Georgi Smirnov

Introduction to the Theory of Differential Inclusions

Foundations: Convex analysis Set-valued analysis Nonsmooth analysis Differential inclusions: Existence theorems Viability and invariance Controllability Optimality Stability Stabilization Comments Bibliography Index.

Transversality condition for infinite-horizon problems

We present necessary conditions of optimality for an infinitehorizon optimal control problem. The transversality condition is derived with the help of stability theory and is formulated in terms of the Lyapunov exponents of solutions to the adjoint equation. A problem without an exponential factor in the integral functional is considered. Necessary and sufficient conditions of optimality are proved for linear quadratic problems with conelike control constraints.

Stationary Configurations of a Tetrahedral Tethered Satellite Formation

O RBITAL dynamics analysis of a connected multibody system is relevant for several missions, including tethered systems, space robots and manipulators, telescopes, space stations, etc. It is important to determine stationary motions of such structures due to their possible use as nominal motions in energy-saving mode. The study of the behavior of multibody systems in the orbital environment continues to grow since its beginning in the 1960s. Sarychev [1] and Wittenburg [2] investigate the equilibria of two connected rigid bodies in circular orbit with respect to the orbital reference frame. Cheng and Liu [3] consider the in-plane dynamics of a three-body systemwith two equal extreme bodies, examining the equilibrium orientations, bifurcations, and stability with respect to in-plane perturbations. Lavagna and Ercoli Finzi describe planar equilibrium configurations [4] and study their stability [5] for an extended rigid body with two pendulums attached either in a sequence or in a parallel scheme. Quite frequently, the orbital dynamics of multibody systems is studied for models that approximate connected satellites by an open chain of material points linked by weightless straight rods with spherical hinges. Misra and Modi [6] develop the general threedimensional formulation for an n-link chain. For twoand three-link systems, they determine the libration frequencies and study the control laws. In [7,8] this model is used to determine in-plane equilibrium configurations of a two-link chain and to study their stability. Sarychev [9] finds all spatial equilibrium orientations of a two-link chain in a circular orbit and describes their number as a function of problem parameters. Continued interest in such studies was confirmed recently by Correa and Gomez [10], who numerically study the equilibrium configurations of a three-link chain in the plane of the orbit and analyze their stability with respect to in-plane perturbations. An analytical study of chains with arbitrary number of links can be found in [11,12]. The center ofmass of the chain (CMC)moves along a circular orbit. All in-plane equilibrium configurations are described in [11]. All spatial equilibria are listed in [12], in which it is shown that each connecting rod can be oriented with respect to the tangent, normal and binormal to the CMC orbit in one of the following ways: 1) A rod is aligned with the tangent axis. 2) A rod belongs to an NBN group, that is, to a subchain of rods that lies in the plane parallel to normal and binormal so that its center of mass is on the tangent axis. 3) A rod links two NBN groups or an end of an NBN group with the tangent axis. Some of these equilibrium configurations are actually twodimensional, with all the system lying in one of the coordinate planes of the orbital reference frame. Yet, there exist essentially threedimensional configurations, for example, with masses placed at the vertices of a tetrahedron. Tetrahedral satellite formations are of significant interest due to numerous applications and many projects under development, including NASA research programs in formation flying [13–15]. Tetrahedral configuration has been successfully implemented in the Cluster mission to study three-dimensional structure of the Earth’s magnetosphere [16]. Being the simplest spatial configuration, a tetrahedral formation is a natural tool in experiments of this kind, because it enables one to execute simultaneous measurements at points of a large-span three-dimensional basis. In the present article, we apply the general results of [12] to study tetrahedral equilibrium configurations of a tethered satellite system. We identify the spectrum of spatial equilibrium configurations that can be achieved by varying the masses of the bodies and lengths of the rods.We determine the links that can be replacedwith tethers.We study the possibility of controlling such a system and of stabilizing its orientation.

Stabilization by constrained controls

A stabilization problem for a general nonlinear control system is considered. In particular the control corresponding to the equilibrium position may belong to the boundary of the control set. A linear control system is considered as a first approximation for the original problem. The right-hand side of the linear system generates a set-valued map of a special type known as a convex process. This set-valued map has a number of properties similar to those of a linear operator. They allow one to establish necessary and sufficient conditions for solvability of the regulator design problem for the first approximation and to construct a Lyapunov function. Based on these results the nonlinear stabilization problem is investigated. Different statements of the regulator design problem are studied. Stabilization problems for some mechanical systems are considered to illustrate the regulator design techniques. The properties of transient characteristics (the “peak” effect) are discussed for a linear stabilization p...

