On unimodular graphs

We study graphs whose adjacency matrices have determinant equal to 1 or −1, and characterize certain subclasses of these graphs. Graphs whose adjacency matrices are totally unimodular are also characterized. For bipartite graphs having a unique perfect matching, we provide a formula for the inverse of the corresponding adjacency matrix, and address the problem of when that inverse is diagonally similar to a nonnegative matrix. Special attention is paid to the case that such a graph is unicyclic.     

Positive real control for 2-D discrete delayed systems via output feedback controllers

This paper considers the problem of positive real control for two-dimensional (2-D) discrete delayed systems in the Fornasini–Marchesini second local state-space model. Attention is focused on the design of dynamic output feedback controllers, which guarantee that the closed-loop system is asymptotically stable and the closed-loop transfer function is extended strictly positive real. We first present a sufficient condition for extended strictly positive realness of 2-D discrete delayed systems. Based on this, a sufficient condition for the solvability of the positive real control problem is obtained in terms of a linear matrix inequality (LMI). When the LMI is feasible, an explicit parametrization of a desired output feedback controller is presented. Finally, we provide a numerical example to demonstrate the application of the proposed method.     

Digitization of nonautonomous control systems

Methods of the theory of nonautonomous differential equations are used to study the extent to which the properties of local null controllability and local feedback stabilizability are preserved when a control system with time-varying coefficients is digitized, e.g., approximated by piecewise autonomous systems on small time subintervals.     

Digitization of nonautonomous control systems

Methods of the theory of nonautonomous differential equations are used to study the extent to which the properties of local null controllability and local feedback stabilizability are preserved when a control system with time-varying coefficients is digitized, e.g., approximated by piecewise autonomous systems on small time subintervals.     

On entropy-preserving stochastic averages

When an n×n doubly stochastic matrix A acts on Rⁿ on the left as a linear transformation and P is an n-long probability vector, we refer to the new probability vector AP as the stochastic average of the pair (A,P). Let Γ_n denote the set of pairs (A,P) whose stochastic average preserves the entropy of P:H(AP)=H(P). Γ_n is a subset of B_n×Σ_n where B_n is the Birkhoff polytope and Σ_n is the probability simplex.We characterize Γ_n and determine its geometry, topology,and combinatorial structure. For example, we find that (A,P)∈Γ_n if and only if A^tAP=P. We show that for any n, Γ_n is a connected set, and is in fact piecewise-linearly contractible in B_n×Σ_n. We write Γ_n as a finite union of product subspaces, in two distinct ways. We derive the geometry of the fibers (A,·) and (·,P) of Γ_n. Γ₃ is worked out in detail. Our analysis exploits the convexity of xlogx and the structure of an efficiently computable bipartite graph that we associate to each ds-matrix. This graph also lets us represent such a matrix in a permutation-equivalent, block diagonal form where each block is doubly stochastic and fully indecomposable.     

Qualitative aspects for the cubic nonlinear Schrödinger equations with localized damping: Exponential and polynomial stabilization

Results of exponential/polynomial decay rates of the energy in L²-level, related to the cubic nonlinear Schrödinger equation with localized damping posed on the whole real line, will be established in this paper. We use Kato's theory and a priori estimates to obtain the result of global well-posedness and we determine exponential/polynomial stabilization combining the ideas of unique continuation due to Zhang, semigroup property and Komornik's approach.     

Optimal control of the primitive equations of the ocean with state constraints

We investigate in this article Pontryagin’s maximum principle for a class of control problems associated with the primitive equations (PEs) of the ocean. These optimal problems involve a state constraint similar to that considered in Wang (2002) [7] for the three-dimensional Navier-Stokes (NS) equations. The main difference between this work and Wang (2002) [7] is that the nonlinearity in the PEs is stronger than in the three-dimensional NS systems.     

