Alexander P. Krishchenko

• Dr. (Dr.phys.-math. sci.)
• Bauman Moscow State Technical University

In this paper, we study various types of ultimate dynamics for the 5-dimensional pancreatic cancer model without treatment. This model has been created by Hu et al. and describes interactions between pancreatic cancer cells (PCCs); pancreatic stellate cells (PSCs); immune cells and two types of cytokines, tumour–promoting and tumour–suppressing. Us...

A functional method of localization has proved to be good in solving the qualitative analysis problems of dynamic systems. Proposed in the 90s, it was intensively used when studying a number of well-known systems of differential equations, both of autonomous and of non-autonomous discrete systems, including systems that involve control and / or dis...

In this paper, using the localization method of compact invariant sets, we examine the ultimate dynamics of the 3D prey–predator model containing two subpopulations of susceptible and infected predators. Our attention is focused to finding ultimate sizes of interacting populations, and, in addition, we show the existence of a global attracting set....

Two approaches to the use of families of functions when solving localization problems for autonomous systems are considered. The one approach, based on the so-called iteration procedure, is more difficult to implement than another, but gives a more accurate estimate of compact invariant sets. Application of the iteration procedure is illustrated on...

In this paper we study the phenomena of the extinction and persistence of predator populations of the three-dimensional Kooi et al. model in the global formulation of the problem. This model contains three populations: prey, susceptible predators and infected predators. We compute ultimate sizes of interacting populations and establish that all bio...

The paper is devoted to the qualitative analysis of a nonautonomous Duffing equation with nonlinearity in the form of a monomial of odd degree. For all values of the parameters, compact localizing sets containing all compact invariant sets of the system are constructed. The behavior of the trajectories of the system outside the localizing set is an...

The method of localization of invariant compact sets is used to study the properties of solutions of the Levinson-Smith equation with or without bounded disturbances. Necessary and sufficient conditions for the existence of localizing functions with a bounded universal section are obtained. Conditions for the existence of a bounded localizing set a...

This paper deals with trajectory tracking control of a quadcopter in a horizontal plane. A full rigid body model of the flying vehicle that doesn’t assume smallness of the Euler angles is considered. For synthesis of the tracking control the nonlinear dynamics inversion approach is used. Three different control strategies with restrictions on the o...

In this paper we consider the Duffing equation with bounded unknown external forcing. To examine long-time behavior of its solutions we apply the localization method of compact invariant sets. Obtained results are illustrated by two numerical examples. In the first example the Duffing equation is an autonomous one because it has no external forcing...

In this note we deal with control of a quadrocopter in a horizontal plane under state constraints in the form of a labyrinth. An algorithm to construct the programmed motion of the quadrocopter in a flat labyrinth is proposed. A full rigid body model of the flying vehicle that doesn’t assume smallness of the Euler angles is considered. For synthesi...

This paper deals with missile longitudinal dynamics control. The angle of attack tracking problem is considered for an unpowered flight phase of a short range tail-controlled missile with the zero engine thrust. For the control synthesis, a simplified missile dynamics model that includes the actuator dynamics is used. The tracking control law is de...

In this paper, we deal with the problem of aircraft take-off control in the presence of a windshear. A simplified point mass nonlinear model of aircraft dynamics assuming the flight in a vertical plane is used. A reference trajectory is constructed to satisfy the altitude, relative path inclination and relative velocity state constraints. An integr...

Finitely many embedded localizing sets are constructed for invariant compact sets of a time-invariant differential system. These localizing sets are used to divide the state space into three subsets, the least localizing set and two sets called sets of the first kind and the second kind. We prove that the trajectory passing through a point of the s...

In this paper we examine ultimate dynamics of the four-dimensional model describing interactions between tumor cells, effector immune cells, interleukin -2 and transforming growth factor-beta. This model was elaborated by Arciero et al. and is obtained from the Kirschner-Panetta type model by introducing two various treatments. We provide ultimate...

In this paper we examine ultimate dynamics of the four-dimensional model describing interactions between tumor cells, effector immune cells, interleukin -2 and transforming growth factor-beta. This model was elaborated by Arciero et al. and is obtained from the Kirschner-Panetta type model by introducing two various treatments. We provide ultimate...

It is well known that simple and complex dynamics of a nonlinear system are separated by a localizing set that contains all compact invariant sets and corresponds to a function in the phase space of the system. This separation means that, in the complement of the localizing set, the trajectory behavior of the system admits a standard description in...

