Constants of Strong Unicity of Minimal Projections onto some Two-Dimensional Subspaces of l∞

In this paper the constants of strong unicity of minimal projections onto some two-dimensional subspaces in l_∞⁽⁴⁾ will be calculated.     

Actions that characterize I∞

It is well known that, if an identity operator on an n-dimensional Banach space V can be extended to any Banach space with the same norm, then V is isometric to l_∞⁽ⁿ⁾. We show that the identity is the only such operator.     

H∞ estimation for fuzzy membership function optimization

Given a fuzzy logic system, how can we determine the membership functions that will result in the best performance? If we constrain the membership functions to a specific shape (e.g., triangles or trapezoids) then each membership function can be parameterized by a few variables and the membership optimization problem can be reduced to a parameter optimization problem. The parameter optimization problem can then be formulated as a nonlinear filtering problem. In this paper we solve the nonlinear filtering problem using H_∞ state estimation theory. However, the membership functions that result from this approach are not (in general) sum normal. That is, the membership function values do not add up to one at each point in the domain. We therefore modify the H_∞ filter with the addition of state constraints so that the resulting membership functions are sum normal. Sum normality may be desirable not only for its intuitive appeal but also for computational reasons in the real time implementation of fuzzy logic systems. The methods proposed in this paper are illustrated on a fuzzy automotive cruise controller and compared to Kalman filtering based optimization.     

L∞-boundedness of L2-projections on splines for a geometric mesh

In this paper we complete the investigations started by [14.]. We study C. de Boor's conjecture of the L_∞-boundedness of the L₂-projection P on smooth splines in the special case of a geometric mesh x = {q^v}. A connection to the interpolation projection is established and the uniform boundedness of ¦|P¦|_π with respect to q is proved.     

L∞ convergence of interpolation and associated product integration for exponential weights

We investigate convergence in a weighted L_∞ norm of Hermite–Fejér, Hermite, and Grünwald interpolations at zeros of orthogonal polynomials with respect to exponential weights such as Freud, Erd s, and exponential weight on (−1,1). Convergence of product integration rules induced by the various approximation processes is deduced. We also give more precise weight conditions for convergence of interpolations with respect to above three types of weights, respectively.     

L∞-upper bound of L2-projections onto splines at a geometric mesh

For an integer k 1 and a geometric mesh (qⁱ)_−∞^∞ with q ε (0, ∞), let M_i,k(x): = k[q^{i + k}](· − x)₊^{k − 1}, N_i,k(x): = (q^{i + k} − qⁱM_i,k(x)/k, and let A_k(q) be the Gram matrix (∝M_i,kN_j,k)_i,jεz. It is known that A_k(q)⁻¹_∞ is bounded independently of q. In this paper it is shown that A_k(q)⁻¹_∞ is strictly decreasing for q in [1, ∞). In particular, the sharp upper bound and lower bound for A_k (q)⁻¹ are obtained: for all q ε (0, ∞).     

Robust stochastic stability and H∞ performance for a class of uncertain impulsive stochastic systems

In this paper, we discuss the problem of robust stochastic stability and H_∞ performance for a class of uncertain impulsive stochastic systems under sampled measurements. The parameter uncertainties are assumed to be time-varying and value-bounded. We give a sufficient condition in terms of certain linear matrix inequalities (LMIs) to guarantee the uncertain impulsive stochastic system to be robustly stochastically stable. Furthermore, we discuss a stochastically stable filter, using the locally sampled measurements, which ensures both the stochastic stability and a prescribed level of H_∞ performance for the filtering error system for all admissible uncertainties. We give a sufficient condition for the existence of such a filter and an explicit expression of a desired filter if relevant conditions are satisfied.     

On the optimal quadrature least L∞-norm of monosplines with free knots on the real axis

In this paper, the exact solutions of the L_∝ norm on some classes of monosplines with free knots on the real axis are obtained, and the optimal quadrature formula and optimal error on the convolution class taking a PF density as its kernal are given.     

Synchronization of FitzHugh–Nagumo systems in EES via H∞ variable universe adaptive fuzzy control

In this paper, H_∞ variable universe adaptive fuzzy control law is derived and applied to synchronize two FitzHugh—Nagumo neurons. Firstly, the different dynamical behavior of the nonlinear cable model based on the FitzHugh–Nagumo model responding to the various external electrical simulations (EES) is studied. Next, using the result of the analysis, a robust adaptive variable universe fuzzy control scheme is designed to synchronize two neurons. The computer simulations is provided for verify the efficiency of the designed synchronization scheme.     

Synchronization of Ghostburster neuron in external electrical stimulation via H∞ variable universe fuzzy adaptive control

In this paper, H_∞ variable universe fuzzy adaptive control law is derived and applied to synchronize two unidirectional coupled Ghostbursters neurons under external electrical stimulation. Firstly, the dynamical behavior of the nonlinear Ghostburster model responding to various external electrical stimulations is studied. Then, using the results of the analysis, a robust adaptive variable universe fuzzy control scheme is designed to synchronize two unidirectional coupled neurons. Numerical simulations demonstrate the validity and feasibility of the proposed method.     

Approximation algorithms for the bi-criteria weighted MAX-CUT problem

We consider a generalization of the classical MAX-CUT problem where two objective functions are simultaneously considered. We derive some theorems on the existence and the non-existence of feasible cuts that are at the same time near optimal for both criteria. Furthermore, two approximation algorithms with performance guarantee are presented. The first one is deterministic while the second one is randomized. A generalization of these results is given for the bi-criteria MAX-k-CUT problem.     

