The quickest transfer of a multidimensional dynamic object onto the surface of an ellipsoid

The time-optimal control of motion of a multidimensional dynamic object is investigated. Both open-loop and feedback controls are considered. It is assumed that the force vector is constrained by a non-degenerate ellipsoid. The terminal set is described by the surface of an ellipsoid that allows degeneracy, and the terminal velocity is not fixed. The initial position of the object can be either outside or inside the ellipsoid. Using the maximum principle, the necessary and sufficient conditions for the optimality of the control are established in the form of a system of polynomial equations, the orders of which depend on the dimension of the problem and the degree of degeneracy of the terminal ellipsoid. Iterative procedures and a technique for continuation with respect to the parameters of the approximating ellipsoid in the admissable region of change of the phase vector are proposed. Control problems for the limiting shapes of the terminal ellipsoid are investigated. The qualitative effect of the discontinuity of the functional and the control defined as functions of the phase vector is established.     

Minimum-order observers for discrete-time systems

We consider the synthesis of a minimum-order state or functional observer for a linear dynamical system. The synthesis problem is solved for completely certain systems of general form and for some classes of uncertain systems. Various approaches are described, which ultimately lead to the same task: finding a minimum-dimension Hurwitz solution for a system of linear equations with a Hankel matrix. For scalar and vector linear systems, prior upper and lower bounds on the observer dimension are derived, which makes it possible to switch to an iterative procedure of finding an optimal solution. The discussion is set out for discrete-time dynamical systems.     

Estimates of reachable sets of multidimensional control systems with nonlinear interconnections

The paper is devoted to the problem of constructing external estimates for the reachable set of a multidimensional control system by means of vector estimators. A system is considered that permits a decomposition into several independent subsystems with simple structure (for example, linear subsystems), which are connected to each other by means of nonlinear interconnections. For each of the subsystems, an external estimate of the reachable set is assumed to be known; this estimate is representable in the form of a level set of some function satisfying a differential inequality. An estimate for the reachable set of the combined system is constructed with the use of estimates for subsystems. The method of deriving the estimates is based on constructing comparison systems for analogs of vector Lyapunov functions (value functions).     

On a class of control problems with incomplete information

For a class of nonlinear control systems with constrained control, we consider a terminal control problem of reaching a target point in which the initial point of the process and the vector of parameters of the system belong to known sets and there is no information on which point from the set of initial states is true and which parameter of the system from the set of parameters is true. We establish sufficient conditions for the existence of a solution to the problem in the class of guaranteeing program packages of Yu.S. Osipov and A.V. Kryazhimskiy. We also present the results of calculation of a model example.     

Pattern recognition with the help of quadratic discriminant functions

The problem of pattern recognition with the help of spherical and elliptic discriminant functions is studied; in so doing the pattern of an object is assumed to be a vector of its characters from a finite-dimensional Euclidean space. Using a conformal mapping of a punctured sphere onto the plane as well as the inversion transformation, a criterion for the error-free recognition of two sets containing a finite number of points of training samples is obtained with the help of spherical discriminant functions. An algorithm for solving approximately a problem of construction of an ellipsoid of the minimal volume containing a given finite set of points is described. Bibliography: 5 titles. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 80, 1996, pp. 90–105.     

On the solution of the trajectory survival problem for a nonlinear dynamical system

Nonlinear control systems possessing the flatness property are encountered in many applied mathematical models. In this paper, a trajectory survival problem is considered for a specific nonlinear system that possesses the above property. A method based on the properties of the system is proposed for constructing a control that solves the trajectory survival problem when the controlled object moves to the goal set within a bounding set containing an obstacle. Results of numerical calculations of the control and the trajectory of a system with a given initial position are presented.     

Feedback Stabilization Methods for the Solution of Nonlinear Programming Problems

In this work, we show that, given a nonlinear programming problem, it is possible to construct a family of dynamical systems, defined on the feasible set of the given problem, so that: (a) the equilibrium points are the unknown critical points of the problem, which are asymptotically stable, (b) each dynamical system admits the objective function of the problem as a Lyapunov function, and (c) explicit formulas are available without involving the unknown critical points of the problem. The construction of the family of dynamical systems is based on the Control Lyapunov Function methodology, which is used in mathematical control theory for the construction of stabilizing feedback. The knowledge of a dynamical system with the previously mentioned properties allows the construction of algorithms, which guarantee the global convergence to the set of the critical points.     

