Sufficient conditions for viability under imperfect measurement

Viability theory gives a necessary and sufficient condition for the existence of a (set-valued) state feedback control such that all trajectories of the closed-loop system starting from the graph of a given tube in the state space remain in the tube. Here we investigate the same problem in the case where only incomplete and inexact measurement of the state is available. In the time-invariant case, we give a sufficient condition for the existence of anoutput feedback regulation map. The condition is shown to be equivalent to Haddad's viability condition if the measurement is perfect.     

The viability property of controlled jump diffusion processes

In this paper, we first give a comparison theorem of viscosity solution to some nonlinear second order integrodifferential equation. And then using the comparison theorem, we obtain a necessary and sufficient condition for the viability property of some controlled jump diffusion processes which can keep the solution within a constraint K.     

A viability theorem for set-valued states in a Hilbert space

Applications in robust control problems and shape evolution motivate the mathematical interest in control problems whose states are compact (possibly non-convex) sets rather than vectors. This leads to evolutions in a basic set which can be supplied with a metric (like the well-established Pompeiu–Hausdorff distance), but it does not have an obvious linear structure. This article extends differential inclusions with state constraints to compact-valued states in a separable Hilbert space H. The focus is on sufficient conditions such that a given constraint set (of compact subsets) is viable a.k.a. weakly invariant. Our main result extends the tangential criterion in the well-known viability theorem (usually for differential inclusions in a vector space) to the metric space of non-empty compact subsets of H.     

Robustness of viability controllers under small perturbations

In this note, a perturbed control problem with state constraints depending on a parameteru is considered. Assuming that, for a certain value ofu, there exists a viability controller, we explicitly estimate the range of variations ofu for which the same controller gives viable solutions.This work was supported by the Academy of Sciences of the Czech Republic and by EC, Cooperation in Science and Technology with Central and Eastern European Countries.     

On the semicontinuity problem of fibers and global F-regularity

In this article, we discuss the semicontinuity problem of certain properties on fibers for a morphism of schemes. One aspect of this problem is local. Namely, we consider properties of schemes at the level of local rings, in which the main results are established by solving the lifting and localization problems for local rings. In particular, we obtain the localization theorems in the case of seminormal and F-rational rings, respectively. Another aspect of this problem is global, which is often related to the vanishing problem of certain higher direct image sheaves. As a test example, we consider the deformation of the global F-regularity.     

Weak convergence to a Markov process: The martingale approach

Summary In this article, we obtain some sufficient conditions for weak convergence of a sequence of processes {X _n } toX, whenX arises as a solution to a well posed martingale problem. These conditions are tailored for application to the case when the state space for the processesX _n ,X is infinite dimensional. The usefulness of these conditions is illustrated by deriving Donsker's invariance principle for Hilbert space valued random variables. Also, continuous dependence of Hilbert space valued diffusions on diffusion and drift coefficients is proved.Research supported by National Board for Higher Mathematics, Bombay, IndiaPart of the work was done at University of California, Santa Barbara, USA     

A Geometric Approach to Maximum Likelihood Estimation of the Functional Principal Components From Sparse Longitudinal Data

In this article, we consider the problem of estimating the eigenvalues and eigenfunctions of the covariance kernel (i.e., the functional principal components) from sparse and irregularly observed longitudinal data. We exploit the smoothness of the eigenfunctions to reduce dimensionality by restricting them to a lower dimensional space of smooth functions. We then approach this problem through a restricted maximum likelihood method. The estimation scheme is based on a Newton–Raphson procedure on the Stiefel manifold using the fact that the basis coefficient matrix for representing the eigenfunctions has orthonormal columns. We also address the selection of the number of basis functions, as well as that of the dimension of the covariance kernel by a second-order approximation to the leave-one-curve-out cross-validation score that is computationally very efficient. The effectiveness of our procedure is demonstrated by simulation studies and an application to a CD4+ counts dataset. In the simulation studies, our method performs well on both estimation and model selection. It also outperforms two existing approaches: one based on a local polynomial smoothing, and another using an EM algorithm. Supplementary materials including technical details, the R package fpca, and data analyzed by this article are available online.     

控制系统生存核的逼近算法

生存核的计算是控制理论中的一个重要研究方向．给出了一种计算一般离散控制系统生存核的新算法．基于机器学习的方法,给出了逼近生存核的算法．并在一定条件下,证明了此算法的收敛性．此算法在一定程度上避免了计算量随控制空间的维数增长而指数增长的问题．最后,给出具体的实际例子来说明算法的有效性．     

A new alternating‐direction finite element method for hyperbolic equation

A modified backward difference time discretization is presented for Galerkin approximations for nonlinear hyperbolic equation in two space variables. This procedure uses a local approximation of the coefficients based on patches of finite elements with these procedures, a multidimensional problem can be solved as a series of one‐dimensional problems. Optimal order H₀¹ and L² error estimates are derived. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

A viability theorem of stochastic semilinear evolution equations

A viability theorem of stochastic semilinear evolution equations is discussed under a dissipative condition in terms of uniqueness functions and a stochastic subtangential condition. Our strategy is to interpret a stochastic viability problem into a characterization problem of evolution operators associated with stochastic semilinear evolution equations. The main theorem is a generalization of the results due to Aubin and Da Prato in the case of stochastic differential equations in ℝ ^d .     

