Applications of time scale symplectic systems without normality

Recently, the authors obtained new characterizations of the positivity and nonnegativity of a time scale quadratic functional F with separable endpoints related to a time scale symplectic system (S). In these results, the assumption of normality is absent. In this paper we present applications of such results. Namely, without assuming normality we derive Sturmian comparison theorems, results for general jointly varying endpoints, and characterizations of the positivity of F via the corresponding time scale Riccati equation, a certain perturbed quadratic functional, and a time scale Riccati inequality. These results generalize and unify many recent as well as classical ones.     

Equivalent conditions to the nonnegativity of a quadratic functional in discrete optimal control

In this paper we provide a characterization of the nonnegativity of a discrete quadratic functional ? with fixed right endpoint in the optimal control setting. This characterization is closely related to the kernel condition earlier introduced by M. Bohner as a part of a focal points definition for conjoined bases of the associated linear Hamiltonian difference system. When this kernel condition is satisfied only up to a certain critical index m, the traditional conditions, which are the focal points, conjugate intervals, implicit Riccati equation, and partial quadratic functionals, must be replaced by a new condition. This new condition is determined to be the nonnegativity of a block tridiagonal matrix, representing the remainder of ? after the index m, on a suitable subspace. Applications of our result include the discrete Jacobi condition, a unification of the nonnegativity and positivity of ? into one statement, and an improved result for the special case of the discrete calculus of variations. Even when both endpoints of ? are fixed, this paper provides a new result. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Coupled Intervals in the Discrete Optimal Control

In this paper, we investigate the nonnegativity and positivity of a quadratic functional ? with variable (i.e. separable and jointly varying) endpoints in the discrete optimal control setting. We introduce a coupled interval notion, which generalizes (i) the conjugate interval notion known for the fixed right endpoint case and (ii) the coupled interval notion known in the discrete calculus of variations. We prove necessary and sufficient conditions for the nonnegativity and positivity of ? in terms of the nonexistence of such coupled intervals. Furthermore, we characterize the nonnegativity of ? in terms of the (previously known notions of) conjugate intervals, a conjoined basis of the associated linear Hamiltonian system, and the solvability of an implicit Riccati equation. This completes the results for the nonnegativity that are parallel to the known ones on the positivity of ?. Finally, we define partial quadratic functionals associated with ? and a (strong) regularity of ?, which we relate to the positivity and nonnegativity of ?.     

Time scale symplectic systems without normality

We present a theory of the definiteness (nonnegativity and positivity) of a quadratic functional F over a bounded time scale. The results are given in terms of a time scale symplectic system (S), which is a time scale linear system that generalizes and unifies the linear Hamiltonian differential system and discrete symplectic system. The novelty of this paper resides in removing the assumption of normality in the characterization of the positivity of F, and in establishing equivalent conditions for the nonnegativity of F without any normality assumption. To reach this goal, a new notion of generalized focal points for conjoined bases (X,U) of (S) is introduced, results on the piecewise-constant kernel of X(t) are obtained, and various Picone-type identities are derived under the piecewise-constant kernel condition. The results of this paper generalize and unify recent ones in each of the discrete and continuous time setting, and constitute a keystone for further development in this theory.     

Riccati equations for abnormal time scale quadratic functionals

This paper focuses on developing new Riccati type conditions for an abnormal time scale symplectic system (S). These conditions provide characterizations of the nonnegativity (with and without a certain “image condition”) and positivity of the quadratic functionals associated with such a system. The novelty of these conditions rely on the natural conjoined basis (Xa,Ua) of (S) in which Xa(t) is not necessarily invertible, and thus the system (S) could be abnormal. These results are new even in the special case of continuous time, as are some of them in the discrete time setting.     

Discrete Optimal Control: Second Order Optimality Conditions

In this paper, we present a survey and refinement of our recent results in the discrete optimal control theory. For a general nonlinear discrete optimal control problem (P) , second order necessary and sufficient optimality conditions are derived via the nonnegativity ( I S 0) and positivity ( I >0) of the discrete quadratic functional I corresponding to its second variation. Thus, we fill the gap in the discrete-time theory by connecting the discrete control problems with the theory of conjugate intervals, Hamiltonian systems, and Riccati equations. Necessary conditions for I S 0 are formulated in terms of the positivity of certain partial discrete quadratic functionals, the nonexistence of conjugate intervals, the existence of conjoined bases of the associated linear Hamiltonian system, and the existence of solutions to Riccati matrix equations. Natural strengthening of each of these conditions yields a characterization of the positivity of I and hence, sufficiency criteria for the original problem (P) . Finally, open problems and perspectives are also discussed.     