We suggest a new method for constructing Lyapunov functions for autonomous systems of differential equations. The method is based on the construction of a family of sets whose boundaries have the properties typical of the level surfaces of Lyapunov functions. These sets are found by the method of localization of invariant compact sets. For the resu...

In this paper we consider the ultimate dynamics of the Kirschner-Panetta model which was created for studying the immune response to tumors under special types of immunotherapy. New ultimate upper bounds for compact invariant sets of this model are given, as well as sufficient conditions for the existence of a positively invariant polytope. We esta...

In this paper, we examine the problem of construction of Lyapunov functions for asymptotically stable equilibrium points. We exploit conditions of asymptotic stability in terms of compact invariant sets and positively invariant sets. Our results are methods of verification of these conditions and construction of Lyapunov functions by the localizati...

In this paper, we examine the problem of compact invariant sets localization for continuous-time nonlinear dynamical systems with uncertainties. The uncertainties in the system can reflect an approximate character of a mathematical model. In this context there arises the problem of constructing the localizing sets which don’t depend on possible ina...

We consider the terminal control problem for affine dynamical systems that are differentially flat. Two different analytical approaches are proposed in the presence of state constraints. One is based on the parametric set of functions that satisfy an integral equation. The other one utilizes time polynomials that are monotonic on the relevant time...

This paper deals with asymptotic stability criteria for equilibrium points of nonlinear autonomous systems formulated in terms of compact invariant sets and positively invariant sets. To verify these criteria the functional method of compact invariant sets localization is used. The obtained results can be applied to show the asymptotic stability pr...

The method of localization of invariant compact sets was proposed to study for asymptotic stability the equilibrium points of an autonomous system of differential equations. This approach relies on the necessary and sufficient conditions for asymptotic stability formulated in terms of positive invariant sets and invariant compact sets, and enables...

We analyze the method of localization of invariant compact sets in the case where the right-hand side of an autonomous system includes an uncertainty. We consider two cases depending on the current state of an autonomous system: the uncertainty on the right-hand side of the system is expressed by a constant parameter or a parameter varying in time....

We study the asymptotic stability and the global asymptotic stability of equilibria of autonomous systems of differential equations. We prove necessary and sufficient conditions for the global asymptotic stability of an equilibrium in terms of invariant compact sets and positively invariant sets. To verify these conditions, we use some results of t...

The localization method, which makes it possible to find regions in the phase space that contain all attractors of the system is used to analyze a phase system. Systems of inequalities describing such sets have been obtained. Phase-lock systems of the fourth and third order, which allow existence of chaotic attractors of various types, have been in...

The problem is considered of finding domains in the phase space in which the trajectories of a system have a fairly simple behavior determined by a typical scenario. The problem is solved by applying the method of localization of compact invariant sets of the system. It is proved that localizing sets separate simple and complex dynamics of nonlinea...

The asymptotic stability and global asymptotic stability of equilibria in autonomous systems of differential equations are analyzed. Conditions for asymptotic stability and global asymptotic stability in terms of compact invariant sets and positively invariant sets are proved. The functional method of localization of compact invariant sets is propo...

The Gray — Scott system used as a model of three-component autocatalytic reaction X + 2Y → Y, Y → Z, demonstrates the complex behavior and was studied in a number of papers. In this work we refine the known results of the bifurcation analysis of the Gray — Scott system and show them by diagram in parameter space. In addition we construct localizing...

The localization method of the compact positively invariant sets for the nonautonomous discrete-time systems is proposed. The features of the corresponding localizing sets are formulated. By means of this method the localizing sets are obtained for the compact invariant sets of the nonautonomous variant of discrete-time Cathala system.

Conditions for asymptotic stability and global asymptotic stability of equilibrium points of nonlinear time-invariant systems are obtained in terms of compact invariant sets and positively invariant sets. To verify these conditions the functional localization method of compact invariant sets is proposed. The obtained results can be applied to inves...

In this paper, we consider the problem of compact invariant sets localization for the Chua system. To obtain our results we develop and apply a localization method. This method allows us to find two types of subsets in the phase space of a nonlinear system. The first type consists of Poincaré sections having a nonempty intersection with any compact...

In this paper we analyze some features of global dynamics of a three-dimensional chronic myelogenous leukemia (CML) model with the help of the stability analysis and the localization method of compact invariant sets. The behavior of CML model is defined by concentrations of three cellpopulations circulating in the blood: naive T cells, effector T c...

The functional method of localization of invariant compact sets developed for continuous- and discrete-time dynamical systems is extended to families of discrete-time systems. Positively invariant compact sets are considered. As an example, the method is applied to the Hénon system with uncertain parameters.

