Statistical characteristics of attainability set of controllable systems with random coefficients

For controllable systems with random coefficients we study a property of statistical invariance, satisfied with given probability. We obtain sufficient conditions for invariance of a set with respect to controllable system expressed in terms of Lyapunov functions and shift dynamic system. We study the statistical characteristics of attainability set of a controllable system which is parameterized by metric dynamic system.     

Estimates for attainability sets of control systems

We consider an approach to the estimation of attainability sets of a control system for solving various control problems. The approach is based on two types of extension of the system, by weakening the finite constraints describing the set of velocities and by diminishing the order of the differential constraint. By combining extensions of both types, one can obtain a sufficiently wide class of estimates from which one can then choose the most effective estimates from the viewpoint of accuracy or simplicity.     

Error bounds for attainability sets of control systems with phase constraints

The problem of the error bounds for attainability sets of control systems described by ordinary differential equations under discretization of the phase constraints is studied. The peculiarity of the problems investigated in this paper is the phase constraints in the form of equality.     

Internal ellipsoidal estimates of attainability sets for linear systems under conditions of uncertainty

The problem of constructing internal ellipsoidal estimates of the geometric difference between two ellipsoids and applying the estimated results for the attainability sets of linear systems with a disturbance is considered. An addition to the existing method of constructing the difference between two ellipsoids is presented, and the previous constraints are removed. In the process of validating the addition, some relationships between certain properties of constructed ellipsoidal estimations and set convexity are given, being the data for the problem. A method for estimating the attainability sets for linear systems with a disturbance, equivalent to the existing approach to systems without disturbances, are given. The disturbances are considered using the obtained results.     

Approximate construction of attainability sets of control systems with integral constraints on the controls

The approximate construction of attainability sets of control systems with quadratic integral constraints on the controls is considered. It is assumed that a control system is non-linear with respect to the phase variable and linear with respect to the variable which describes the controlling action. The approximation of the attainability sets of a control system is accomplished in several stages. The latter class of controls generates a finite number of trajectories of the system. The trajectories of the system are then replaced by Euler broken lines. An estimate of the accuracy of the Hausdorff distance between the attainability set and the set which has been approximately constructed is obtained.     

Reachable and controllable sets for two-dimensional,linear, discrete-time systems

We consider two-dimensional discrete-time linear systems with constrained controls. We propose a simple polynomial time procedure to give an exact external representation of theN-step reachable set and controllable set. The bounding hyperplanes are explicitly derived in terms of the data of the problem. By using a result in computational geometry, all the calculations are made in polynomial time in contrast to classical methods. The limit case asN is also investigated.     

On the closure of sets of attainability in ℝ2

We display a class of control systems in the plane for which it is generically true that the set attainable from any point in the plane is a closed Whitney prestratified set. That is, even if the attainable set of some given system is not closed, we can approximate the system as closely as we wish by another system whose attainable sets are closed.     

Statistical characteristics of continuous functions and statistically weakly invariant sets of controllable system

We continue the investigation of expansion of a concept of invariance for sets which consists in studying statistically invariant sets with respect to control systems and differential inclusions. We consider the statistical characteristics of continuous functions: Upper and lower relative frequency of containing for graph of a function in a given set. We obtain conditions under which statistical characteristics of two various asymptotical equivalent functions coincide; then by the value of one of them it is possible to calculate the value of another one. We adduce the equality for finding relative frequencies of hitting functions the given set in the case when the distance from the graph of one of functions to the given set is a periodic function. A consequence of these statements are conditions of statistically weak invariance of a set with respect to controlled system. For some almost periodic functions we obtain the formulas by which we can calculate the mean values and the statistical characteristics. We also consider the following problem. Let the number λ₀ ∈ [0, 1] be given. It is necessary to find the value c(λ₀) such that the upper solution z(t) of the Cauchy problem does not exceed c(λ₀) with the relative frequency being equal λ₀. Depending on statement of the problem, a value z(t) can be interpreted as the size of population, energy of a particle, concentration of substance, size of manufacture or the price of production.     

Ellipsoidal approximation of attainability sets of a linear system with indeterminate matrix

Linear dynamical systems described by finite-difference or differential equations are considered. It is assumed that the matrix of the system is either completely known or is subject to uncontrollable perturbations, so that each element is known only to within a certain possible interval. Outer approximations, by means of ellipsoids, are constructed for the attainability sets of such systems. The equations of evolution of the approximating ellipsoids are obtained. An example is presented.     

Attraction sets in abstract attainability problems: Equivalent representations and basic properties

We consider attainability problems in topological spaces under asymptotic constraints.     

On the intersection of controllable and reachable sets

In this paper, we prove that the intersection of the controllable set to and the reachable set from a point target is connected, and that points within the intersection have the same controllable and reachable sets. We further demonstrate that the intersection is positive invariant whenever the intersection is not a single point. We then introduce the concept of a maneuverable set which allows us to generalize the results by replacing the point target with a maneuverable set. Use of the theory is illustrated.This paper is based on research supported by NSF Grant INT-82-10803 and the University of Western Australia (Visiting Fellowship, Department of Mathematics, 1983).     

The region of attainability of nonlinear systems with unbounded controls

In this work, we give a characterization of the topological boundary and the interior of the reachable set for a family of vector fields of the formX ⁰+aX ⁱ , wherea andi belongs to an arbitrary set of indices. The methods considered in this article are applied to various problems in control theory.The authors wish to express their gratitude to Professor Felix Albrecht for his precious comments on the first version of this work.     

Structure of the attainability set for controlled systems with inertia

Translated from Matematicheskie Zametki, Vol. 49, No. 2, pp. 150–151, February, 1991.     

An algebraic characterization of closed small attainability subspaces of delay systems

If small attainability subspaces of linear time delay systems are closed in a certain Sobolev space, the existence of Lagrange multipliers for optimal control to small solutions is guaranteed. This paper characterizes the required closedness property using an algebraic approach due to B. Jakubczyk. As a main result it turns out that closedness is—in an algebraic sense—generic in the variety of system matrices (A₀,A₁, B₀) with rank A₁ not greater than the dimension of the control space. This is in contrast to known results on closedness of attainability subspaces playing an analogous role for optimal control to fixed final states instead of small solutions.     

Globally controllable linear systems

In this paper, we collect main results referring to sufficient and necessary global controllability conditions of nonstationary linear systems and present generalizations of these results. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 23, Optimal Control, 2005.     

Fuzzy sets and statistical data

Specific features of probability and possibility theories are discussed with emphasis on semantical aspects. Instead of putting forward the acknowledged usefulness of possibility theory for the non-statistical modelling of subjective categories, we try to figure out how statistical data and possibility theory could be matched. As a result, procedures for constructing weak possibilistic substitutes of probability measures, and for processing imprecise statistical data are outlined. They provide new insights on relationship between fuzzy sets and probability theories.