On an estimate of the domain of attraction for systems with aftereffect

We consider the problem of construction of the domain of attraction for systems with aftereffect. The structure of solutions of the Cauchy problem in the vector space of states of a system is clarified with examples. We describe a constructive method for the estimation of the domain of attraction from the inside with the use of the Lyapunov functions. This method is used for the estimation of the effect of delay in a control device in the process of solution of the problem of uniaxial orientation of a spacecraft. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 5, pp. 671–678, May, 1998. The present work was partially supported by the Center for Fundamental Research in Economics.     

The stability and stabilization of the motion of non-conservative mechanical systems

Two stability problems are solved. In the first, the stability of mechanical systems, on which dissipative, gyroscopic, potential and positional non-conservative forces (systems of general form) act, is investigated. The stability is considered in the case when the potential energy has a maximum at equilibrium. The condition for asymptotic stability is obtained by constructing Lyapunov's function. In the second problem, the possibility of stabilizing a gyroscopic system with two degrees of freedom up to asymptotic stability using non-linear dissipative and positional non-conservative forces is investigated. Stability of the gyroscopic system is achieved by gyroscopic stabilization. The stability conditions are obtained in terms of the system parameters. Cases when the gyroscopic stabilization is disrupted by these non-linear forces are indicated.     

The stability of mechanical systems with positional non-conservative forces

A classical problem is discussed, namely, the influence of the structure of the applied forces on the stability of the equilibrium position of an autonomous mechanical system. Several propositions extending the Thomson-Tate-Chetayev theorems to systems with non-conservative positional forces are proved.     

The exponential stability and stabilization of non-autonomous mechanical systems with non-conservative forces

The exponential stability of the unperturbed motion of a non-autonomous mechanical system with a complete set of forces, that is, dissipative, gyroscopic, potential and non-conservative positional forces, is investigated. The problem of stabilizing a non-autonomous system with specified non-conservative forces is considered with and without the use of potential forces. The problem of stabilizing a non-autonomous system with specified potential forces by the action of the forces of another structure is studied. The domain of stabilizability of the relative equilibrium position of a satellite in a circular orbit is found.     

An estimate of the attraction domain with a specified exponential stability index in a wheeled robot control problem

The problem of synthesizing a law for the control of the plane motion of a wheeled robot is investigated. The rear wheels are the drive wheels and the front wheels are responsible for the turning of the platform. The aim of the control is to steer a target point to a specified trajectory and to stabilize the motion along it. The trajectory is assumed to be specified by a smooth curve. The actual curvature of the trajectory of the target point, which is related to the angle of rotation of the front wheels by a simple algebraic relation, is considered as the control. The control is subjected to bilateral constraints by virtue of the fact that the angle of rotation of the front wheels is limited. The attraction domain in the distance to trajectory - orientation space, is investigated for the proposed control law. Arrival at a trajectory with a specified exponential stability index is guaranteed in the case of initial conditions belonging to the given domain. An estimate of the attraction domain in the form of an ellipse is given.     

The domain of attraction and the stability region for stochastic partial differential equations with delays

In this paper, a stochastic partial differential equation with delays is considered. On the basis of the properties of nonnegative matrices, stochastic convolution and the inequality technique, sufficient conditions for determining the domain of p$p$th-moment attraction and the p$p$th-moment asymptotic stability region are obtained. An example is also discussed, to illustrate the efficiency of the results obtained.     

Qualitative analysis of systems with an ideal non-conservative constraint

The Hamiltonian form developed in /1/ for the equations of motion of systems with ideal non-conservative constraints enables familiar methods of classical and celestial mechanics to be used to analyse the dynamics of such systems. When this is done certain difficulties arise, due to the fact that the Hamiltonian is not analytic. In this paper one of the possible algorithms applying KAM theory /2/ and Poincaré's theory of periodic motions /3/ to the analysis of systems in which the Hamiltonian is non-analytic in one of the phase variables is described. As an example, some results of /4/ concerning the dynamics of a rigid body colliding with a fixed, absolutely smooth, horizontal plane are refined.     

Estimation of the domain of attraction for a class of hybrid systems

The problem of the estimation of the domain of attraction for Impulsive Dynamical Systems (IDSs) is tackled in this paper. IDSs are a special class of hybrid systems that exhibit jumps in the state trajectory, which can be either time-driven (time-dependent IDSs), or driven by specific state values (state-dependent IDSs). Sufficient conditions to determine whether a polytope belongs to the domain of attraction of the zero equilibrium point are provided for both time-dependent and state-dependent IDS, when a nonlinear quadratic continuous-time dynamic is considered. The proposed results are stated in terms of Linear Matrix Inequalities problems. The effectiveness of the proposed results is shown by means of the analysis of a biological model for tumor progression.     

Asymptotic stability and estimates for the attraction domains of monotone difference systems

We generalize asymptotic stability criteria and estimates for attraction domains in the nonnegative cone for systems of autonomous difference equations with monotone discontinuous right-hand side.     