On comparison of approximate solutions in vector optimization problems

The problem of comparison of approximations (approximate solutions to a vector optimization problem) obtained using different numerical methods is considered. In the absence of a priori information about the set of weakly efficient vectors, a scalar function is introduced that enables pair-wise comparison of approximations and establishes a binary preference relation according to which the approximations close (in the sense of the Hausdorff distance) to the set containing all possible efficient vectors are preferable to other approximations.     

Control of a family of nonlinear dynamic systems under measurements with bounded disturbances

We consider a control synthesis problem for nonlinear dynamic systems under parametric uncertainty and bounded measurement noises. Because of bounded disturbances in measurements of the state vector and the nonlinearity in the control object, the initially formulated control synthesis problem for a family of nonlinear systems as a generalized Zubov problem is transformed into a symbiosis of generalized Zubov–Bulgakov problems. The main result of the paper is the analytic solution of a minimax synthesis problem, which yields a constructive method for finding an invariant set.     

Order Reduction of General Nonlinear DAE Systems by Automatic Tearing

The object-oriented approach to modelling has recently made possible to build models of large-scale real systems. However, the resulting system of equations is generally a nonlinear DAE (Differential Algebraic Equations) system of large dimension, which must be reduced in some way to make it tractable for numerical solutions. A way to do this is automatic symbolic tearing, which aims at splitting the DAE system into two parts: a core consisting of a reduced implicit DAE system and a set of explicit assignments. The problem is here dealt with by a graph theoretic approach, first proving the NP-completeness in the general case, then formulating the problem with reference to a bipartite graph and finally defining an efficient and flexible algorithm for automatic tearing. It is also shown how the proposed algorithm can easily incorporate both general and domain-specific heuristic rules, and can also be used to deal with equations in vector form. The application to serial multibody systems is considered as a significant example.     

Robust inversion algorithms for vector linear systems

We consider the inversion problem for linear systems, which involves estimation of the unknown input vector. The inversion problem is considered for a system with a vector output and a vector input assuming that the observed output is of higher dimension than the unknown input. The problem is solved by using a controlled model in which the control stabilizes the deviations of the model output from the system output. The stabilizing model control or its averaged form may be used as the estimate of the unknown system input. __________ Translated from Nelineinaya Dinamika i Upravlenie, No. 4, pp. 17–22, 2004.     

Relative controllability of linear stationary completely integrable Pfaff systems

We consider a process described by stationary completely integrable Pfaff systems linear in the input and the control as well as in the output and the state. We study the state controllability properties of the Pfaff system for the case in which an arbitrary initial state (a constant vector) and an arbitrary terminal state (either a constant vector, or an analytic vector function, or a constant vector and an analytic vector function) are given. The terminal states belong to a subspace of the state space of the system. The entire set of Pfaff systems splits into three classes, in each of which we obtain conditions for the controllability of Pfaff systems in the class of analytic controls. The conditions are stated in terms of the rank of some matrices formed from the given matrices of the Pfaff system and the matrix that specifies the subspace of terminal states.     

Thermoelectric problem for an axisymmetric ellipsoid particle in the liquid metal: Analytical solution and numerical modeling

A thermo-electric problem is solved analytically for an electrically conducting particle in a form of an ellipsoid of revolution immersed in the liquid metal and subjected to a temperature gradient. It is shown that the density of the thermoelectric current is constant inside the particle and its value depends on the eccentricity of the ellipse in the meridian plane of the ellipsoid, but does not depend on the size of the particle. Another parameter which affects the value of the thermoelectric current is the orientation of the ellipsoid with respect to the imposed temperature gradient. The vector of the thermoelectric current inside the particle and the vector of the imposed thermal gradient are co-planar, but a planar angle between these vectors exist and its value is also a function of the eccentricity of the ellipse and its orientation in a thermal field. Limiting minimal and maximal value of the thermoelectric current inside a very elongated particle are found and compared with values obtained in simulations for a dendrite grain. Numerical simulation performed with FEM software for two orientations of an elongated ellipsoid with respect to the imposed thermal gradient provided results similar to analytical solutions with the relative error less than 0.1%.     