Effective graph resistance

This paper studies an interesting graph measure that we call the effective graph resistance. The notion of effective graph resistance is derived from the field of electric circuit analysis where it is defined as the accumulated effective resistance between all pairs of vertices. The objective of the paper is twofold. First, we survey known formulae of the effective graph resistance and derive other representations as well. The derivation of new expressions is based on the analysis of the associated random walk on the graph and applies tools from Markov chain theory. This approach results in a new method to approximate the effective graph resistance. A second objective of this paper concerns the optimisation of the effective graph resistance for graphs with given number of vertices and diameter, and for optimal edge addition. A set of analytical results is described, as well as results obtained by exhaustive search. One of the foremost applications of the effective graph resistance we have in mind, is the analysis of robustness-related problems. However, with our discussion of this informative graph measure we hope to open up a wealth of possibilities of applying the effective graph resistance to all kinds of networks problems.     

Growing perfect cubes

An (n,a,b)-perfect double cube is a b×b×b sized n-ary periodic array containing all possible a×a×a sized n-ary array exactly once as subarray. A growing cube is an array whose c_j×c_j×c_j sized prefix is an (n_j,a,c_j)-perfect double cube for , where and n₁<n₂<?. We construct the smallest possible perfect double cube (a 256×256×256 sized 8-ary array) and growing cubes for any a.     

Error bounds in the gap metric for dissipative balanced approximations

We derive an error bound in the gap metric for positive real balanced truncation and positive real singular perturbation approximation. We prove these results by working in the context of dissipative driving-variable systems, as in behavioral and state/signal systems theory. In such a framework no prior distinction is made between inputs and outputs. Dissipativity preserving balanced truncation of dissipative driving-variable systems is addressed and a gap metric error bound is obtained. Bounded real and positive real input–state–output systems are manifestations of a dissipative driving-variable system through particular decompositions of the signal space. Under such decompositions the existing bounded real and positive real balanced truncation schemes can be seen as special cases of dissipative balanced truncation and the new positive real error bounds follow.     

Multidimensional constant linear systems

A continuous resp. discrete r-dimensional (r1) system is the solution space of a system of linear partial differential resp. difference equations with constant coefficients for a vector of functions or distributions in r variables resp. of r-fold indexed sequences. Although such linear systems, both multidimensional and multivariable, have been used and studied in analysis and algebra for a long time, for instance by Ehrenpreis et al. thirty years ago, these systems have only recently been recognized as objects of special significance for system theory and for technical applications. Their introduction in this context in the discrete one-dimensional (r=1) case is due to J. C. Willems. The main duality theorem of this paper establishes a categorical duality between these multidimensional systems and finitely generated modules over the polynomial algebra in r indeterminates by making use of deep results in the areas of partial differential equations, several complex variables and algebra. This duality theorem makes many notions and theorems from algebra available for system theoretic considerations. This strategy is pursued here in several directions and is similar to the use of polynomial algebra in the standard one-dimensional theory, but mathematically more difficult. The following subjects are treated: input-output structures of systems and their transfer matrix, signal flow spaces and graphs of systems and block diagrams, transfer equivalence and (minimal) realizations, controllability and observability, rank singularities and their connection with the integral respresentation theorem, invertible systems, the constructive solution of the Cauchy problem and convolutional transfer operators for discrete systems. Several constructions on the basis of the Gröbner basis algorithms are executed. The connections with other approaches to multidimensional systems are established as far as possible (to the author).Partially supported by US Air Force Grant AFOSR-87-0249 and by Office of Naval Research Grant N 00014-86-K-0538 through the Center for Mathematical System Theory, University of Florida, Gainesville, Florida, U.S.A.     

Commute times for a directed graph using an asymmetric Laplacian

The expected commute times for a strongly connected directed graph are related to an asymmetric Laplacian matrix as a direct extension to similar well known formulas for undirected graphs. We show the close relationships between the asymmetric Laplacian and the so-called Fundamental matrix. We give bounds for the commute times in terms of the stationary probabilities for a random walk over the graph together with the asymmetric Laplacian and show how this can be approximated by a symmetrized Laplacian derived from a related weighted undirected graph.     