INVESTIGATING THE VIABILITY OF SQUIRREL POPULATIONS; A MODELING APPROACH FOR THE ISLAND OF LESVOS,GREECE

ABSTRACT. Habitat loss and fragmentation are considered to be the most important factors responsible for population decreases in small mammal populations. Particularly important is also the effect of insularity that can act syn‐ergistically with the previous factors. Population Viability Analysis (PVA) combines the spatial component of the problem with the species population structure offering an integrated platform for testing and assessing the effects of critical parameters upon the population viability. Various management options can also be quantified and tested. In the case of Sciurus anomalous, a vulnerable squirrel species endemic in Lesvos, a series of threats and management problems were identified and assessed. A stochastic simulation model was developed and parameterized with field data for the species using the program Ramas/GIS. The results suggested that special attention has to be paid to the planning of road system networks and to stopping illegal hunting, especially when extinction risks for vulnerable populations are higher with the above threats.     

Systems with distributions and viability theorem

The approach to the consideration of the ordinary differential equations with distributions in the classical space D^′ of distributions with continuous test functions has certain insufficiencies: the notations are incorrect from the point of view of distribution theory, the right-hand side has to satisfy the restrictive conditions of equality type. In the present paper we consider an initial value problem for the ordinary differential equation with distributions in the space of distributions with dynamic test functions T^′, where the continuous operation of multiplication of distributions by discontinuous functions is defined [V. Derr, D. Kinzebulatov, Distributions with dynamic test functions and multiplication by discontinuous functions, preprint, arXiv: math.CA/0603351, 2006], and show that this approach does not have the aforementioned insufficiencies. We provide the sufficient conditions for viability of solutions of the ordinary differential equations with distributions (a generalization of the Nagumo Theorem), and show that the consideration of the distributional (impulse) controls in the problem of avoidance of encounters with the set (the maximal viability time problem) allows us to provide for the existence of solution, which may not exist for the ordinary controls.     

Viability in dynamic social networks

Two examples, Sampson's monks and Padgett and Ansell’ Florentines, illustrate the viability approach of dynamic networks. Notably, the relationship with centrality is studied. Historical processes involving networks are discussed.

Networks are presented as controls in controlled dynamic systems. Viability is the property for a state x that there exists a trajectory starting from x and satisfying the constraints until the time horizon. To obtain this, connection matrices must be selected at each time and each visited state among a specific set, the regulation map, which is carefully defined and built.     

Local well‐posedness for a system of quadratic nonlinear Schrödinger equations in one or two dimensions

In this article, the local well‐posedness of Cauchy's problem is explored for a system of quadratic nonlinear Schrödinger equations in the space L^p( R ⁿ). In a special case of mass resonant 2 × 2 system, it is well known that this problem is well posed in H^s(s≥0) and ill posed in H^s(s < 0) in two‐space dimensions. By translation on a linear semigroup, we show that the general system becomes locally well posed in L^p( R ²) for 1 < p < 2, for which p can arbitrarily be close to the scaling limit p_c=1. In one‐dimensional case, we show that the problem is locally well posed in L¹( R ); moreover, it has a measure valued solution if the initial data are a Dirac function. Copyright © 2016 John Wiley & Sons, Ltd.     

Single cell discretization of O(kh2 + h4) for the estimates of ${\partial u}\over{\partial n}$ for the two‐space dimensional quasi‐linear parabolic equation

In this article, new stable two‐level explicit difference methods of O(kh² + h⁴) for the estimates of for the two‐space dimensional quasi‐linear parabolic equation are derived, where k > 0 and h > 0 are grid sizes in time and space directions, respectively. We use a single computational cell for the methods, which are applicable to the problems both in cartesian and polar coordinates. The proposed methods have the simplicity in nature and employ the same marching type technique of solution. Numerical results obtained by the proposed methods for several different problems were compared with the exact solutions. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 250–261, 2001     

Numerical solution to a one‐dimensional thermoplastic problem with unilateral constraint

In this article, we present a numerical simulation of one‐dimensional problem of quasi‐static contact with an elastic obstacle. A finite difference scheme is derived by the method of reduction of order on uniform meshes. The stability and convergence are proved. The convergence order is of O(τ² + h²), where τ and h are the time step size and the space step size, respectively. Some numerical examples demonstrate the theoretical results. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Multi-Target Control Problems

We study control problems with several targets in the case of nonlinear dynamic systems. The map associating with every initial condition the minimal time to reach successively two given targets is characterized in the framework of differential inclusions through the notion of viability kernel. This approach allows one to treat the problem without assumptions of regularity and to build numerical schemes computing the minimal time. We also study the problem where an order of visit of the targets is required. The statements are also extended to the case of p targets under state constraints. Equivalent formulations in terms of Hamilton–Jacobi equations are also provided.     

Convergence analysis of a balancing domain decomposition method for solving a class of indefinite linear systems

A variant of balancing domain decomposition method by constraints (BDDC) is proposed for solving a class of indefinite systems of linear equations of the form (K?σ²M)u=f, which arise from solving eigenvalue problems when an inverse shifted method is used and also from the finite element discretization of Helmholtz equations. Here, both K and M are symmetric positive definite. The proposed BDDC method is closely related to the previous dual–primal finite element tearing and interconnecting method (FETI‐DP) for solving this type of problems (Appl. Numer. Math. 2005; 54 :150–166), where a coarse level problem containing certain free‐space solutions of the inherent homogeneous partial differential equation is used in the algorithm to accelerate the convergence. Under the condition that the diameters of the subdomains are small enough, the convergence rate of the proposed algorithm is established, which depends polylogarithmically on the dimension of the individual subdomain problems and which improves with a decrease of the subdomain diameters. These results are supported by numerical experiments of solving a two‐dimensional problem. Copyright © 2009 John Wiley & Sons, Ltd.