Closed relative trajectories for formation flying with single-input control

We study the problem of formation shape control under the constraints on the thrust direction. Formations composed of small satellites are usually subject to serious limitations for power consumption, mass, and volume of the attitude and orbit control system (AOCS). If the purpose of the formation flying mission does not require precise tracking of a given relative trajectory, AOCS of satellites may be substantially simplified; however, the capacity of AOCS to ensure a bounded or even periodic relative motion has to be studied first. We consider a formation of two satellites; the deputy one is equipped with a passive attitude control system that provides one-axis stabilization and a propulsion system that consists of one or two thrusters oriented along the stabilized axis. The relative motion of the satellites is modeled by the Schweighart-Sedwick linear equations taking into account the effect of perturbations. We prove that both in the case of passive magnetic attitude stabilization and spin stabilization for all initial relative positions and velocities of satellites there exists a control guaranteeing their periodic relative motion.

Nonparametric Estimation for Non-Homogeneous Semi-Markov Processes: An Application to Credit Risk

We propose procedures for estimating the time-dependent transition matrices for the general class of finite nonhomogeneous continuous-time semi-Markov processes. We prove the existence and uniqueness of solutions for the system of Volterra integral equations defining the transition matrices, therefore showing that these empirical transition probabilities can be estimated from window censored event-history data. An implementation of the method is presented based on nonparametric estimators of the hazard rate functions in the general and separable cases. A Monte Carlo study is performed to assess the small sample behavior of the resulting estimators. We use these new estimators for dealing with a central issue in credit risk. We consider the problem of obtaining estimates of the historical corporate default and rating migration probabilities using a dataset on credit ratings from Standard & Poors.

Renormalization of nanoparticle polarizability in the vicinity of a graphene-covered interface

Financial support from the Portuguese Foundation for Science and Technology (FCT) through Projects PTDC-FIS-113199-2009 and PEst-C/FIS/UI0607/2013 is acknowledged. We also acknowledge support from the European Commission under Graphene Flagship (Contract No. CNECT-ICT-604391). J.E.S.s work contract is financed in the framework of the Program of Recruitment of Post Doctoral Researchers for the Portuguese Scientific and Technological System, within the Operational Program Human Potential (POPH) of the QREN, participated by the European Social Fund (ESF) and national funds of the Portuguese Ministry of Education and Science (MEC).

Attitude determination and stabilization of a spherically symmetric rigid body in a magnetic field

Observation and stabilization problems for a rigid body rotating about its mass centre in a time-periodic magnetic field is considered. The moment of the applied forces about the mass centre is a vector product of the field and the vector of control. The only measurement is of the field and its time derivatives in the body co-ordinates. The model describes the angular motion of a satellite in the geomagnetic field with the control carried out by a magnetic moment of current coils (magnetorquers) mounted on the satellite body. It is shown that generically the knowledge of the field and its derivatives is sufficient for the attitude determination. A problem of stabilization along one direction is considered. The stabilizer constructed in this work drives the system to a given motion from all initial points that do not belong to an unstable manifold. It is shown that all stabilizers have such a manifold of initial conditions.

Effect of clustering on the surface plasmon band in thin films of metallic nanoparticles

Abstract. We theoretically investigate the optical response of ensembles of polarizable metallic nanoparticles (NPs) that form (1) submonolayer films of particles adsorbed on a dielectric substrate, considered as two-dimensional (2-D) systems, and (2) thin three-dimensional (3-D) films, where NPs are embedded in a dielectric matrix. For system (1), the effect of NPs’ distance to the substrate is taken into account. In both cases, we find that short-range clustering leads to a broadening and a spectral shift of the absorption band related to the surface plasmon resonance (SPR) in individual NPs. We show that the clustering can help in achieving spectrally broad SPR bands, especially if NPs aggregate into fractal clusters, which can be interesting for some applications such as surface-enhanced Raman scattering. In particular, submonolayer films on NPs generated using the diffusion-limited aggregation algorithm produce sizable and spectrally broad absorption, which can be tuned to the visible range by choosing an appropriate capping and/or substrate material. Calculated results for thin 3-D films are compared with experimental data obtained for Au/TiO2 nanocomposite layers produced by reactive cosputtering.

Probing spatial correlations in a system of polarizable nanoparticles via measuring its optical extinction spectrum

It is shown that quantitative information on spatial correlations in a system of polarizable particles can be extracted directly from its experimentally measurable optical spectra. For a collection of metallic nanoparticles (NPs), it is demonstrated that the degree of short-range correlation in NPs positions can be evaluated by an appropriate numerical analysis of the extinction spectrum in the surface plasmon resonance region, given the polarizability of an individual NP. The spectrum analysis consists in the evaluation of a single number, which is the derivative of the ensemble response function in the vanishing polarizability limit, using the Tikhonov regularization method. This number has to be compared to pre-calculated values for a model NP system with given density and correlation parameters.