The mathematical theory of control, essentially developed during the last decades, is used for solving many problems of practical importance. The efficiency of its applications has increased in connec
For unperturbed and perturbed Kolmogorov models of population dynamics, new sufficient conditions of stability and boundedness with respect to two criteria (measures) are provided. An approach to treating Kolmogorov models of population dynamics via two measures was suggested. This approach worked out for both ordinary and partial differential equations allows conditions to obtain sufficient for the Kolmogorov models to possess various dynamical properties. This chapter continues the investigation in this direction and provides new stability and boundedness conditions for Kolmogorov-type ordinary differential equations. It introduces unperturbed and perturbed Kolmogorov models and takes multiplicative and additive perturbations into account. The chapter formulates boundedness and stability conditions with respect to two measures. It applies these conditions for the analysis of a generalized Lotka-Volterra model.
In this chapter, the author begins with a discussion of dynamic programming techniques for the viability problem. He deals with two schemes for the calculation of information (consistency) sets in the guaranteed state estimation problem, as well as of viability kernels, with the aim of presenting solution schemes that appear to be somewhat different from those introduced earlier. A similar dynamic programming scheme allows to be applied to the calculation of information (consistency) sets for the set-membership (bounding) approach to the state-estimation problem. It is important to emphasize that the dynamic programming schemes show close connections with the approaches to uncertain systems based on Liapunov functions.
Dynamical game interactions are relevant to both differential and evolutionary game-theoretical models. This chapter introduces a notion of a dynamical Nash equilibrium in a class of feedback controls. Feedbacks driving the coalitions to the classical "punishment" solutions in static bimatrix games give a natural and elementary example of a dynamical Nash equilibrium. The chapter proposes another solution which provides a better (at least not worse) long-term result. Our approach originates from theory of positional differential games and rests on the idea of optimal guaranteeing feedbacks in the associated zero sum games. The chapter defines relevant zero sum games and studies them within the framework of the theory of viscosity (minimax) solutions of Hamilton-Jacobi equations. It shows that the equilibrium trajectories generated under the optimal feedbacks stay, in the long run, within a domain in which the current payoffs to each coalition are better (no worse) than the payoff at a static Nash equilibrium point.
Let us consider a linear time invariant dynamic system described by the state equation 1 d x ( t ) d t = A x ( t ) + B u ( t ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078258/b3a0cd8e-6c4d-405e-8084-bab2fa1242ed/content/eqn10_1_B.tif"/>
This chapter increases our understanding of the cause and nature of the stability present in physical systems. The nonlinear model used here to study the role of coupling in stabilizing and synchronizing nonlinear systems is derived from population dynamics, though several of our results appear valid for general nonlinear maps. The chapter analyzes the effect of migration by studying the interactive dynamics of two subcolonies of a single species. It considers two interacting populations (colonies) of biological organisms, each of whose population dynamics is described by equations. The interaction between the colonies may be thought of as being brought about by migration between the two populations. The chapter investigates the effect of increasing the number of habitats that are coupled. It assumes that more than two habitats can be present on a ring and adjacent habitats are coupled through migration. The stabilization demonstrated is applicable to discrete dynamics in the form of maps.
This chapter aims to relax the requirement of the exponential stability of the nominal uncontrolled part of the system by making use of a version of the receding horizon control method. In a great part of the literature, continuous systems are considered. Stability issues for discrete time uncertain systems without any control constraint have been discussed. The chapter provides a neighbourhood of the origin, in which the controller can be given as the sum of two terms: one of them is a linear feedback for exponential stabilization of the nominal part of the system, the other is a nonlinear feedback for counteracting the uncertainty. It presents a method of stabilization of nonlinear, discrete-time uncertain systems in which the uncertainties are modeled deterministically rather than stochastically. The control has been subject to the hard constraint with a prespecified constant. The nonlinear feedback has been constructed by using Lyapunov stability theory.
The main objective discussed in [5] is the suppression of undesired noise or vibrations in dynamical systems which are modelled by an o.d.e. of the form 1 x ˙ = A x + B ( x ) u + e ( x , t ) with x ( 0 ) = x 0 ∈ IR n . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078258/b3a0cd8e-6c4d-405e-8084-bab2fa1242ed/content/eqn2_1_B.tif"/>
A dynamical model of the climate-biosphere system involving a carbon cycle is designed. A qualitative analysis is given for a model zero-dimensional in space. The climate of our hypothetical planet is described by a single variable, namely, the annual average temperature on its surface. The chapter considers a planet with vegetation and a "two-layer" atmosphere. Clouds and the underlying surface (vegetation and ocean can be taken into consideration) reflect the solar radiation, and the "greenhouse" gases transform it. In the present model we use a very primitive sub-model for the global carbon cycle. The presented climate-vegetation model involving a carbon cycle submodel shows some interesting features. The performed stability analysis of the system's equilibria gives us up to three stable points. The initiation of life on our virtual planet depends on the value of two bifurcation parameters, the total amount of carbon in the system and a combination of biotic characteristics of the vegetation.