Recent exact algorithms for solving the vehicle routing problem under capacity and time window constraints

This paper provides a review of the recent developments that had a major impact on the current state-of-the-art exact algorithms for the vehicle routing problem (VRP). The paper reviews mathematical formulations, relaxations and recent exact methods for two of the most important variants of the VRP: the capacitated VRP (CVRP) and the VRP with time windows (VRPTW). The paper also reports a comparison of the computational performances of the different exact algorithms for the CVRP and VRPTW.     

Stress L∞-estimates and the uniqueness problem for the quasistatic equations to the model of Bodner-Partom

This paper proves the uniqueness result for global in time large solutions of quasistatic equations to an inelastic model of material behavior of metals, provided that an a priori L^∞-estimation for the Cauchy stress tensor holds. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Robust reliable L2 – L∞ control for continuous‐time systems with nonlinear actuator failures

This article examines the reliable L₂ – L_∞ control design problem for a class of continuous‐time linear systems subject to external disturbances and mixed actuator failures via input delay approach. Also, due to the occurrence of nonlinear circumstances in the control input, a more generalized and practical actuator fault model containing both linear and nonlinear terms is constructed to the addressed control system. Our attention is focused on the design of the robust state feedback reliable sampled‐data controller that guarantees the robust asymptotic stability of the resulting closed‐loop system with an L₂ – L_∞ prescribed performance level γ > 0, for all the possible actuator failure cases. For this purpose, by constructing an appropriate Lyapunov–Krasovskii functional (LKF) and utilizing few integral inequality techniques, some novel sufficient stabilization conditions in terms of linear matrix inequalities (LMIs) are established for the considered system. Moreover, the established stabilizability conditions pave the way for designing the robust reliable sampled‐data controller as the solution to a set of LMIs. Finally, as an example, a wheeled mobile robot trailer model is considered to illustrate the effectiveness of the proposed control design scheme. © 2016 Wiley Periodicals, Inc. Complexity 21: 309–319, 2016     

Finite‐time l2 − l∞ synchronization for discrete‐time nonlinear chaotic systems via information‐constrained delayed feedback

This article is concerned with the problem of finite‐time synchronization control for a class of discrete‐time nonlinear chaotic systems under unreliable communication links. Our aim is to design a delayed feedback controller such that the resulting synchronization error system is stochastically finite‐time bounded with a guaranteed performance level over a finite time interval. Some sufficient conditions for the solvability of the above problem are established. A delayed feedback control scheme involving constrained information about the past state is presented. Finally, the Fold chaotic system is used to demonstrate the effectiveness of our proposed approach. © 2014 Wiley Periodicals, Inc. Complexity 21: 138–146, 2015     

Convexity and all‐time C∞‐regularity of the interface in flame propagation

We consider the following one‐phase free boundary problem: Find (u, Ω) such that Ω = {u > 0} and with Q_T = ?ⁿ × (0, T). Under the condition that Ω_o is convex and log u_o is concave, we show that the convexity of Ω(t) and the concavity of log u(·, t) are preserved under the flow for 0 ≤ t ≤ T as long as ?Ω(t) and u on Ω(t) are smooth. As a consequence, we show the existence of a smooth‐up‐to‐the‐interface solution, on 0 < t < T_c, with T_c denoting the extinction time of Ω(t). We also provide a new proof of a short‐time existence with C^2,α initial data on the general domain. © 2002 John Wiley & Sons, Inc.     

Exponential H∞ synchronization for discrete‐time chaotic neural networks with time delays and stochastic perturbations via output feedback control

This paper investigates the problem of exponential H_∞ synchronization of discrete‐time chaotic neural networks with time delays and stochastic perturbations. First, by using the Lyapunov‐Krasovskii (Lyapunov) functional and output feedback controller, we establish the H_∞ performance of exponential synchronization in the mean square of master‐slave systems, which is analyzed using a matrix inequality approach. Second, the parameters of a desired output feedback controller can be achieved by solving a linear matrix inequality. Finally, 2 simulated examples are presented to show the effectiveness of the theoretical results.     

Improved dynamic programming algorithms for bandwidth minimization and the MinCut Linear Arrangement problem

The dynamic programming algorithm of [12.] for the bandwidth minimization problem is improved. It is shown that, for all k > 1, BANDWIDTH(k) can be solved in O(n^k) steps and simultaneous O(n^k) space, where n is the number of vertices in the graph, and that each such problem is in NSPACE(log n). The same improved dynamic programming algorithm approach works to show that the MINCUT LINEAR ARRANGEMENT problem restricted to the fixed value k, denoted by MINCUT(k), is solvable in O(n^k) steps and simultaneous O(n^k) space and is in the class NSPACE(log n).     

Branch and bound based solution algorithms for the budget constrained discrete time/cost trade-off problem

The time/cost trade-off models in project management aim to reduce the project completion time by putting extra resources on activity durations. The budget problem in discrete time/cost trade-off scheduling selects a time/cost mode for each activity so as to minimize the project completion time without exceeding the available budget. There may be alternative modes that solve the budget problem optimally and each solution may have a different total cost value. In this study we consider the budget problem and aim to find the minimum cost solution among the minimum project completion time solutions. We analyse the structure of the problem together with its linear programming relaxation and derive some mechanisms for reducing the problem size. We solve the reduced problem by branch and bound based optimization and heuristic algorithms. We find that our branch and bound algorithm finds optimal solutions for medium-sized problem instances in reasonable times and the heuristic algorithms produce high quality solutions very quickly.     

Explicit solution of the finite time L2-norm polynomial approximation problem

The aim of this paper is to present a new approach to the finite time L₂-norm polynomial approximation problem. A new formulation of this problem leads to an equivalent linear system whose solution can be investigated analytically. Such a solution is then specialized for a polynomial expressed in terms of Laguerre and Bernstein basis.