The influence of a supporting canopy on the dynamics of opening of a parachute

A mathematical model of the opening of a circular parachute under the action of a supporting canopy was executed on a computer. The proposed model is based on the simultaneous integration of the equations of nonlinear aerodynamics and the theory of soft membranes. It is shown that the supporting canopy may both improve the characteristics of the parachute system and make them worse. The numerical results are presented in the form of graphs. One figure.Translated fromDinamicheskie Sistemy, No. 6, 1987, pp. 23–26.     

Mathematical model of the dynamics of the canopy of an axisymmetric parachute

The article presents a mathematical model of the dynamics of the canopy of an axisymmetric parachute taking into account the biaxiality of the state of complex stress of the tissue, the effect of the rigging lines of the annular carcass, viscous deformation, riffling. For the solution a threelayered symmetric implicit difference schema is suggested. The method of accelerated attainment of a stable solution without distortion of the dynamic pattern is explained.Translated from Dinamicheskie Sistemy, No. 5, pp. 25–32, 1986.     

On the zero dynamics of one class of distributed discrete systems

The problem of optimal control of a linear dynamical system under set-membership uncertainty is studied: it is required to steer the system to the terminal set with a guarantee and to maximize the guaranteed value of the quality criterion. The sets of the initial and current a preposteriori distributions of the states of the dynamical system are introduced; they are used to determine a positional solution of the problem of optimal a preposteriori³ observation with the help of inaccurate measurements of input and output signals of the observation object by two measuring devices. The obtained solution is used for determining a positional solution of the optimal control problem under uncertainty. Depending on the amount of the information used, optimal closable and closed output feedbacks are determined. The method of quasiimplementation of optimal feedbacks by means of optimal estimators and a regulator producing real-time control actions is described. The results are illustrated by examples.     

Invariant states for the time dynamics of a class of multidimensional lattice quantum Fermi systems

The study of invariant states of fermionic lattice systems begun earlier is contined. Under the assumption that the time dynamics corresponds to a (formal) HamiltonianH ₀ and the invariant state is a KMS state for some HamiltonianH [1], one-dimensional lattice Fermi systems were considered in the earlier work. In particular, the case whenH ₀ is not a quadratic form in the creation and annihilation operators and all nonquadratic terms inH ₀ are diagonal was studied. In this case, it was shown that up to an arbitrary diagonal quadratic formN the HamiltonianH is proportional toH ₀, i.e., that is a KMS state of H ₀+N. In this paper, we obtain a similar result for Fermi systems of arbitrary dimension by a somewhat different method to the one used earlier [1].Chelyabinsk State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 94, No. 1, pp. 76–83, January, 1993.     

On structural transformations of equations of perturbed motion for a certain class of dynamical systems

We consider a general method for structural transformations of one class of dynamical systems with gyroscopic forces, which enables us to remove gyroscopic terms from the original equations of perturbed motion. Without changing the qualitative properties of these equations, this method simplifies their investigation.     

Global dynamics for a model of a class of continuous‐time dynamical systems

In this paper, the global asympotic behavior of solutions of a class of continuous‐time dynamical system is studied. Not only do we obtain the ultimate boundedness of solutions of the system but we also obtain the rate of the trajectories of the system going from the exterior of the trapping set to the interior of the trapping set, which can be applied to study chaotic control and chaotic synchronization of the system. Copyright © 2015 John Wiley & Sons, Ltd.     

The use of the simplex method in the design of parachute systems with use of a prototype

We develop a formal mathematical apparatus that makes it possible using the simplex method (linear programming) to carry out the design of parachute systems with use of a prototype, the experimental development of parachute systems using a computer, the analysis of a technical problem, and the identification of mathematical models of an automated parachute design system.Translated fromDinamicheskie Sistemy, No. 6, 1987, pp. 46–49.     

The computation of stability boundaries in state space for a class of biochemical engineering systems

The stability of a class of biochemical processes defined by a set of m biochemical reactions involving n components is analysed. The processes operate in a continuous mode and possess at least two stable equilibrium states: the normal operating point and a biological wash out state. Using a canonical state representation of the process dynamics the geometric structure of the operating point's stability boundary is characterized. Numerical algorithms are developed to evaluate this boundary and to visualize its extent in state space. The proposed technique is illustrated with a representative engineering example.     

