Linear-quadratic optimal control for time-varying descriptor systems via space decompositions

This paper aims at solving the linear-quadratic optimal control problems(LQOCP)for time-varying descriptor systems in a real Hilbert space.By using the Moore-Penrose inverse theory and space decomposition technique,the descriptor system can be rewritten as a new differential-algebraic equation(DAE),and then some novel sufficient conditions for the solvability of LQOCP are obtained.Especially,the methods proposed in this work are simpler and easier to verify and compute,and can solve LQOCP without the range inclusion condition.In addition,some numerical examples are shown to verify the results obtained.

On the Convergence Rate of a Space Decomposition Method

Domain decomposition method and multigrid method can be unified in the framework of the space decomposition method. This paper has obtained a new result on the convergence rate of the space decomposition method, which can be applied to some nonuniformly elliptic problems.

Triple reverse order law for the Drazin inverse

In this paper,we investigate the reverse order law for Drazin inverse of three bound-ed linear operators under some commutation relations.Moreover,the Drazin invertibility of sum is also obtained for two bounded linear operators and its expression is presented.

Zoning Search With Adaptive Resource Allocating Method for Balanced and Imbalanced Multimodal Multi-Objective Optimization

Maintaining population diversity is an important task in the multimodal multi-objective optimization.Although the zoning search(ZS)can improve the diversity in the decision space,assigning the same computational costs to each search subspace may be wasteful when computational resources are limited,especially on imbalanced problems.To alleviate the above-mentioned issue,a zoning search with adaptive resource allocating(ZS-ARA)method is proposed in the current study.In the proposed ZS-ARA,the entire search space is divided into many subspaces to preserve the diversity in the decision space and to reduce the problem complexity.Moreover,the computational resources can be automatically allocated among all the subspaces.The ZS-ARA is compared with seven algorithms on two different types of multimodal multi-objective problems(MMOPs),namely,balanced and imbalanced MMOPs.The results indicate that,similarly to the ZS,the ZS-ARA achieves high performance with the balanced MMOPs.Also,it can greatly assist a“regular”algorithm in improving its performance on the imbalanced MMOPs,and is capable of allocating the limited computational resources dynamically.

On Hereditarily Indecomposable Banach Spaces

This paper shows that every non-separable hereditarily indecomposable Banach space admits an equivalent strictly convex norm, but its bi-dual can never have such a one; consequently, every non-separable hereditarily indecomposable Banach space has no equivalent locally uniformly convex norm.

Extracting outer function part from Hardy space function

Any analytic signal fa(e^(it)) can be written as a product of its minimum-phase signal part(the outer function part) and its all-phase signal part(the inner function part). Due to the importance of such decomposition, Kumarasan and Rao(1999), implementing the idea of the Szeg?o limit theorem(see below),proposed an algorithm to obtain approximations of the minimum-phase signal of a polynomial analytic signal fa(e^(it)) = e^(iN0t)M∑k=0a_k^(eikt),(0.1)where a_0≠ 0, a_M≠ 0. Their method involves minimizing the energy E(f_a, h_1, h_2,..., h_H) =1/(2π)∫_0^(2π)|1+H∑k=1h_k^(eikt)|~2|fa(e^(it))|~2dt(0.2) with the undetermined complex numbers hk's by the least mean square error method. In the limiting procedure H →∞, one obtains approximate solutions of the minimum-phase signal. What is achieved in the present paper is two-fold. On one hand, we rigorously prove that, if fa(e^(it)) is a polynomial analytic signal as given in(0.1),then for any integer H≥M, and with |fa(e^(it))|~2 in the integrand part of(0.2) being replaced with 1/|fa(e^(it))|~2,the exact solution of the minimum-phase signal of fa(e^(it)) can be extracted out. On the other hand, we show that the Fourier system e^(ikt) used in the above process may be replaced with the Takenaka-Malmquist(TM) system, r_k(e^(it)) :=((1-|α_k|~2e^(it))/(1-α_ke^(it))^(1/2)∏_(j=1)^(k-1)(e^(it)-α_j/(1-α_je^(it))^(1/2), k = 1, 2,..., r_0(e^(it)) = 1, i.e., the least mean square error method based on the TM system can also be used to extract out approximate solutions of minimum-phase signals for any functions f_a in the Hardy space. The advantage of the TM system method is that the parameters α_1,..., α_n,...determining the system can be adaptively selected in order to increase computational efficiency. In particular,adopting the n-best rational(Blaschke form) approximation selection for the n-tuple {α_1,..., α_n}, n≥N, where N is the degree of the given rational analytic signal, the minimum-phase part of a rational analytic signal can be accurately and efficiently extracted out.

ON SOLVABLE COMPLETE LIE SUPERALGEBRAS

The authors discuss the properties of solvable complete Lie superalgebra, proving that solvable Lie superalgebras of maximal rank are complete.

A New and Efficient Approach to Determining the Rate of the Second-Order Symmetric Tensor and Its Isotropic Function

When the rate of a symmetric second-order symmetric tensor is discussed,the spin of the principal axis is involved.This paper proposes a method to establish the basis-free expression of the spin in terms of tensor and its rate by making use of the tensor function representation theorem.The proposed method is simple and the expression of the spin established is compact.To obtain the rate of the isotropic function of a second-order symmetric tensor,the fourth-order tangent tensor needs to be derived,which is the derivative of the tensor function to the second-order tensor.By decomposing the second-order symmetric tensor space into two orthogonal subspaces,the closed-form fourth-order tangent tensor is decomposed into two parts,which are linear mappings in these two orthogonal subspaces,respectively.These two linear mappings are derived in an extremely simple way.Finally,the method proposed in this paper is applied to obtain the expression of the relationship between material logarithmic strain rate and deformation rate.The whole process is simple and avoids tedious operations.

An Algebraic Approach to the Schwarz Alternating Methods

In this paper, the choice of the optimal parameters for a relaxation additive Schwarz alternating method in two subregions case is obtained by an algebraic method, which shows that the arithmetic average is the best. A counterexample illustrates that the same result is not true for many subregions case. In the last, this technique is applied to demonstrate some well known results ,, simply and intuitively.

Homogeneous wavelets and framelets with the refinable structure

Homogeneous wavelets and framelets have been extensively investigated in the classical theory of wavelets and they are often constructed from refinable functions via the multiresolution analysis. On the other hand, nonhomogeneous wavelets and framelets enjoy many desirable theoretical properties and are often intrinsically linked to the refinable structure and multiresolution analysis. In this paper, we provide a comprehensive study on connecting homogeneous wavelets and framelets to nonhomogeneous ones with the refinable structure. This allows us to understand better the structure of homogeneous wavelets and framelets as well as their connections to the refinable structure and multiresolution analysis.