Arie Leizarowitz

Tracking periodic signals with the overtaking criterion

The overtaking criterion is suggested to evaluate the tracking of periodic signals on an infinite horizon. Existence, uniqueness, and a feedback form are established. The asymptotic dynamics of the solution and its relation to the asymptotic performance of the finite time optimizers are examined.

Tracking Nonperiodic Trajectories with the Overtaking Criterion

A tracking problem on an infinite time interval is studied, where the plant is linear with quadratic cost, and the tracked trajectory is not necessarily period. Optimal solutions with respect to the overtaking criterion are studied. Existence and uniqueness of such optimal solutions are proved and they are shown to be given by a linear feedback law which is the same as in the periodic case. A close relation between the solutions of tracking problems where the tracked trajectories are different only for very large times is established.

Nonlinear Analysis and Optimization II: Optimization

We prove a necessary optimality condition for isoperimetric problems on time scales in the space of delta-differentiable functions with rd-continuous derivatives. The results are then applied to Sturm-Liouville eigenvalue problems on time scales.

Existence of Overtaking Optimal Trajectories for Problems with Convex Integrands

In this work we study the existence of optimal overtaking trajectories for autonomous control systems on an infinite time interval. We extend, in two directions, the research initiated by Brock and Haurie (Brock, W. A., A. Haurie. 1976. On existence of overtaking optimal trajectories over an infinite time horizon. Math. Oper. Res. 1 337--346.). First we relax the technical assumptions needed to establish the existence of overtaking optimal trajectories. The assumptions in our existence result are basically those made in (Brock, W. A., A. Haurie. 1976. On existence of overtaking optimal trajectories over an infinite time horizon. Math. Oper. Res. 1 337--346.) for the existence of weakly overtaking optimal trajectories, with the additional assumption that a certain linear differential equation determined by the integrand has no nontrivial periodic solutions. A discussion and an example concerning this assumption are provided. In particular we show that this property is generic in the epi-convergence topology. Secondly, we consider convex integrands which grow uniformly to infinity rather than integrands with compact domains.

Mining for misconfigured machines in grid systems

Grid systems are proving increasingly useful for managing the batch computing jobs of organizations. One well-known example is Intel, whose internally developed NetBatch system manages tens of thousands of machines. The size, heterogeneity, and complexity of grid systems make them very difficult, however, to configure. This often results in misconfigured machines, which may adversely affect the entire system.We investigate a distributed data mining approach for detection of misconfigured machines. Our Grid Monitoring System (GMS) non-intrusively collects data from all sources (log files, system services, etc.) available throughout the grid system. It converts raw data to semantically meaningful data and stores this data on the machine it was obtained from, limiting incurred overhead and allowing scalability. Afterwards, when analysis is requested, a distributed outliers detection algorithm is employed to identify misconfigured machines. The algorithm itself is implemented as a recursive workflow of grid jobs. It is especially suited to grid systems, in which the machines might be unavailable most of the time and often fail altogether.

Frequency domain criterion for robust stability of interval time-delay systems

Abstract In this paper we characterize the boundary ∂f ( B ) of the image f ( B ) of a box B in R m under a nonlinear mapping f : R m → C . We generalize results recently reported by Polyak and Kogan (1993) [Necessary and Sufficient Conditions for Robust Stability of Multiaffine Systems. Mathematics Research Report 93-06, University of Maryland Baltimore County] for multiaffine mappings, and provide computationally tractable necessary and sufficient robust stability conditions for quasipolynomials with interval coefficients and interval delays. A numerical stability verification for a quasipolynomial family with two interval delays is presented.

Order Reduction Is Invalid for Singularly Perturbed Control Problems with a Vector Fast Variable

Abstract. The order reduction method for singularly perturbed optimal control systems consists of employing the system obtained while setting the small parameter to zero. In many situations the differential-algebraic system thus obtained indeed provides an appropriate approximation to the singularly perturbed optimal control problem under consideration. In this paper we show, however, that the set of singularly perturbed optimal control systems for which the order reduction approach is invalid is dense (in the L∞ norm) in the class of systems which we consider. This is established under the assumption that the fast variable in the singularly perturbed system is not a scalar.

Overtaking optimal regulation and tracking of piecewise diffusion linear systems

The infinite horizon optimal control of linear stochastic systems with quadratic cost integrand is studied, and the tracking of a periodic signal on an infinite time interval is considered. The system is exposed to three types of noises and is modeled by a nonhomogeneous linear stochastic control plant with modal and diffusion disturbances, where the dynamics switch at random times within a finite number of descriptions according to a Markov chain.The overtaking optimality criterion is employed. Considering the expected cost, the existence of a unique optimal control is established for the above noisy control systems with partial information. This is realized as an affine feedback control of the best estimate of the state. Moreover, in the case of partial information, this feedback control is proved to be also almost surely overtaking optimal.

Optimal Solutions of Linear Control Systems with Nonperiodic Convex Integrands

In this work we study the existence and asymptotic behaviour of overtaking optimal trajectories for a linear system with a nonperiodic convex cost function. We extend the results obtained by Leizarowitz Leizarowitz, A. 1986. Tracking nonperiodic trajectories with the overtaking criterion. Appl. Math. Opt.14 155--171. for a problem of tracking a prescribed nonperiodic trajectory with a quadratic cost function and establish the existence and uniqueness of optimal trajectories on an infinite horizon. The asymptotic dynamics of finite time optimizers is examined.

Overtaking and almost-sure optimality for infinite horizon Markov decision processes

We consider infinite horizon optimal control of Markov chains on complete metric spaces. We employ the overtaking optimality criterion, which is either applied to the expected cost-flow, or to the individual sample paths, yielding almost-sure optimality results. We use the existence of a solution pair Φ·, λ to the optimality equation LΦx = λ to establish and characterize optimal strategies. For finite state-spaces we derive sufficient, as well as necessary conditions for overtaking optimality.