G. This report in a summary form included a section dedicated to the research of scientists of TsAGI in the field of automation of full life-cycle (i.e. engineering-design-manufacturing, or CAE/CAD/CAM, or CALS-technologies) of wind tunnel models [21]. Within this framework, methods of geometric modeling [1,11] were intensively developed, new classes of optimal splines have been built, including the Pontryagin splines and the Chebyshev splines [12-13,19,37]. This paper reviews some results on the Chebyshev splines. We also give brief remarks about the new applications of Chebyshev splines (outside the usual scope of CALS-technologies in design and manufacturing), namely to the actual problem of air traffic management (ATM) within the Free Flight concept.

2013, 3(3): 491-518
doi: 10.3934/naco.2013.3.491

*+*[Abstract](455)*+*[PDF](421.3KB)**Abstract:**

A deflation technique is presented for an irreducible singular $M$-matrix Algebraic Riccati Equation (MARE). The technique improves the rate of convergence of a doubling algorithm, especially for an MARE in the critical case for which without deflation the doubling algorithm converges linearly and with deflation it converges quadratically. The deflation also improves the conditioning of the MARE in the critical case and thus enables its minimal nonnegative solution to be computed more accurately.

2013, 3(3): 519-548
doi: 10.3934/naco.2013.3.519

*+*[Abstract](645)*+*[PDF](5659.2KB)**Abstract:**

We investigate and analyze a computational method for the approximation of reachable sets for nonlinear dynamic systems. The method uses grids to cover the region of interest and the distance function to the reachable set evaluated at grid points. A convergence analysis is provided and shows the convergence of three different types of discrete set approximations to the reachable set. The distance functions can be computed numerically by suitable optimal control problems in combination with direct discretization techniques which allows adaptive calculations of reachable sets. Several numerical examples with nonconvex reachable sets are presented.

2013, 3(3): 549-555
doi: 10.3934/naco.2013.3.549

*+*[Abstract](463)*+*[PDF](280.4KB)**Abstract:**

The recent paper [H. Ding, Y. Zhang, S. Wang, X. Yang, A note on some quadrature based three-step iterative methods for non-linear equations,

*A*ppl. Math. Comput.

**215**(1): (2009) 53--57] shows that the Algorithm 2.2 and Algorithm 2.3 in the article [N.A. Mir, T. Zaman, Some quadrature based three-step iterative methods for non-linear equations,

*A*ppl. Math. Comput.

**193**(2): (2007) 366--373] have twelfth-order and ninth order convergence respectively, not seventh-order as claimed in the later article. In this work; we propose two simple modifications, without increasing the computational cost or functional evaluations, of the Algorithms 2.3 and 2.3. The first modification improves the convergence order of the Algorithm 2.3 from ninth-order to tenth order. In the second modification, we remove evaluation of the second derivative from the Algorithm 2.2 while preserving its twelfth-order convergent nature.

2013, 3(3): 557-565
doi: 10.3934/naco.2013.3.557

*+*[Abstract](440)*+*[PDF](330.7KB)**Abstract:**

In this paper, we introduce a countably infinite iterative scheme and consider a sufficient and necessary condition for the existence of common fixed points of a countably infinite family of asymptotically quasi-$f_i$-expansive mappings in convex metric spaces.

2013, 3(3): 567-581
doi: 10.3934/naco.2013.3.567

*+*[Abstract](506)*+*[PDF](385.8KB)**Abstract:**

The notions of $\alpha$-well-posedness and generalized $\alpha$-well-posedness for a system of constrained variational inequalities involving set-valued mappings (for short, (SCVI)) are introduced in Hilbert spaces. Existence theorems of solutions for (SCVI) are established by using penalty techniques. Metric characterizations of $\alpha$-well-posedness and generalized $\alpha$-well-posedness, in terms of the approximate solutions sets, are presented. Finally, the equivalences between (generalized) $\alpha$-well-posedness for (SCVI) and existence and uniqueness of its solutions are also derived under quite mild assumptions.

2013, 3(3): 583-599
doi: 10.3934/naco.2013.3.583

*+*[Abstract](717)*+*[PDF](442.5KB)**Abstract:**

Let $n$ be a positive integer and $m$ be a positive even integer. Let ${\mathcal A}$ be an $m^{th}$ order $n$-dimensional real weakly symmetric tensor and ${\mathcal B}$ be a real weakly symmetric positive definite tensor of the same size. $\lambda \in \mathbb{R}$ is called a ${\mathcal B}_r$-eigenvalue of ${\mathcal A}$ if ${\mathcal A} x^{m-1} = \lambda {\mathcal B} x^{m-1}$ for some $x \in \mathbb{R}^n \backslash \{0\}$. In this paper, we introduce two unconstrained optimization problems and obtain some variational characterizations for the minimum and maximum ${\mathcal B}_r$--eigenvalues of ${\mathcal A}$. Our results extend Auchmuty's unconstrained variational principles for eigenvalues of real symmetric matrices. This unconstrained optimization approach can be used to find a Z-, H-, or D-eigenvalue of an even order weakly symmetric tensor. We provide some numerical results to illustrate the effectiveness of this approach for finding a Z-eigenvalue and for determining the positive semidefiniteness of an even order symmetric tensor.

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