A computer realization of control-with-guide procedures

Positional differential games of pursuit with target for-conflict-control systems, non-linear with respect to the phase vector, are considered. The problems investigated are approximate construction of the set of positional absorption and construction of control procedures guaranteeing guidance to the target. Issues relating to the development of algorithms for the approximate construction of the positional absorption set and control-with-guide procedures [1–4] are also examined. As an example to illustrate the possibilities of the algorithms considered, a pursuit game of the Homicidal Chauffeur type [5] is considered, with approximate computation of positional absorption sets in the problem of pursuit over a fixed time interval for several parameter sets. Motions of a conflict-control system that generate a control-with-guide procedure are computed for several specific initial values of the phase vector and several choices of the evader's control.     

The Canonical Theory of the Impulse Process Optimality

The paper is devoted to the development of the canonical theory of the Hamilton–Jacobi optimality for nonlinear dynamical systems with controls of the vector measure type and with trajectories of bounded variation. Infinitesimal conditions of the strong and weak monotonicity of continuous Lyapunov-type functions with respect to the impulsive dynamical system are formulated. Necessary and sufficient conditions of the global optimality for the problem of the optimal impulsive control with general end restrictions are represented. The conditions include the sets of weak and strong monotone Lyapunov-type functions and are based on the reduction of the original problem of the optimal impulsive control a finite-dimensional optimization problem on an estimated set of connectable points.     

New estimations for ultimate boundary and synchronization control for a disk dynamo system

We consider globally exponentially attractive sets and synchronization control for a disk dynamo system. First, based on generalized Lyapunov function theory and the extremum principle of function, we derive some new 4D ellipsoid estimations and a polydisk domain estimation of the globally exponentially attractive set of a 4D disk dynamo system without existence assumptions. Our results improve existing results on the globally exponentially attractive set as special cases and can lead to a series of new estimations. Second, we propose linear feedback control with a single input or two inputs to realize globally exponential synchronization of two 4D disk dynamo systems using inequality techniques. Some new sufficient algebraic criteria for the globally exponential synchronization of two 4D disk dynamo systems are obtained analytically. The controllers designed here have a simple structure and less conservation. Finally, numerical simulations are presented to show the effectiveness of the proposed chaos synchronization scheme.     

Conditions for unique solvability of inverse problems for controllable dynamical systems

Problems arising in the applications of control theory of dynamical systems, when part of the variables of the mathematical model of the system is unknown and is subject to determination from information on the output of the system, are considered. Fundamental among them are the observation and identification problems, when unknown are the state of the system and its parameters, respectively, and also the problem of inverting the system, in which one seeks the control. Based on an analysis of mappings that are generated by an extended measurement vector, conditions for the unique solvability of the afore-mentioned problems with respect to a single trajectory or a set of trajectories are obtained.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 10, pp. 1359–1366, October, 1992.     

The Neumann problem on ellipsoids

The Neumann problem on an ellipsoid in \(\mathbf {R}^n\) asks for a function harmonic inside the ellipsoid whose normal derivative is some specified function on the ellipsoid. We solve this problem when the specified function on the ellipsoid is a normalized polynomial (a polynomial divided by the norm of the normal vector arising from the definition of the ellipsoid). Specifically, we give a necessary and sufficient condition for a solution to exist, and we show that if a solution exists then it is a polynomial whose degree is at most the degree of the polynomial giving the specified function. Furthermore, we give an algorithm for computing this solution. We also solve the corresponding generalized Neumann problem and give an algorithm for computing its solution.     

On the Optimal Stabilisation for a set of Programmed Motions of the Bilinear Control System

During recent years there has been considerable interest in using bilinear systems [1, 2] as mathematical models to represent the dynamic behavior of a wide class of engineering, biological and economic systems. Moreover, there are some methods [3] which may approximate nonlinear control systems by bilinear systems. For the first time Zubov has proposed a method of stabilization control synthesis for a set of programmed motions in linear systems [4]. In papers [5, 6] this method has been developed and used to solve the same problem for bilinear systems. In the present paper the following problems are considered. First, synthesis of nonlinear control as feedback under which the bilinear control system has a given set of programmed and asymptotic stable motions. Because this control is not unique, the second problem concerns optimal stabilization. In this paper a method for the design of nonlinear optimal control is suggested. This control is constructed in the form of a convergent series. The theorem on the sufficient conditions to solve this problem is represented.     

To the question of the minimal number of inputs for linear differential algebraic systems

We examine a control linear system of ordinary differential equations with an identically degenerate matrix coefficient of the derivative of the unknown vector function. We study the question of the minimal dimension of the control vector when the system could be fully controllable on any segment in the domain of definition. The problem is investigated in the cases of stationary systems and the systems with real analytic and smooth coefficients for which some structural forms can be defined.