Sufficient conditions are given for planar cooperative maps to have the qualitative global dynamics determined solely on local stability information obtained from fixed and minimal period-two points. The results are given for a class of strongly cooperative planar maps of classC1on an order interval. The maps are assumed to have a finite number of strongly ordered fixed points, and also the strongly ordered minimal period-two points. Some applications are included.

It is shown that locally asymptotically stable equilibria of planar cooperative or competitive maps have basin of attraction \begin{document}$ \mathcal{B} $\end{document} with relatively simple geometry: the boundary of each component of \begin{document}$ \mathcal{B} $\end{document} consists of the union of two unordered curves, and the components of \begin{document}$ \mathcal{B} $\end{document} are not comparable as sets. The boundary curves are Lipschitz if the map is of class \begin{document}$ C^1 $\end{document}. Further, if a periodic point is in \begin{document}$ \partial \mathcal{B} $\end{document}, then \begin{document}$ \partial\mathcal{B} $\end{document} is tangential to the line through the point with direction given by the eigenvector associated with the smaller characteristic value of the map at the point. Examples are given.

In this paper we present results on the existence of invariant curves for planar maps that are monotone with respect to either the south-east or north-east ordering. Some of these curves are the stable or unstable manifolds of hyperbolic fixed points (saddle points) or non-hyperbolic fixed points, and are also the boundary of basins of attraction of such points.

We present a global attractivity result for maps generated by systems of autonomous difference equations. It is assumed that the map of the system leaves invariant a box, is monotone in a coordinate-wise sense (but not necessarily monotone with respect to a standard cone), and satisfies certain algebraic condition. It is shown that there exists a unique equilibrium, and that it is a global attractor. As an application, it is shown that a discretized version of the Lotka-Volterra system of differential equations of order k has a global attractor in the positive orthant for certain range of parameters.

We investigate the global attractivity of the equilibrium of second-order difference equation[graphics]where the parameters p , q , q < p are nonnegative for all n . We prove that the unique equilibrium of this equation is global attractor which gives the affirmative answer to a conjecture of Kulenovic and Ladas.The method of proof is innovative, and it has the potential to be used in the proof of global attractivity of equilibria of many similar equations.

We investigate the unbounded solutions of the second order difference equation where all parameters and C and initial conditions are nonnegative and such that for all n. We give a characterization of unbounded solutions for this equation showing that whenever an unbounded solution exists the subsequence of even indexed (resp. odd) terms tends to and the subsequence of odd indexed (resp. even) terms tends to a nonnegative number. We also show that two sets in the plane of initial conditions corresponding to the two cases are separated by the global stable manifold of the unique positive equilibrium. Our result answers two open problems posed by Kulenović and Ladas (2001, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Boca Raton/London: Chapman and Hall/CRC).

We consider the problem of finding an analytic spectral factor g of a given rank k with k<n, positive semidefinite n×n matrix valued function f on the unit circle of the complex plane. The values of g are k×n matrices. We present an operator equation that must be satisfied by a function g to be a spectral factor. The condition is also sufficient. We also prove that Newton's method can be applied successfully to calculate spectral factors g in the Wiener algebra, thus the method works for all rational and many instances of non-rational data function f. A numerical example is given

Inequalities involving matrix polynomials and associated optimization problems have become very important in engineering. Commonplace in design problems are performance functions Gamma(X,Y) which are convex in X and convex in Y but which are not jointly convex, and the problem is to minimize the highest eigenvalue of Gamma. In a previous paper (CDC97) we derived first order tests for coordinate optimization (the most common approach to these problems) and to first order optimality tests for a true optimum. This article treats second order optimality conditions for optimization of matrix functions. Second order tests are important because optimization of matrix valued Gamma based on linearization or coordinate descent will often produce critical points which are not local solutions to the problem. Sufficient conditions, especially if they include second order information, become very valuable in these cases, as they can be used to tell which among the critical points correspond to true local optimal points, and to provide good update directions.Also we introduce and characterize a strong notion of matrix convexity which appears suited to many well behaved engineering problems.

The fundamental H∞ problem of control is that of finding the stable frequency response function that best fits worst case frequency domain specifications. This is a non-smooth optimization problem that underlies the frequency domain formulation of the H∞ problem of control; it is the main optimization problem in qualitative feedback theory for example. It is shown in this article how the fundamental H∞ optimization problem of control can be naturally treated with modern primal–dual interior point (PDIP) methods. The theory introduced here generalizes and unifies approaches to solving large classes of optimization problems involving matrix-valued functions, a subclass of which are commonly treated with linear matrix inequalities techniques. Also, in this article new optimality conditions for H∞ optimization problems over matrix-valued functions are proved, and numerical experience on natural (PDIP) algorithms for these problems is reported. In experiments we find the algorithms exhibit (local) quadratic convergence rate in many instances. Finally, H∞ optimization problems with an uncertainty parameter are considered. It is shown how to apply the theory developed here to obtain optimality conditions and derive algorithms. Numerical tests on simple examples are reported. © 1998 John Wiley & Sons, Ltd.