This paper deals with the zero equilibrium stabilization problem for affine systems that have control input singularities. We consider a class of scalar input systems written in a canonical form with the input coefficient vanishing to zero on a set of points in the state space that includes the origin. The necessary and sufficient conditions are ob...

We consider problems of finding subsets (localizing sets) that contain all compact invariant sets in the state spaces of time-invariant and time-varying systems of differential equations. We describe the behavior of trajectories outside localizing sets corresponding to localizing functions. We obtain conditions under which the phase portrait of the...

A new method for solving terminal control problems for dynamical systems is announced. In the method the initial system is supplemented with equations for the derivatives of the control and the terminal problem is restated as two associated Cauchy problems. In the case of flat systems, the method generalizes a previously used approach. By choosing...

The paper considers a problem of the time-specified control terminal for the second order system with restrictions on the state variables.Most developed methods for solving problems of the terminal [1, 2, 3, 4, 5] do not allow us to take into account the restrictions on the system condition. To solve such problems are widely used methods based on t...

We consider the problem of practical stabilization of bilinear third-order dynamical systems by stationary state feedback. For bilinear systems in canonical form, we suggest to generalize the method of feedback linearization on the basis of a feedback of variable structure. This generalization is used for the derivation of sufficient conditions for...

Anti-angiogenesis therapy is an alternative and successfully employed method for treatment of cancerous tumour. However, this therapy isn't widely used in medicine because of expensive drugs. It leads naturally to elaboration of such treatment regimens which use minimum amount of drugs.The aim of the paper is to investigate the model of development...

Terminal control problem with fixed finite time for the second order affine systems with state constraints is considered. A solution of such terminal problem is suggested for the systems with scalar control of regular canonical form.In this article it is shown that the initial terminal problem is equivalent to the problem of auxiliary function sear...

This paper deals with the zero equilibrium stabilization problem for affine systems that have control input singularities. We consider a class of scalar input second-order systems written in a canonical form with the input coefficient vanishing to zero on a set of points in the state space that includes the origin. The necessary and sufficient cond...

One of the approaches to solving terminal control problems for affine dynamical systems is based on the use of polynomials of degree 2n − 1, where n is the order of the system in question. In this paper, we investigate the terminal control problem for which the final state of the system coincides with the origin in the phase space. We seek a set of...

In this paper, we propose a new method and algorithms that allow us to design complex spatial trajectories for an unmanned aerial vehicle (UAV) passing through a given sequence of waypoints in the three-dimensional space.The nonlinear six-dimensional model of the UAV center-of-mass motion given in the trajectory frame is used for calculations. The...

In the last 15 years one way for a qualitative analysis of dynamical systems was formed i.e. the localization of invariant compact sets of a dynamical system. Here the localization means creating a system of such sets, which contain all invariant compact sets of a dynamic system [1], in the phase space.Invariant compact sets are closely connected w...

In this paper, we examine the localization problem of compact invariant sets of Raychaudhuri equations with nonzero parameters. The main attention is attracted to the localization of periodic/homoclinic orbits and homoclinic cycles: we prove that there are neither periodic/homoclinic orbits nor homoclinic cycles; we find heteroclinic orbits connect...

We consider three types of nonsingular smooth transformations of nonlinear systems with control. These transformations are induced by changes of variables in the state space and in the control space and by changes of the independent variable. We introduce the notions of integrable and nonintegrable changes of the independent variable (time scalings...

In this paper we study some features of global behavior of one three-dimensional tumour growth model obtained by de Pillis and Radunskaya in 2003, with dynamics described in terms of densities of three cells populations: tumour cells, healthy host cells and effector immune cells. We find the upper and lower bounds for the effector immune cells popu...

For affine systems that cannot be linearized by feedback, conditions are proved for transforming them into equivalent linear stationary control systems by applying changes of the independent variable and feedback linearization.

We suggest a method for solving the terminal control problem for multidimensional affine systems that are not linearizable by feedback. We prove a sufficient condition for the existence of a solution and present a numerical procedure for its construction.

One approach to the solution of terminal problems is to transform the system under consideration to a special form for which solution methods are available for the terminal problem. Affine systems that can be feedback linearized were considered by the first author and A. A. Zhevnin [Sov. Phys., Dokl. 26, 559–561 (1981); translation from Dokl. Akad....

Given a linear stationary system x ˙=Ax+Bu, the condition for its controllability is known to coincide with the condition for its equivalence to a system in the Brunovsky canonical form, namely, the rank of the controllability matrix is equal to the dimension of the state vector. The concepts of the controllability matrix A.A. Zhevnin and A.P. Kris...