Coupled intervals in the discrete calculus of variations: necessity and sufficiency

In this work we study nonnegativity and positivity of a discrete quadratic functional with separately varying endpoints. We introduce a notion of an interval coupled with 0, and hence, extend the notion of conjugate interval to 0 from the case of fixed to variable endpoint(s). We show that the nonnegativity of the discrete quadratic functional is equivalent to each of the following conditions: The nonexistence of intervals coupled with 0, the existence of a solution to Riccati matrix equation and its boundary conditions. Natural strengthening of each of these conditions yields a characterization of the positivity of the discrete quadratic functional. Since the quadratic functional under consideration could be a second variation of a discrete calculus of variations problem with varying endpoints, we apply our results to obtain necessary and sufficient optimality conditions for such problems. This paper generalizes our recent work in [R. Hilscher, V. Zeidan, Comput. Math. Appl., to appear], where the right endpoint is fixed.     

Nonnegativity and positivity of quadratic functionals in discrete calculus of variations: survey

In this paper we provide a survey of characterizations of the nonnegativity and positivity of discrete quadratic functionals which arise as the second variation for nonlinear discrete calculus of variations problems. These characterizations are in terms of (i) (strict) conjugate and (strict) coupled intervals, (ii) the conjoined bases of the associated Jacobi difference equation, and (iii) the solution of the corresponding Riccati difference equation. The results depend on the form of the boundary conditions of the quadratic functional and, basically, we distinguish three types: (a) separable endpoints with zero right endpoint (this of course includes the simplest case of both zero endpoints), (b) separable endpoints, and (c) jointly varying endpoints.     

Certificates for nonnegativity of polynomials with zeros on compact semialgebraic sets

We prove a criterion for an element of a commutative ring to be contained in an archimedean subsemiring. It can be used to investigate the question whether nonnegativity of a polynomial on a compact semialgebraic set can be certified in a certain way. In case of (strict) positivity instead of nonnegativity, our criterion simplifies to classical results of Stone, Kadison, Krivine, Handelman, Schmüdgen et al. As an application of our result, we give a new proof of the following result of Handelman: If an odd power of a real polynomial in several variables has only nonnegative coefficients, then so do all sufficiently high powers.Partially supported by the DFG project 214371 “Darstellung positiver Polynome”.     

Primitive stable representations of free Kleinian groups

In this paper, we give a complete criterion for a discrete faithful representation ρ: F _n →PSL(2, ?) to be primitive stable. This will answer Minsky’s conjectures about geometric conditions on ?³/ρ(F _n ) regarding the primitive stability of ρ.     

A kind of nonnegative matrices and its application on the stability of discrete dynamical systems

In this paper we introduce a new kind of nonnegative matrices which is called (sp) matrices. We show that the zero solutions of a class of linear discrete dynamical systems are asymptotically stable if and only if the coefficient matrices are (sp) matrices. To determine that a matrix is (sp) matrix or not is very simple, we need only to verify that some elements of the coefficient matrices are zero or not. According to the result above, we obtain the conditions for the stability of several classes of discrete dynamical systems.     

Decentralized stabilization of a class of interconnected systems

Abstract. This paper is concerned with the decentralized stabilization of continuous and discretelinear interconnected systems with the structural constraints about the interconnection matri-ces. For the continuous case,the main improvement in the paper as compared with the corre-sponding results in the literature is to extend the considered class of systems from S to S“ (bothwill be defined in the paper) without resulting in high decentralized gain and difficult numericalcomputation. The algorithm for obtaining decentralized state feedback control to stable theoverall system is presented. The discrete case and some very useful results are discussed aswell.     

Manifolds with pinched 2-positive curvature operator

In this paper, we show that any complete Riemannian manifold of dimension great than 2 must be compact if it has positive complex sectional curvature and ??-pinched 2-positive curvature operator, namely, the sum of the two smallest eigenvalues of curvature operator are bounded below by ??·scal > 0. If we relax the restriction of positivity of complex sectional curvature to nonnegativity, we can also show that the manifold is compact under the additional condition of positive asymptotic volume ratio.     

