<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> Control and Exponential Stability of Nonlinear Nonautonomous Systems with Time-Varying Delay

This paper addresses the design of H _∞ state feedback controllers for a class of nonlinear time-varying delay systems. The interesting features here are that the system in consideration is nonautonomous with fast-varying delays, the delay is also involved in the observation output, and the controllers to be designed satisfy some exponential stability constraints on the closed-loop poles. By using the proposed Lyapunov functional approach, neither a controllability assumption nor a bound restriction on nonlinear perturbations is required to obtain new sufficient conditions for the H _∞ control. The conditions are derived in terms of a solution to the standard Riccati differential equations, which allows for simultaneous computation of the two bounds that characterize the stability rate of the solution. This work was supported by the National Foundation for Science and Technology Development, Vietnam and the Center of Excellence for Autonnomous Systems funded by the Australian Research Council, Australia.     

Parameter-dependent <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> Control for Time-varying Delay Polytopic Systems

This paper addresses the robust stabilization and H _∞ control problem for a class of linear polytopic systems with continuously distributed delays. The control objective is to design a robust H _∞ controller that satisfies some exponential stability constraints on the closed-loop poles. Using improved parameter-dependent Lyapunov Krasovskii functionals, new delay-dependent conditions for the robust H _∞ control are established in terms of linear matrix inequalities.     

Strict L<Subscript>∞</Subscript> Isotonic Regression

Given a function f and weights w on the vertices of a directed acyclic graph G, an isotonic regression of (f,w) is an order-preserving real-valued function that minimizes the weighted distance to f among all order-preserving functions. When the distance is given via the supremum norm there may be many isotonic regressions. One of special interest is the strict isotonic regression, which is the limit of p-norm isotonic regression as p approaches infinity. Algorithms for determining it are given. We also examine previous isotonic regression algorithms in terms of their behavior as mappings from weighted functions over G to isotonic functions over G, showing that the fastest algorithms are not monotonic mappings. In contrast, the strict isotonic regression is monotonic.     

Delay-Dependent Robust Stabilization and <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> Control for Nonlinear Stochastic Systems with Markovian Jump Parameters and Interval Time-Varying Delays

This paper considers the problems of delay-dependent robust stabilization and H _∞ control for nonlinear stochastic systems with Markovian jump parameters and interval time-varying delays. Based on the Lyapunov method and introducing some appropriate free-weighting matrices, sufficient conditions for the solvability of above problems have been investigated in terms of linear matrix inequalities (LMIs). Furthermore, the desired state feedback controller has also been designed by solving these LMIs. Finally, a numerical example is provided to demonstrate the potential of the proposed techniques.     

Homology and cohomology of <Emphasis Type="Italic">E</Emphasis><Subscript>∞</Subscript> ring spectra

Every homology or cohomology theory on a category of E ring spectra is Topological André–Quillen homology or cohomology with appropriate coefficients. Analogous results hold more generally for categories of algebras over operads.Mathematics Subject Classification (2000): 55P43, 55P48, 55U35The second author was supported in part by NSF grant DMS-0203980.     

Distance-Regular Graph with <Emphasis Type="Italic">c</Emphasis><Subscript>2</Subscript> > 1 and <Emphasis Type="Italic">a</Emphasis><Subscript>1</Subscript> = 0 < <Emphasis Type="Italic">a</Emphasis><Subscript>2</Subscript>

Let Γ be a distance-regular graph of diameter d ≥ 3 with c ₂ > 1. Let m be an integer with 1 ≤ m ≤ d − 1. We consider the following conditions:

Restrictions of Modular Representations of Special Linear Groups to A<Subscript>1</Subscript>×A<Subscript>1</Subscript> Subgroups

We study the restrictions of infinitesimally irreducible modular representations of algebraic groups of type A_n to subsystem subgroups of type A₁ × A₁. Under some restrictions on the highest weight of a representation we find their composition factors.     

<Emphasis Type="Italic">L</Emphasis><Subscript>∞</Subscript>-Locality of Three-Dimensional Peano Curves

A theory and corresponding algorithms are developed for fast and exact calculation of the L_∞-locality (i.e., the greatest cube-to-linear ratio in the maximum metric) for polyfractal three-dimensional Peano curves.     

Moduli spaces ℳ<Subscript>2,1</Subscript> and ℳ<Subscript>3,1</Subscript>

The cell structure of the spaces ℳ_2,1 and ℳ_3,1 is considered. These are the spaces of complex curves of genus 2 and 3 with one marked point. For the space ℳ_2,1, nine cells of the highest dimension 8 are described and their adjacency is studied. For the space ℳ_3,1, a list of all 1726 cells of the highest dimension 14 (with orientation) is obtained. The list of adjacent couples of cells is also obtained. These lists can be found on the web.     

BMO Solvability and the <Emphasis Type=Italic>A</Emphasis><Subscript>∞</Subscript> Condition for Elliptic Operators

We establish a connection between the absolute continuity of elliptic measure associated with a second order divergence form operator with bounded measurable coefficients with the solvability of an end-point BMO Dirichlet problem. We show that these two notions are equivalent. As a consequence we obtain an end-point perturbation result, i.e., the solvability of the BMO Dirichlet problem implies L ^p solvability for all p>p ₀.     

