Boundary feedback stabilization of a Schr$\＂{o}$dinger equation interconnected with a heat equation

In this paper, we study stabilization for a Schoedinger equation, which is interconnected with a heat equation via boundary coupling. A direct boundary feedback control is adopted. By a detailed spectral analysis, it is found that there are two branches of eigenvalues： one is along the negative real axis, and the other is approaching to a vertical line, which is parallel to the imagine axis. Moreover, it is shown that there is a set of generalized eigenfunctions, which forms a Riesz basis for the energy state space. Finally, the spectrum-determined growth condition is held and the exponential stability of the system is then concluded.     

Local stabilization of the Burgers–Fisher equation via boundary control

Locally exponential stabilization for the Burgers–Fisher system is addressed by boundary control in this paper. For the nonlinear partial differential equation, a linear boundary feedback control law is applied to control the Burgers–Fisher system. Locally exponential stabilization of the closed loop system is established based on the relationship between operator theories and relations of different norms. Finally, the theory is validated through numerical simulations.     

Comments on ‘ On decentralized feedback stabilization of large-scale interconnected systems ’

A proposition presented by Mahalanabis and Singh, which asserts that, the stability of each aggregated subsystem implies the stability of a large scale system, is shown to be erroneous.     

Fisher’s decision tree

Univariate decision trees are classifiers currently used in many data mining applications. This classifier discovers partitions in the input space via hyperplanes that are orthogonal to the axes of attributes, producing a model that can be understood by human experts. One disadvantage of univariate decision trees is that they produce complex and inaccurate models when decision boundaries are not orthogonal to axes. In this paper we introduce the Fisher’s Tree, it is a classifier that takes advantage of dimensionality reduction of Fisher’s linear discriminant and uses the decomposition strategy of decision trees, to come up with an oblique decision tree. Our proposal generates an artificial attribute that is used to split the data in a recursive way.The Fisher’s decision tree induces oblique trees whose accuracy, size, number of leaves and training time are competitive with respect to other decision trees reported in the literature. We use more than ten public available data sets to demonstrate the effectiveness of our method.     

Constant feedback stabilization of discrete-time systems with random-coefficients†

A method is proposed to effectively stabilize discrete-time systems involving random coefficients. The stability results are both of sample solutions and second moments of system states. The method is handy in that any finite number of steps of a Riccati-like difference equation is sufficient to guarantee the construction of a stabilizing feedback gain. The results are given for cases both with and without the presence of an additive noise term. An easy-to-test stabilizability condition accompanies the method, enhancing its utility.     

Comments on ‘Algorithm for simultaneous stabilization of single-input systems via dynamic feedback’

Wu et al. (1990) presented a sufficient condition for the simultaneous stabilizability of single-input systems. Their condition is shown in this note by a counterexample to be incorrect. The flaw leading to this error is pointed out and the remedy for their error is provided.     

Transformation of feature space based on Fisher’s linear discriminant

Linear transformation of data in multidimensional feature space based on Fisher’s criterion is considered. The case of two classes with arbitrary distributions is studied. We derived expressions for recurrent calculation of weight vectors which form new features. Example offered shows that the newly found features which represent the data more accurately make it possible to achieve linear separability of classes which remains impossible using the technique of principal components and the classic Fisher’s linear discriminant.     

Wirtinger’s inequality and Lyapunov-based sampled-data stabilization

Discontinuous Lyapunov functionals appeared to be very efficient for sampled-data systems (Fridman, 2010, Naghshtabrizi et al., 2008). In the present paper, new discontinuous Lyapunov functionals are introduced for sampled-data control in the presence of a constant input delay. The construction of these functionals is based on the vector extension of Wirtinger’s inequality. These functionals lead to simplified and efficient stability conditions in terms of Linear Matrix Inequalities (LMIs). The new stability analysis is applied to sampled-data state-feedback stabilization and to a novel sampled-data static output-feedback problem, where the delayed measurements are used for stabilization.     

Non-Reflecting Boundary Conditions for Maxwell’s Equations

A new discrete non-reflecting boundary condition for the time-dependent Maxwell equations describing the propagation of an electromagnetic wave in an infinite homogenous lossless rectangular waveguide with perfectly conducting walls is presented. It is derived from a virtual spatial finite difference discretization of the problem on the unbounded domain. Fourier transforms are used to decouple transversal modes. A judicious combination of edge based nodal values permits us to recover a simple structure in the Laplace domain. Using this, it is possible to approximate the convolution in time by a similar fast convolution algorithm as for the standard wave equation.     

Stabilisation of Schrödinger equation in dynamic boundary feedback with a memory-typed heat equation

In this work, we study the dynamic behaviour for a heat equation with exponential polynomial kernel memory to be a controller for a Schrödinger system. By introducing some new variables, the time-variant system is transformed into a time-invariant one. Remarkably, the resolvent of the closed-loop system operator is not compact anymore. The residual spectrum is shown to be empty and the continuous spectrum consisting of finite isolated points are obtained. It is shown that the sequence of generalised eigenfunctions forms a Riesz basis for the Hilbert state space. This deduces the spectrum-determined growth condition for the C ₀-semigroup, and the exponential stability is then established.     

On Sontag’s formula for the input-to-state practical stabilization of retarded control-affine systems

In this paper input-to-state practically stabilizing control laws for retarded, control-affine, nonlinear systems with actuator disturbance are investigated. The developed methodology is based on Artstein’s theory of control Lyapunov functions and related Sontag’s formula, extended to retarded systems. If the actuator disturbance is bounded, then the controller yields the solution of the closed-loop system to achieve an arbitrarily fixed neighborhood of the origin, by increasing a control tuning parameter. The considered systems can present an arbitrary number of discrete as well as distributed time-delays, of any size, as long as they are constant and, in general, known.     