The frequency domain design of robust feedback control systems involves the fitting of a designable complex function of frequency to specification/uncertainty derived constraints. At present there are two basically different approaches to these problems: the several gain vs. frequency approaches and the two gain-phase vs. frequency approaches. Unfortunately, the commonalities, advantages and disadvantages of these approaches are not widely appreciated. In this paper the authors develop a common framework for examining these alternatives and use this framework to reveal some of their similarities, strengths and weaknesses.

We consider optimization of the largest eigenvalue of a smooth selfadjoint matrix valued function Gamma(X, Y) of two vector or matrix variables X and Y. We shall assume that. Gamma is concave or convex in Y and separately in X, but possibly has bad joint behavior. A typical problem one faces in control design are matrix versions of minimizing in Y and maximizing in X. Also minimizing in X and Y is an important problem. When joint behavior in X and Y is bad existing commercial software must be applied to each coordinate separately, and so can be used only to give a coordinate optimization algorithm. We give strong evidence in this article that, on "well behaved Gamma" coordinate optimization always gives a local optimum for the min(Y) max(X) problem and that it almost never gives a local solution to the min(Y) min(X) problem.

An approach to H-infinity optimization using modern primal-dual and semidefinite programming techniques was introduced by the authors in their CDC95 paper.In this paper the authors report results of numerical experiments with some of the algorithms introduced in their CDC95 paper. These experiments indicate that in many cases it is possible to obtain second order convergence rate to local solutions. Also, experiments suggest that convergence rate is related to properties of optimal dual functions.On a more theoretical note, one consequence of the general theory, which we present here, is optimality conditions for mu-synthesis and D-K iteration.

This article shows how the fundamental HOO optimiza­ tion problem of control can be naturally treated with modern primal-dual interior point (PDIP) methods. The fundamental H= problem of control is that of find­ ing the stable frequency response function which best fits worst case frequency domain specifications. This is a non smooth optimization problem which underlies the frequency domain formulation of the HOO problem of control; it is the main optimization problem in QFT for example. Also, in this article we present new optimality conditions for matrix valued HOO problems, and com­ pare natural (PDIP) algorithms for these problems, as well as fit them into the context of classical lIDO theory. The method which we expect will be very effective on Hoo optimization problems might be thought of as a hy­ brid between primal-dual methods and those for tradi­ tional optimization of smooth objectives functions. For example, a simple Hoo optimization problem is, given g, find

The main result presented here gives the first order (necessary) conditions for solutions to ℋ∞ optimization problems where there is an uncertainty parameter. These conditions can be stated as a set of equations which when solved produce excellent candidates for solutions, thus our result should lead to concrete algorithms in the spirit of the highly successful ones found by the authors for the certain plant case. How one crafts such algorithms for this uncertain case is indicated, but not analysed or tested

The fundamental H∞ optimization problem of control has become fairly well understood at both the qualitative and computational level. In many mature optimization theories one has a characterization of an optimizer which is a set of equations the optimizer must satisfy. Solution of these equations (say by Newton's method) then gives the optimizer. The trick is getting optimality equations which are nondegenerate with respect to Newton's method. This was done for pure H∞ optimization in previous papers by the authors, and in this article they extend the theory to include time domain constraints. Also an algorithm is described, a numerical example is presented, and a method for solving control problems is outlined

The fundamental H/sup /spl infin// optimization problem of control has become fairly well understood at both the qualitative and computational level. In many mature optimization theories one has a characterization of an optimizer which is a set of equations the optimizer must satisfy. Solution of these equations (say by Newton's method) then gives the optimizer. The trick is getting optimality equations which are nondegenerate with respect to Newton's method. This was done for pure H/sup /spl infin// optimization in previous papers by the authors, and in this article they extend the theory to include time domain constraints. Also an algorithm is described, a numerical example is presented, and a method for solving control problems is outlined.<>

We present algorithms for solving general sup-norm minimization problems over spaces of analytic functions, such as those arising inH∞ control. We also give an analysis and some theory of these algorithms. Part of this is specific to analytic optimization, while part holds for general sup-norm optimization. In particular, we are proposing a type of Newton-type algorithm which actually uses very high-order terms. The novel feature is that higher-order terms can be chosen in many ways while still maintaining a second-order convergence rate. Then, a clever choice of higher-order terms greatly reduces computation time. Conceivably this technique can be modified to accelerate Newton algorithms in some other circumstances. Estimates of order of convergence as well as results of numerical tests are also presented.