Interrelations between mixed moduli of smoothness in metrics of <Emphasis Type="Italic">L</Emphasis><Subscript>p</Subscript> and <Emphasis Type="Italic">L</Emphasis><Subscript>∞</Subscript>

Interrelations between mixed fractional moduli of smoothness considered in the metrics of L _p and L _∞ are studied on the paper.     

Motivic Milnor fibre of cyclic <Emphasis Type="Italic">L</Emphasis><Subscript>∞</Subscript>-algebras

We define the motivic Milnor fiber of cyclic L _∞-algebras of dimension three using the method of Denef and Loeser of motivic integration. It is proved by Nicaise and Sebag that the topological Euler characteristic of the motivic Milnor fiber is equal to the Euler characteristic of the étale cohomology of the analytic Milnor fiber. We prove that the value of Behrend function on the germ moduli space determined by a cyclic L _∞-algebra L is equal to the Euler characteristic of the analytic Milnor fiber. Thus we prove that the Behrend function depends only on the formal neighborhood of the moduli space.     

Interrelations between full moduli of smoothness in the metrics of <Emphasis Type="Italic">L</Emphasis> <Subscript>1</Subscript> and <Emphasis Type="Italic">L</Emphasis> <Subscript>∞</Subscript>

Interrelation between full moduli of smoothness of natural orders in the metrics L₁ and L_∞ is considered in the paper.     

Path <Emphasis Type="Italic">A</Emphasis><Subscript>∞</Subscript> algebras of positively graded quivers

Let A be a path A_∞-algebra over a positively graded quiver Q: We prove that the derived category of A is triangulated equivalent to the derived category of kQ; which is viewed as a DG algebra with trivial differential. The main technique used in the proof is Koszul duality for DG algebras.     

Some properties of the <Emphasis Type="Italic">ɛ</Emphasis><Subscript>∞</Subscript>-product of quotient bornological spaces

Some properties of the ? _∞-product defined in [4] are obtained by a study of a kind of isomorphism between the computation of this ? _∞-product and the ordinary ?-product of L. Schwartz [9]. The paper contains several corollaries.     

Combined ℓ<Subscript>2</Subscript> data and gradient fitting in conjunction with ℓ<Subscript>1</Subscript> regularization

We are interested in minimizing functionals with ℓ₂ data and gradient fitting term and ℓ₁ regularization term with higher order derivatives in a discrete setting. We examine the structure of the solution in 1D by reformulating the original problem into a contact problem which can be solved by dual optimization techniques. The solution turns out to be a ’smooth’ discrete polynomial spline whose knots coincide with the contact points while its counterpart in the contact problem is a discrete version of a spline with higher defect and contact points as knots. In 2D we modify Chambolle’s algorithm to solve the minimization problem with the ℓ₁ norm of interacting second order partial derivatives as regularization term. We show that the algorithm can be implemented efficiently by applying the fast cosine transform. We demonstrate by numerical denoising examples that the ℓ₂ gradient fitting term can be used to avoid both edge blurring and staircasing effects.      

Robust generalized <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript>-suboptimal control

We consider an optimal perturbation damping problem taking into account not only an external perturbation with bounded L ₂-norm and an initial perturbation caused by unknown initial conditions in the system but also unknown bounded parametric perturbations. We synthesize a robust generalized H _∞-suboptimal control minimizing the upper bound, expressed in terms of solutions of linear matrix inequalities, for the perturbation damping level under uncertainty in the closed system.     

Projecting <Emphasis Type="Italic">l</Emphasis><Subscript>∞</Subscript> onto Classical Spaces

We describe an explicit construction of a linear projection of a symmetric conical section of the n-dimensional cube onto a (1+ε)-isomorphic version of the Euclidean ball of proportional dimension, or more generally onto a (1+ε)-isomorphic image of an l _p ^m -ball. Allowing non-linear projections (of logarithmic polynomial nonlinearity) we may even project the full n-dimensional cube onto the same images. This is done by gluing together explicit projections onto two-dimensional spaces, interpreting and modifying a paper of Ben-Tal and Nemirowski on polynomial reductions of conic quadratic programming problems to linear programming problems in terms of Banach spaces.      

Multiplicative <Emphasis Type="Italic">A</Emphasis><Subscript>∞</Subscript>-structure in terms of spectral sequences of fibrations

In the present paper, the technique of spectral sequences with A _∞ -structures in their terms is developed for differential algebras with filtrations. Applications of this technique to the multiplicative spectral sequences of fibrations are given. We show that if the base of fibration is connected and simply connected, then the structure graded A _∞ -algebra in the second term of the spectral sequence of a fibration is the tensor product of the cohomology A _∞ -algebra of the base and the cohomology A _∞ -algebra of the fibre of this fibration.     

Homotopy properties of ∞-simplicial coalgebras and homotopy unital supplemented <Emphasis Type="Italic">A</Emphasis> <Subscript>∞</Subscript>-algebras

The homotopy theory of ∞-simplicial coalgebras is developed; in terms of this theory, an additional structure on the tensor bigraded coalgebra of a graded module is described such that endowing the coalgebra with this structure is equivalent to endowing the given graded module with the structure of a homotopy unital A_∞-algebra.