Optimal control of systems governed by a class of nonlinear evolution equations in a reflexive Banach space

The paper presents a closure theorem for the attainable trajectories of a class of control systems governed by a large class of nonlinear evolution equations in reflexive Banach spaces. Several existence theorems for optimal controls are proven that include a terminal control problem, a time-optimal control problem, and a special Bolza problem. Some results of independent interest are also presented.This work was supported in part by the National Research Council of Canada under Grant No. 7109.The authors would like to thank Professor L. Cesari for pointing out that joint continuity off is required for the setsG andR to satisfy the upper semicontinuity property (Theorems 5.1 and 5.2).     

On a class of operators on Hilbert space with applications to factorization and systems theory

A class of operators is defined in a Hilbert resolution space setting that offers a new perspective on problems of causal invertibility, special factorization, and the theory of quadratic cost optimization problems for dynamical systems. The major results include an extension of the special factorization to a class of noncompact operators and the definition of an abstract state space. These results are then used to obtain an optimal feedback solution to an abstract linear regular-quadratic cost problem.     

Existence of optimal controls for a class of systems governed by differential inclusions on a Banach space

Using Cesari's approach, we prove the existence of optimal controls for a class of systems governed by differential inclusions on a Banach space having the Radon-Nikodym property. Theorem 3.1 gives the existence result for optimal relaxed controls under fairly general assumptions on the system and the admissible controls. This result depends on a fundamental result (Theorem 2.1) that proves the existence of mild solutions of differential inclusions on a Banach space, which has also independent interest. Further, the preparatory results, such as Lemma 3.1 and Lemma 3.2, are also useful in the study of time-optimal and terminal control problems.For illustration of the results, we present two examples, one on distributed controls for a class of systems governed by nonlinear parabolic equations and the other on boundary controls with discontinuous boundary operator.This work is supported in part by the National Science and Engineering Council of Canada under Grant No. 7109.     

Global solutions to a class of multi-species reaction-diffusion systems with cross-diffusions arising in population dynamics

In this paper, an n$n$-species strongly coupled cooperating diffusive system is considered in a bounded smooth domain, subject to homogeneous Neumann boundary conditions. Employing the method of energy estimates, we obtain some conditions on the diffusion matrix and inter-specific cooperatives to ensure the global existence and uniform boundedness of a nonnegative solution. The globally asymptotical stability of the constant positive steady state is also discussed. As a consequence, all the results hold true for multi-species Lotka–Volterra type competition model and prey–predator model.     

Shape of the parachute canopy

The article presents new equations of equilibrium of an axisymmetric parachute canopy generalizing Kh. A. Rakhmatulin's well-known equations. The equations take the biaxial nature of the state of stress near the pole of the canopy into account, and this leads to a finite value of the tension of the tissue and to a finite value of the radius of curvature at the pole.Translated from Dinamicheskie Sistemy, No. 5, pp. 21–25, 1986.     

The motion of a material point in friedmann-lobachevsky space

Some simple instances of the motion of a material point in Friedmann-Lobachevsky space /1/ are constructed and investigated, on the assumption that the space, like Galilean space, is empty (i.e., contains no matter) and that the forces acting on the point, including the gravitational forces, constitute a factor extraneous to the space. Thus, the problem is being considered in the context of a rather unusual “relativistic” mechanics, distinct from relativisitc mechanics proper. As will be seen later, the difference is quantitatively small and can be regulated by slowly varying cosmological factors in the pseudo-Euclidean metric of the space of special relativity theory.     

Phase space transport in a map of asteroid motion

IntroductionIt is well known that a complex phase space of a Hamiltonian system containing largemeasures of both regular and chaotic orbits is often partitioned by such partial obstructionas canton or Arnold web, ac.hich although not serving as absolute barriers, can significantlyimpede the motion of a chaotic orbit through a connected phase space region. This "stickiness"effect makes the phase space transport complicated. In fact, the chaotic transport or diffusionphenomenon can be met in ma…     

Global dynamics of one class of nonlinear nonautonomous systems with time-varying delays

A class of nonautonomous systems of nonlinear delay differential equations was studied via construction of matrix inequalities and comparison techniques. The results for the nonautonomous systems with time-varying delays are novel, e.g., the global stability of differential equations with nonlinear (casual) Volterra operators is considered for the first time in the literature. Criteria obtained for permanence and global attractivity are explicit and hence are convenient for applying/verifying in practice. We illustrate applications of the results obtained to the nonautonomous and asymptotically autonomous Nicholson-type models.