Sufficiency and duality in nondifferentiable multiobjective programming involving generalized type I functions

Recently, Hachimi and Aghezzaf defined generalized (F,α,ρ,d)-type I functions, a new class of functions that unifies several concepts of generalized type I functions. In this paper, the generalized (F,α,ρ,d)-type I functions are extended to nondifferentiable functions. By utilizing the new concepts, we obtain several sufficient optimality conditions and prove mixed type and Mond-Weir type duality results for the nondifferentiable multiobjective programming problem.     

New Constrained Optimization Reformulation of Complementarity Problems

We suggest a reformulation of the complementarity problem CP(F) as a minimization problem with nonnegativity constraints. This reformulation is based on a particular unconstrained minimization reformulation of CP(F) introduced by Geiger and Kanzow as well as Facchinei and Soares. This allows us to use nonnegativity constraints for all the variables or only a subset of the variables on which the function F depends. Appropriate regularity conditions ensure that a stationary point of the new reformulation is a solution of the complementarity problem. In particular, stationary points with negative components can be avoided in contrast to the reformulation as unconstrained minimization problem. This advantage will be demonstrated for a class of complementarity problems which arise when the Karush–Kuhn–Tucker conditions of a convex inequality constrained optimization problem are considered.     

One-variable and multi-mariable calculus on a non-Archimedean field extension of the real numbers

New elements of calculus on a complete real closed non-Archimedean field extension F of the real numbers ? will be presented. It is known that the total disconnectedness of F in the topology induced by the order makes the usual (topological) notions of continuity and differentiability too weak to extend real calculus results to F. In this paper, we introduce new stronger concepts of continuity and differentiability which we call derivate continuity and derivate differentiability [2, 12]; andwe show that derivate continuous and differentiable functions satisfy the usual addition, product and composition rules and that n-times derivate differentiable functions satisfy a Taylor formula with remainder similar to that of the real case. Then we generalize the definitions of derivate continuity and derivate differentiability to multivariable F-valued functions and we prove related results that are useful for doing analysis on F and F ⁿ in general.     

Comparison of two repairable systems

Any repairable system improves (deteriorates) with time if the interarrival times of failure tend to get larger (smaller) in some sense. In this paper we consider two such repairable systems, and their performance in terms of several partial orderings of their respective interarrival times of failure are compared. The comparison of two systems’ improvement/deterioration under minimal repair policy has been characterized in terms of s-FR orders and also in terms of their shifted and dispersive versions. These results generalize some of the existing results in the literature and also provide some new results in this direction.     

Efficient Recognition of Totally Nonnegative Matrix Cells

The space of m×p totally nonnegative real matrices has a stratification into totally nonnegative cells. The largest such cell is the space of totally positive matrices. There is a well-known criterion due to Gasca and Peña for testing a real matrix for total positivity. This criterion involves testing mp minors. In contrast, there is no known small set of minors for testing for total nonnegativity. In this paper, we show that for each of the totally nonnegative cells there is a test for membership which only involves mp minors, thus extending the Gasca and Peña result to all totally nonnegative cells.     

The Ordered Gradual Covering Location Problem on a Network

In this paper we develop a network location model that combines the characteristics of ordered median and gradual cover models resulting in the Ordered Gradual Covering Location Problem (OGCLP). The Gradual Cover Location Problem (GCLP) was specifically designed to extend the basic cover objective to capture sensitivity with respect to absolute travel distance. The Ordered Median Location problem is a generalization of most of the classical locations problems like p-median or p-center problems. The OGCLP model provides a unifying structure for the standard location models and allows us to develop objectives sensitive to both relative and absolute customer-to-facility distances. We derive Finite Dominating Sets (FDS) for the one facility case of the OGCLP. Moreover, we present efficient algorithms for determining the FDS and also discuss the conditional case where a certain number of facilities is already assumed to exist and one new facility is to be added. For the multi-facility case we are able to identify a finite set of potential facility locations a priori, which essentially converts the network location model into its discrete counterpart. For the multi-facility discrete OGCLP we discuss several Integer Programming formulations and give computational results.