Unit-distance graphs,graphs on the integer lattice and a Ramsey type result

Summary Let (R ², 1) denote the graph withR ² as the vertex set and two vertices adjacent if and only if their Euclidean distance is 1. The problem of determining the chromatic number(R ², 1) is still open; however,(R ², 1) is known to be between 4 and 7. By a theorem of de Bruijn and Erdös, it is enough to consider only finite subgraphs of (R ², 1). By a recent theorem of Chilakamarri, it is enough to consider certain graphs on the integer lattice. More precisely, forr > 0, let (Z ²,r, ) denote a graph with vertex setZ ² and two vertices adjacent if and only if their Euclidean distance is in the closed interval [r – ,r + ]. A simple graph is faithfully -recurring inZ ² if there exists a real numberd > 0 such that, for arbitrarily larger, G is isomorphic to a subgraph of (Z ²,r, ) in which every pair of vertices are at least distancedr apart. Chilakamarri has shown that, ifG is a finite simple graph, thenG is isomorphic to a subgraph of (R ², 1) if and only ifG is faithfully -recurring inZ ². In this paper we prove that(Z ²,r, ) 5 for integersr 1. We also prove a Ramsey type result which states that for any integerr > 1, and any coloring ofZ ² either there exists a monochromatic pair of vertices with their distance in the closed interval [r – ,r + ] or there exists a set of three vertices closest to each other with three distinct colors.     

Controllability of a quantum particle in a moving potential well

We consider a nonrelativistic charged particle in a 1D moving potential well. This quantum system is subject to a control, which is the acceleration of the well. It is represented by a wave function solution of a Schrödinger equation, the position of the well together with its velocity. We prove the following controllability result for this bilinear control system: given ψ₀ close enough to an eigenstate and ψ_f close enough to another eigenstate, the wave function can be moved exactly from ψ₀ to ψ_f in finite time. Moreover, we can control the position and the velocity of the well. Our proof uses moment theory, a Nash-Moser implicit function theorem, the return method and expansion to the second order.     

Local exact controllability to the trajectories of the 1-D Kuramoto-Sivashinsky equation

We are concerned with the boundary controllability to the trajectories of the Kuramoto-Sivashinsky equation. By using a Carleman estimate, we obtain the null controllability of the linearized equation around a given solution. From a local inversion theorem we get the local controllability to the trajectories of the nonlinear system.     

The time-varying discrete four block Nehari problem: A generalized Popov-Yakubovich type approach

The paper provides necessary and sufficient solvability conditions for the time-variant discrete four block Nehari problem in terms of the existence of the stabilizing solutions to two coupled Riccati equations. A parametrization of the class of all solutions is also given. The results are easily obtained from a signature condition — a generalized Popov Yakubovich type argument-imposed on an appropiate rational node. The present development may be seen as an alternative of the theory developed by Gohberg, Kaashoek and Woerdeman [15].     

Redheffer's theorem revisited

Based on the Ben Artzi-Gohberg result concerning the equivalence between the invertibility of theL ²-operator and the exponential dichotomic evolution defined byA(t), the time-varying counterpart of the Redheffer theorem is considered under relaxed conditions.     

Weak maximum principle and accessory problem for control problems on time scales

In this paper we derive the first and second variations for a nonlinear time scale optimal control problem with control and state-endpoints equality constraints. Using the first variation, a first order necessary condition for weak local optimality is obtained under the form of a weak maximum principle generalizing the Dubois–Reymond Lemma to the optimal control setting and time scales. A second order necessary condition in terms of the accessory problem is derived by using the nonnegativity of the second variation at all admissible directions. The control problem is studied under a controllability assumption, and with or without the shift in the state variable. These two forms of the problem are shown to be equivalent.     

The copositive completion problem: Unspecified diagonal entries

In [L. Hogben, C.R. Johnson, R. Reams, The copositive matrix completion problem, Linear Algebra Appl. 408 (2005) 207-211] it was shown that any partial (strictly) copositive matrix all of whose diagonal entries are specified can be completed to a (strictly) copositive matrix. In this note we show that every partial strictly copositive matrix (possibly with unspecified diagonal entries) can be completed to a strictly copositive matrix, but there is an example of a partial copositive matrix with an unspecified diagonal entry that cannot be completed to a copositive matrix.