Near-Optimal Controls of Discrete-Time Dynamic Systems Driven by Singularly-Perturbed Markov Chains

This work is devoted to near-optimal controls of large-scale discrete-time nonlinear dynamic systems driven by Markov chains; the underlying problem is to minimize an expected cost function. Our main goal is to reduce the complexity of the underlying systems. To achieve this goal, discrete-time control models under singularly-perturbed Markov chains are introduced. Using a relaxed control representation, our effort is devoted to finding near-optimal controls. Lumping the states in each irreducible class into a single state gives rise to a limit system. Applying near-optimal controls of the limit system to the original system, near-optimal controls of the original system are derived.     

线性互补问题的并行多分裂松弛迭代算法

运用矩阵多重分裂理论,同时考虑并行计算与松弛迭代法,得到一类求解线性互补问题的高效数值算法．当问题的系数矩阵为对角元为正的H-矩阵或对称半正定矩阵时,证明了算法的全局收敛性；该算法与已有算法相比,具有计算量小、计算速度快等特点,因而特别适于求解大规模问题．数值试验的结果说明了算法的有效性．     

An explicit algorithm for monotone variational inequalities

We introduce a fully explicit method for solving monotone variational inequalities in Hilbert spaces, where orthogonal projections onto the feasible set are replaced by projections onto suitable hyperplanes. We prove weak convergence of the whole generated sequence to a solution of the problem, under only the assumptions of continuity and monotonicity of the operator and existence of solutions.     

Existence of optimal controls for a class of systems governed by differential inclusions on a Banach space

Using Cesari's approach, we prove the existence of optimal controls for a class of systems governed by differential inclusions on a Banach space having the Radon-Nikodym property. Theorem 3.1 gives the existence result for optimal relaxed controls under fairly general assumptions on the system and the admissible controls. This result depends on a fundamental result (Theorem 2.1) that proves the existence of mild solutions of differential inclusions on a Banach space, which has also independent interest. Further, the preparatory results, such as Lemma 3.1 and Lemma 3.2, are also useful in the study of time-optimal and terminal control problems.For illustration of the results, we present two examples, one on distributed controls for a class of systems governed by nonlinear parabolic equations and the other on boundary controls with discontinuous boundary operator.This work is supported in part by the National Science and Engineering Council of Canada under Grant No. 7109.     

Substituting CF2 for O4′ in Components of Nucleic Acids: Towards Systems with Reduced Propensity to Form Abasic Lesions

Intrinsic structural features and energetics of nucleotides containing variously fluorinated sugars as potential building blocks of DNA duplexes and quadruplexes are explored systematically using the modern methods of density functional theory (DFT) and quantum chemical topology (QCT). Our results suggest that fluorination at the 2′‐β or 2′‐α,β positions somewhat stabilizes in vacuo the AI relative to the BI conformations. In contrast, substitution of the CF₂ group for the O4′ atom (O4′‐CF₂ modification) leads to a preference of the BI relative to AI DNA‐like conformers. All the studied modifications result in a noticeable increase in the stability of the glycosidic bond [estimated by the relaxed force constants (RFC) approach], with particularly encouraging results for the O4′‐CF₂ derivative. Consequently, the O4′‐CF₂ modified systems are suggested and explored as promising scaffolds for the development of duplex and quadruplex structures with reduced propensity to form abasic lesions and to undergo DNA damage.     

The Interstitial Carbon of the Nitrogenase FeMo Cofactor is Far Better Stabilized than Previously Assumed

The first quantum-mechanical calculations of all relevant potential constants in both the iron-molybdenum cofactor and the iron-vanadium cofactor of nitrogenase suggest that the carbide is bound to the center of the enzyme much more strongly than hitherto assumed. Previous studies seemed to indicate a dummy function of the interstitial carbon, with a weak force constant (ca. 0.32 N cm⁻¹). Our new investigations confirm a different picture: the central carbon atom binds the iron-sulfur cluster through six covalent C−Fe bonds. With a potential constant of more than 1.3 N cm⁻¹, the interstitial carbon also appears to be dynamically persistent. According to our investigations, the values for the elasticity within the iron-sulfur cluster have to be corrected too. These new details on the mechano-chemical properties of the FeMo cofactor will be important for elucidating the catalytic cycle of nitrogen fixation. By implementing our new algorithm in the freely available COMPLIANCE program, the dependence on the coordinates during the calculation of Hesse matrices is eliminated completely.     

带1-范数约束的分裂可行问题的投影算法

本文主要研究带1-范数约束的分裂可行问题的求解算法.用一种交替投影算法,求得了问题的解,提出松弛交替投影算法,改进了直接往闭凸集上投影这一不足,并证明了该算法的收敛性.     

多集分裂等式问题的逐次松弛投影算法

多集分裂等式问题是分裂可行性问题的拓展问题,在图像重建、语言处理、地震探测等实际问题中具有广泛的应用。为了解决这个问题,提出了逐次松弛投影算法,设计了变化的步长,使其充分利用当前迭代点的信息且不需要算子范数的计算,证明了算法的弱收敛性。数值算例验证了算法在迭代次数与运行时间等方面的优越性。     

Bounded state problem for hereditary control problems

In this paper, we characterize optimal pairs for a hereditary control problem where the state is constrained. We use relaxed controls and the technique of penalization.This research was supported by NSF Grant No. HRD-91-54077.     

Numerical solution of the Boltzmann equation by time relaxed Monte Carlo (TRMC) methods

A new family of Monte Carlo schemes has been recently introduced for the numerical solution of the Boltzmann equation of rarefied gas dynamics (SIAM J. Sci. Comput. 2001; 23 :1253–1273). After a splitting of the equation the time discretization of the collision step is obtained from the Wild sum expansion of the solution by replacing high‐order terms in the expansion with the equilibrium Maxwellian distribution. The corresponding time relaxed Monte Carlo (TRMC) schemes allow the use of time steps larger than those required by direct simulation Monte Carlo (DSMC) and guarantee consistency in the fluid‐limit with the compressible Euler equations. Conservation of mass, momentum, and energy are also preserved by the schemes. Applications to a two‐dimensional gas dynamic flow around an obstacle are presented which show the improvement in terms of computational efficiency of TRMC schemes over standard DSMC for regimes close to the fluid‐limit. Copyright © 2005 John Wiley & Sons, Ltd.