Consider the problem of localizing compact controllability sets of the nonlinear discrete-time system x n+1 =F(x n ,u n ), where F:X×U→x is a continuous mapping on X for any u∈U and u n ∈U is the control. By localization, we man construction of sets in the phase space of the system that contain all its compact controllability invariant sets. A meth...

We consider the stabilization problem for the zero equilibrium of bilinear and affine systems in canonical form. We obtain necessary and sufficient conditions for the stabilizability of second-order bilinear and affine systems in canonical form and generalize these conditions to the case of simultaneous stabilization of a family of bilinear systems...

In the behavior of various dynamical systems, an important role is played by bounded trajectories, which include equilibria, limit cycles, separatrices, and attractors. The estimation of the position of trajectories leads to so-called localization problems. In these problems, the estimation of the position of a bounded trajectory is interpreted as...

We consider the problem of stabilizing the zero equilibrium of canonical bilinear systems. For a second-order bilinear system, necessary and sufficient stabilizability conditions are obtained, which are extended to the simultaneous stabilization of a family of such systems and to an nth-order bilinear system.

A class of problems that may be characterized as localization problems are becoming increasingly popular in qualitative theory of differential equations [1–15]. The specific formulations differ, but geometrically all search for phase space subsets with desired properties, e.g., contain certain solutions of the system of differential equations. Such...

In this paper we examine the existence problem for maximal compact positively invariant sets of discrete-time nonlinear systems. To find these sets we propose the method based on the localization procedure of compact positively invariant sets. Analysis of a location of compact positively invariant sets and maximal compact positively invariant set o...

In this paper, we examine the localization problem of compact invariant sets of discrete-time nonlinear systems. The localization procedure consists in applying the iterative algorithm based on the extremum condition. An analysis of a location of compact invariant sets of the Henon system is realized for all values of its parameters.

Consider the problem of localizing compact invariant sets of the nonlinear discrete-time system x n+1 =f(x n ,w n ), where F:X×W→X is a continuous mapping and w n ∈W are disturbances. By localization, we mean the construction of sets in the phase space of the system that contain all the compact invariant sets. A method for constructing localizing s...

Necessary and sufficient conditions for an autonomous discrete dynamical system to have a maximal positively invariant compact set are presented and a method for determining this set is proposed. An an arbitrary set is considered and a sequence of nested sets is obtained. If the set is compact and contains all the positively invariant compact sets...

In our paper we study the localization problem of compact invariant sets of the system modelling the Rayleigh-B´ enard convection. Our results are based on using the first order extremum conditions, quadratic localizing functions and the symmetrical prolongation constructed between the Lorenz system and this sys- tem.

We generalize the notion of the minimal realization from the linear theory to the nonlinear case. We justify a relationship of this notion with the notion of first integral of a system. We suggest algorithms for the computation of first integrals and minimal realizations.

Nonlinear input-output mappings and their minimal realizations are studied. The state variables are eliminated from the system of differential equations by expressing the state variables in terms of the output and input coordinate functions and their derivatives. Equation for the input-output mapping locally admits a realization if and only if the...

In our paper we study the localization problem of compact invariant sets of nonlinear systems. Methods of a solution of this problem are discussed and a new method is proposed which is based on using symmetrical prolongations and the first-order extremum condition. Our approach is applied to the system modeling the Rayleigh-Bénard convection for wh...

We propose a method for localizing compact invariant sets of discrete dynamical systems. The method is based on the construction of a localizing set, i.e., the set containing all the invariant compact sets of a discrete system. The properties of localizing sets are described. The method is applied to a onedimen� sional discrete system. Of interest...

The problem under consideration is planning of spatial trajectories for a flying vehicle. The methods are based on the six-dimensional model with the longitudinal overload, transversal overload and the roll angle as controls. The class of trajectories with the monotone variation of the mechanical energy of a flying vehicle is considered. The method...

We consider problems of transition between plant descriptions in the form of input-output equations and in terms of state space variables. Related necessary and sufficient conditions and algorithms are obtained. Examples are given.

We generalize the localization method for invariant compact sets of an autonomous dynamical system to the case of a nonautonomous system of differential equations. By using this method, we solve the localization problem for the Vallis third-order dynamical system governing some processes in atmosphere dynamics over the Pacific Ocean. For this syste...

The problem of planning the trajectory of motion of a flying vehicle is considered using a six-dimensional model, in which the state is determined by three spatial coordinates, the value of the velocity, and two angles defining the orientation of the velocity vector. The coordinates of the vector of overloads acting on the flying vehicle are consid...