Improved LMI conditions for H ∞ output feedback stabilization of linear discrete-time systems

The problem of H _∞ output feedback control for discrete-time systems is investigated in this paper. The main contribution is to provide a uniform framework for generating sufficient linear matrix inequality conditions. Those conditions are classified into two parallel parts based on the way of slack variable selection. Moreover, it is shown that the existing result is a special case of the new conditions by taking few of the additional free matrix parameters to be zero. This directly leads to more flexibility and less conservativeness in the H _∞ output feedback control design. Numerical examples are carried out for illustration.     

On boundary feedback stabilization of Timoshenko beam with rotor inertia at the tip

The feedback stabilization problem of a nonuniform Timoshenko beam system with rotor inertia at the tip of the beam is studied. First, as a special kind of linear boundary force feedback and moment control is applied to the beam‘ s tip, the strict mathematical treatment, a suitable state Hilbert space is chosen, and the well-poseness of the corresponding closed loop system is proved by using the semigroup theory of bounded linear operators. Then the energy corresponding to the closed loop system is shown to be exponentially stable. Finally, in the special case of umform beam, some sufficient and necessary conditions for the corresponding closed loop system to be asymptotically stable are derived.     

Boundary Mittag–Leffler stabilization of a class of time fractional-order nonlinear reaction–diffusion systems

Boundary control design of a class of time fractional-order nonlinear reaction–diffusion systems (FNRDSs) is considered in three cases: domain-averaged measurement, collocated boundary point measurement, and anti-collocated boundary point measurement. For domain-averaged measurement and collocated boundary point measurement, boundary controllers are designed directly based on the measurements, respectively. For the anti-collocated boundary point measurement, to overcome the difficulty that the measurement cannot be used for the controller design directly, an observer is constructed firstly, and then, an observer-based boundary controller is designed. Sufficient conditions for Mittag–Leffler (M-L) stability of the closed-loop system are all provided in terms of linear matrix inequalities (LMIs). Numerical simulation results are provided to illustrate the feasibility and effectiveness of the proposed methods.     

Stability analysis of method of fundamental solutions for mixed boundary value problems of Laplace’s equation

Since the stability of the method of fundamental solutions (MFS) is a severe issue, the estimation on the bounds of condition number Cond is important to real application. In this paper, we propose the new approaches for deriving the asymptotes of Cond, and apply them for the Dirichlet problem of Laplace’s equation, to provide the sharp bound of Cond for disk domains. Then the new bound of Cond is derived for bounded simply connected domains with mixed types of boundary conditions. Numerical results are reported for Motz’s problem by adding singular functions. The values of Cond grow exponentially with respect to the number of fundamental solutions used. Note that there seems to exist no stability analysis for the MFS on non-disk (or non-elliptic) domains. Moreover, the expansion coefficients obtained by the MFS are oscillatingly large, to cause the other kind of instability: subtraction cancelation errors in the final harmonic solutions.     

Comments on ‘Delay-independent stabilization of linear systems’

Amemiya (1983) has shown that in some cases a system is delay-independent stabilizable, but the decaying-rate is not freely assignable. In the present correspondence it is shown that in the latter case, it is always possible to assign a decaying-rate.     

Dynamic output feedback stabilization of deterministic finite automata via the semi‑tensor product of matrices approach

This paper deals with the dynamic output feedback stabilization problem of deterministic finite automata (DFA). The static form of this problem is defined and solved in previous studies via a set of equivalent conditions. In this paper, the dynamic output feedback (DOF) stabilization of DFAs is defined in which the controller is supposed to be another DFA. The DFA controller will be designed to stabilize the equilibrium point of the main DFA through a set of proposed equivalent conditions. It has been proven that the design problem of DOF stabilization is more feasible than the static output feedback (SOF) stabilization. Three simulation examples are provided to illustrate the results of this paper in more details. The first example considers an instance DFA and develops SOF and DOF controllers for it. The example explains the concepts of the DOF controller and how it will be implemented in the closed-loop DFA. In the second example, a special DFA is provided in which the DOF stabilization is feasible, whereas the SOF stabilization is not. The final example compares the feasibility performance of the SOF and DOF stabilizations through applying them to one hundred random-generated DFAs. The results reveal the superiority of the DOF stabilization.     

Robust H ∞ static output feedback stabilization of T-S fuzzy systems subject to actuator saturation

This paper proposes a method for designing robust H_?? static output feedback stabilization of Takagi-Sugeno (T-S) fuzzy systems under actuator saturation. In this paper, the input saturation is represented by a polytopic model and the modeling error is assumed a norm-bounded uncertainty. A set invariance condition for robust H_?? static output feedback system under actuator saturation is first established. Then, the estimation of the largest domain of attraction for the system is formulated and solved as a Linear Matrix Inequality (LMI) optimization problem. Two examples are used to demonstrate the effectiveness of the proposed design method.     

Boundary Effect in the Error Estimate of the Finite-Difference Scheme for Two-Dimensional Poisson’s Equation

We obtain the a priori error estimate of the rate of convergence of mesh solution to the generalized solution of two-dimensional Poisson equation in case of mixed boundary condition (boundary conditions of first and third kinds). We prove that the accuracy order is higher near the sides of the domain where the Dirichlet boundary condition is specified.