State constraints in the linear regulator problem: Case study

In this paper, we consider the problem of minimum-norm control of the double integrator with bilateral inequality constraints for the output. We approximate the constraints by piecewise linear functions and prove that the Langrange multipliers associated with the state constraints of the approximating problem are discrete measures, concentrated in at most two points in every interval of discretization. This allows us to reduce the problem to a convex finite-dimensional optimization problem. An algorithm based on this reduction is proposed and its convergence is examined. Numerical examples illustrate our approach. We also discuss regularity properties of the optimal control for a higher-dimensional state-constrained linear regulator problem.The first author was supported by the National Science Foundation, Grant No. DMS-9404431. The second author was supported by a François-Xavier Bagnoud Doctoral Fellowship and by NSF Grants DMS-9404431 and MSS-9114630.     

DSPI strain measurement on an externally reinforced bending beam: A comparison of step-by-step addition and pixel shift correlation

A small-scale concrete beam reinforced with an adhesively bonded carbon fiber reinforced polymer (CFRP) plate was subjected to four-point bending. Finite element analyses (FEA) of the bending deformations were carried out to predict strain gradients near the end of the CFRP plate. In order to measure these strains, phase-stepping 3D-digital speckle pattern interferometry was employed. To avoid speckle decorrelation due to the inevitable rigid body motion of the specimen, the load was increased in small increments. Two evaluation schemes for the electronic speckle pattern interferometry phase maps are compared: summing up the measured displacement components load step-by-load step versus regain of the correlation by shifting the final image by an integer number of pixels. Measured strain values are evaluated using a polynomial fit to the measured in-plane displacements and are compared to the FE predicitions. It can be concluded that pixel shift correlation is preferable to summing up load steps for cases of large rigid body motion.     

The lifespan for general quasilinear wave equation with localized dissipation in exterior domains

In this article, we develop the result of J. Metcalfe and M. Nakamura (General quasi-linear wave equations with localized dissipation in extension domains, J. Differ. Eqns. 233 (2007), pp. 313–344.) into a more general form. By means of the method of commuting vector fields, we get the lifespan for the quasi-linear wave equation with localized dissipations and the non-linear term depends on the unknown function itself in exterior domains.     

Cea's error estimate for strongly monotone variational inequalities

Céa's approximation lemma is extended to variational inequalities which are defined by strongly monotone operators in closed convex subsets of linear normed spaces. This abstract error estimate is applied to the finite element discretization of a nonlinear elliptic two-sided obstacle problem providing an asymptotic error estimate for a smooth enough solution.     

ON LIE ALGEBRAS WHOSE NILRADICAL IS (n — p)-FILIFORM

We prove first that every (n — p)-filiform Lie algebra, p ≤ 3, is the nilradical of a solvable, nonnilpotent rigid Lie algebra. We also analize how this result extends to (n — 4)-filiform Lie algebras. For this purpose, we give a classificaction of these algebras and then determine which of the obtained classes appear as the nilradical of a rigid algebra.     

Rigid Rings and Makar-Limanov Techniques

A ring is rigid if it admits no nonzero locally nilpotent derivation. Although a “generic” ring should be rigid, it is not trivial to show that a ring is rigid. We provide several examples of rigid rings and we outline two general strategies to help determine if a ring is rigid, which we call “parametrization techniques.” and “filtration techniques.” We provide many tools and lemmas which may be useful in other situations. Also, we point out some pitfalls to beware when using these techniques. Finally, we give some reasonably simple rings for which the question of rigidity remains unsettled.     

Generic global rigidity of body–bar frameworks

A basic geometric question is to determine when a given framework G(p)$G(\mathbf{p})$ is globally rigid in Euclidean space R^d${\mathbb{R}}^d$, where G is a finite graph and p is a configuration of points corresponding to the vertices of G  . G(p)$G(\mathbf{p})$ is globally rigid in  R^d${\mathbb{R}}^d$ if for any other configuration q for G   such that the edge lengths of G(q)$G(\mathbf{q})$ are the same as the corresponding edge lengths of G(p)$G(\mathbf{p})$, then p is congruent to q. A framework G(p)$G(\mathbf{p})$ is redundantly rigid, if it is rigid and it remains rigid after the removal of any edge of G.     

A new platform for NCN dimethylamino pincer complexes: Synthesis and structural studies

The reaction of 2,6-(2-{Me₂NCH₂}C₆H₄)₂C₆H₃I (2) with Pd₂(dba)₃ produced the NCN diamine pincer complex [2,6-(2-Me₂{NCH₂}C₆H₄)₂C₆H₃PdI] (3) by an oxidative addition route. The structural analysis of ligand precursor 2 revealed a syn-conformation in the solid state. Single crystal X-ray analysis of complex 3 revealed a conventional square planar geometry about the palladium center and a global C₂ symmetric structure. Variable temperature and concentration NMR spectroscopic studies of complex 3 suggest an equilibrium between 3 and the dinuclear species [{2,6-(2-{Me₂NCH₂}C₆H₄)₂C₆H₃Pd}₂μ²-I]I in CDCl₃ solution. An unusual carbonate complex [{2,6-(2-{Me₂NCH₂}C₆H₄)₂C₆H₃Pd}₃μ³-CO₃]I₃ (4) was also structurally characterized as a minor product during synthesis of 3.     

The number of extremal components of a rigid measure

The Littlewood-Richardson rule can be expressed in terms of measures, and the fact that the Littlewood-Richardson coefficient is one amounts to a rigidity property of some measure. We show that the number of extremal components of such a rigid measure can be related to easily calculated geometric data. We recover, in particular, a characterization of those extremal measures whose (appropriately defined) duals are extremal as well. This result is instrumental in writing explicit solutions of Schubert intersection problems in the rigid case.     

三次多项式样条解一类四阶边值问题

应用三次多项式样条函数解一组四阶单侧、障碍、接触边值问题,取半结点为网格点,并增加了边界方程,应用方法解文献中的数值例子,说明了方法的高效性,数值结果也显示了方法的优越性.