In this paper we examine the localization problem of compact invariant sets of nonlinear time-varying systems with the differentiable right-side. We extend our results respecting the localization problem obtained earlier for time-invariant systems and apply them to a damped driven pendulum and the Vallis model with regard to the seasonal cycle.

2 sin 2y (2) provided that y ∈ (−π/2; π/2). By integrating Eq. (2), we find the general solution y =a rctan(Ce t ). The figure represents the structure of solutions of Eq. (2). All of its solutions except for the identical zero are asymptotically stable, and the zero solution is unstable. Therefore, the stability of nonzero solutions in the variabl...

In this Letter we describe localization results of all compact invariant sets of a system modelling the amplitude of a plasma instability proposed by Pikovski, Rabinovich and Trakhtengerts. We derive ellipsoidal and polytopic localization sets for a number of domains in the 4-dimensional parametrical space of this system. Other localization sets ha...

A planar walking model is considered for a five-link biped robot. The normal form of the system describing the robot’s behavior in the single-support phase is constructed. The periodic motion problem is reduced to analyzing the system of zero dynamics equations. Simulation results are reported.

Conceptions of relative degree and minimum phase are connected to many control problems. To apply these conceptions to nonautonomous nonlinear systems one needs to know an output map which renders an nonautonomous affine system to be uniformly minimum phase. Necessary and sufficient conditions of the existence of such outputs are presented in the m...

In this paper, we examine the localization problem of compact invariant sets of systems with the differentiable right-side. The localization procedure consists in applying the iterative algorithm based on the first order extremum condition originally proposed by one of authors for periodic orbits. Analysis of a location of compact invariant sets of...

The problem of finding domains in the state space of a nonlinear system which contain all compact invariant sets is considered. Such domains are computed for the Lorenz system by using different localizing functions.

First Page of the Article

Consideration was given to asymptotic stabilization of the equilibria of nonlinear dynamic systems using the dynamic output feedbacks, that is, the feedbacks in the estimate of system state made by the asymptotic observer. Presented were the basic methods of constructing the asymptotic observers for the nonlinear dynamic systems with control and th...

In this paper we study the localization problem of periodic orbits of multidimensional continuous-time systems in the global setting. Our results are based on the solution of the conditional extremum problem and using sign-definite quadratic and quartic forms. As examples, the Rikitake system and the Lamb’s equations for a three-mode operating cavi...

Without Abstract

The problem of suppression of chaotic dynamics arises, in particular, if the trajectory of some system has a chaotic dynamics in a bounded domain and this dynamics should be destroyed by a minor intervention in the system. Such an intervention can be implemented by the introduction of small controls in the system. In the present paper, we suggest t...

Conceptions of a relative degree and a minimum phase system are connected with many control problems. In order to apply them it is necessary to know the output map for which the affine system takes on the minimum phaseness property. We present necessary and sufficient conditions of the existence of such outputs. In the case of the relative degree m...

Conceptions of relative degree and minimum phase are connected to many control problems. To apply it, one needs to know an input which renders an affine system minimum phase. We present necessary and sufficient conditions of the existence of such outputs and discuss its relations with backstepping method of stabilization. In the case of more than o...

For the problem of spatial reorientation of spacecraft from an arbitrary initial state to a final rest state, a programme trajectory and its control are found. This trajectory is optimized with respect to a given criterion. The solution is based on the concept of the inverse dynamic problem and on the description of motion in the quaternion form. A...

We analyze a third-order piecewise-linear system that describes the operation of an electronicoscillator. This system can be used to investigate various scenarios for the onset of chaos, in particular, through separatrix loop destruction. Necessary and sufficient conditions are obtained for the existence of an equilibrium separatrix loop. Calculati...

We study classical and higher infinitesimal symmetries of control systems. Defining equations for classical external symmetries are obtained in the general and affine cases. For computing higher symmetries we suggest a simple procedure involving algebraic operations and differentiation but not integration. Relations between clas- sical symmetries a...

For a dynamical system with single output and an affine system with single input the k(x)-duality property is introduced. Necessary and sufficient conditions are given for two above systems to be k(x)-dual. It is shown that controllability properties and observability properties of k(x)-dual systems are closely related to each other. Transformation...

For a dynamical system with single output and an affine system with single input the k(x)-duality property is introduced. Necessary and sufficient conditions are given for two above systems to be k(x)-dual. It is shown that controllability properties and observability properties of k(x)-dual systems are closely related to each other. Transformation...