Estimate of the attraction domain for a class of nonlinear switched systems

We consider a hybrid dynamical system composed of a family of subsystems of nonlinear differential equations and a switching law which determines the order of their operation. It is assumed that subsystems are homogeneous with homogeneity degrees less than one, and zero solutions of all subsystems are asymptotically stable. Using the Lyapunov direct method and the method of differential inequalities, we determine classes of switching laws providing prescribed estimates of domains of attraction for zero solutions of the corresponding hybrid systems. The developed approaches are used for the stabilization of a double integrator.     

Structural transformations of non-conservative systems

The method of structural mappings of gyroscopic systems [1, 2] is developed for systems involving non-conservative positional forces. This technique, considered in the aspect of the legitimate use of the precessional equations of the precessional equations of the applied theory of gyroscopes, enables the difficulties associated with the presence of non-conservative structures in the initial equations to be overcome, and in many cases enables of the Thomson—Tait—Chetayev theorems to be used directly.     

Nonlinear dynamic systems design based on the optimization of the domain of attraction

In this paper an optimization-based methodology for the design of the operating equilibrium of a nonlinear dynamic system based on a measure of the extension of its domain of attraction is proposed. The approach consists in maximizing the radius of a ball in the state space contained in the region of negative definiteness of the time derivative of a quadratic Lyapunov function, using a two level optimization strategy.A deterministic global optimization problem is solved at the inner level to ensure proper estimation of the domain of attraction for each feasible realization of the design variables which are optimized at the outer level. In order to cope with the non-differentiable nature of the inner problem, a stochastic algorithm is applied to manipulate the design variables at the outer level.The methodology is applied to several examples to illustrate different aspects of the approach.     

On the domain of attraction of an operator between supremum and sum

Summary Between the operations which produce partial maxima and partial sums of a sequenceY ₁,Y ₂, ..., lies the inductive operation:X _n =X _n-1(X _n-1+Y _n ),n1, for 0<<1. If theY _n are independent random variables with common distributionF, we show that the limiting behavior of normed sequences formed from {X _n ,n1}, is, for 0<<1, parallel to the extreme value case =0. ForFD() we give a full proof of the convergence, whereas forFD()D(), we only succeeded in proving tightness of the involved sequence. The processX _n is interesting for some applied probability models.     

Stabilization and destabilization in non-conservative gyroscopic systems

Stability of a linear autonomous non-conservative system in presence of potential, gyroscopic, dissipative, and nonconservative positional forces is studied. The cases when the non-conservative system is close to a gyroscopic system or to a circulatory one, are examined. It is known that the marginal stability of gyroscopic and circulatory systems can be destroyed or improved up to asymptotic stability due to action of small non-conservative positional and velocity-dependent forces. The present contribution shows that in both cases the boundary of the asymptotic stability domain of the perturbed system possesses singularities such as “Dihedral angle” and “Whitney umbrella” that govern stabilization and destabilization. Approximations of the stability boundary near the singularities and estimates of the critical gyroscopic and circulatory parameters are found in an analytic form. In case of two degrees of freedom these estimates are obtained in terms of the invariants of matrices of the system. As an example, the asymptotic stability domain of the modified Maxwell-Bloch equations is investigated with an application to the stability problems of gyroscopic systems with stationary and rotating damping, such as the Crandall gyropendulum, tippe top and Jellet's egg. An instability mechanism in a system with two degrees of freedom, originating after discretization of models of a rotating disc in frictional contact and possessing the spectral mesh in the plane ‘frequency’ versus ‘angular velocity’, is described in detail and its role in the disc brake squeal problem is discussed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A stability estimate for an elliptic problem

We investigate the Caucy problem for linear elliptic operators withC ^∞-coefficients at a regular domain ℝ ⊂ ℝ, which is a classical example of an ill-posed problem. The Cauchy data are given at the manifold Γ⊂∂Ω and our goal is to obtain a stability estimate inH ₄(Ω).     

The inverse problem for Lagrangian systems with certain non-conservative forces

We discuss two generalizations of the inverse problem of the calculus of variations, one in which a given mechanical system can be brought into the form of Lagrangian equations with non-conservative forces of a generalized Rayleigh dissipation type, the other leading to Lagrangian equations with so-called gyroscopic forces. Our approach focusses primarily on obtaining coordinate-free conditions for the existence of a suitable non-singular multiplier matrix, which will lead to an equivalent representation of a given system of second-order equations as one of these Lagrangian systems with non-conservative forces.     

The domain of partial attraction of a Poisson law

Groshev gave a characterization of the union of domains of partial attraction of all Poisson laws in 1941. His classical condition is expressed by the underlying distribution function and disguises the role of the mean of the attracting distribution. In the present paper we start out from results of the recent probabilistic approach and derive characterizations for any fixed >0 in terms of the underlying quantile function. The approach identifies the portion of the sample that contributes the limiting Poisson behavior of the sum, delineates the effect of extreme values, and leads to necessary and sufficient conditions all involving . It turns out that the limiting Poisson distributions arise in two qualitatively different ways depending upon whether >1 or <1. A concrete construction, illustrating all the results, also shows that in the boundary case when =1 both possibilities may occur.     

An estimate of the stability of optimal control of certain stochastic and deterministic systems

Translated from:Problemy Ustoichivosti Stokhasticheskikh Modelei, Trudy Seminara, 1989, pp. 26–34.