In this chapter, the author focuses on a situation where the controls base essentially on the history of a motion. The problem is resolved via the construction of upper convex hulls for auxiliary functions in a multidimensional space. A method of reducing multidimensional constructions to operations in smaller dimensions is described. In many typical situations optimal strategies utilize partial information on current histories. Our objective is to provide an effective procedure for computing these values. The author gives a functional interpretation for the control process. The research described in this publication has been supported in part from Grant NMS300 of the International Science Foundation.
This chapter reviews some recent results on the economics of crime and punishment. In particular, a dynamic extension of Becker's static setup is proposed. To determine the optimal amount of enforcement one has to know the damage caused by the offenses and the response of offenders to changes in law enforcement, the cost of apprehending and convicting criminals, and the impact of the nature and the amount of punishments meted out. The 'upper equilibrium' is approached from south-west and north-east, respectively. This means that for a sufficiently large initial number of offenders the corresponding 'optimal' law enforcement rate is relatively high compared with the equilibrium enforcement rate, but gradually decreases. During the past several years, law enforcement in illicit drug markets has developed to growing field in economics and planning. Baveja et al. analyze enforcement programs of finite duration that minimize the total costs of crackdown, subject to constraint that the market is eliminated at the end of the program.
In this chapter, the authors adopt the Lyapunov min-max approach for robust control design for flexible joint manipulators. The main difficulty in directly applying the work in this area to flexible joint manipulators is the lack of the matching condition. The authors propose to overcome this difficulty by first introducing a state transformation. Uniform ultimate boundedness in practical stability means that the response of the system enters and remains within a particular neighborhood of the equilibrium position after some finite time. To design robust control, the authors propose a two-step procedure. This is also similar to Freeman and Kokotovic. A robust control scheme is proposed for flexible joint manipulators. The features of the manipulators, which distinguish the current problem from others, are that the dynamic system is nonlinear, uncertain, and mismatched. The analysis of the original system performance based on that of the transformed system is demonstrated.
In this chapter, the authors solve the definiteness problem for systems with polynomial and quadratic Lyapunov functions using a theorem of Ehlich and Zeller. A small subset of the region of attraction containing the origin is guaranteed with a rough estimation. Using a special bisection method, this subset is improved and inscribing and circumscribing equipotential surfaces which enclose the boundary of the subset of the region of attraction which can be guaranteed with the chosen Lyapunov function are computed. In the chapter, the authors describe the application of the theorems of Ehlich and Zeller and of Gartel to the computation of regions of attraction. They asymptotically stable stationary points of polynomial systems have been investigated. A new algorithm for the computation of a subset of the region of attraction based on a quadratic Lyapunov function has been presented. The work presented here has been extended to Lyapunov functions which are polynomials of degree 3 in the state variables.
The method of analytic centers known in convex programming is implemented for construction of paths leading to equilibrium points in mixed strategy bimatrix games. A bimatrix game is extended to a family of time-parametrized perturbed games in which the payoffs are logarithmically penalized for the approach to the boundary of the strategy space. Homotopy, or path following, optimization methods rest on the idea of approaching an optimum along the family of solutions of time-parametrized perturbed optimization problems. The initial perturbed problem is normally chosen simple enough so that the initial point on a solution path is easily identified, and the final problem is the unperturbed one. In a general path following methodology for finding equilibria — without a detailed analysis of the existence of the solution paths — was presented. In a homotopy method was in the base of a proof of the oddness of the number of equilibrium points in a bimatrix game.
A new approach to building dynamics in repeated 2 × 2 bimatrix games originating from theory of closed-loop differential games is presented, and a natural formalism for viewing typical players' behaviors such as normal, altruistic, aggressive and paradoxical is discussed. To specify behavior patterns incorporating players' objectives, in this chapter, the author invokes some ideas from theory of nonzero-sum closed-loop differential games. The analysis of the global game trajectories under the noncooperative and cooperative dynamics lies beyond the scope of the chapter. Preliminary arguments and simulations show that the cooperative dynamics is much richer in trajectory types than the noncooperative one. The consideration of incomplete and global information adds new dynamical phenomena. In the chapter, the author presents several simulation results. The author is grateful to Mrs. L. V. Kukushkina who created a (C language) program for the simulation of the noncooperative and cooperative game dynamics.
This chapter concerns robust control design of constrained parabolic systems under uncertain disturbances (perturbations) and feedback controllers in the mixed boundary conditions. It develops an effective multi-step approximation procedure to design suboptimal feedback controllers for constrained parabolic systems. The chapter contains new results for the case of mixed boundary controllers. The results obtained include a justification of a suboptimal three-positional control structure with subsequent optimization of its parameters. The chapter formulates the robust feedback control problem of our study and present the main properties of the parabolic dynamics used in the sequel. It devotes to solving first-order ODE approximation problems under maximal perturbations that allow us to justify a suboptimal structure of boundary controls in the parabolic system. The chapter deals with computing a feedback boundary controller that ensures the best system behavior under maximal perturbations and keeps transients within the required state constraint region for any admissible disturbances on a sufficiently large control interval.
Mixed constraints imposed on both state and control variables of a dynamical system, as well as integral constraints imposed on the variables, are often present in applications. Energy and heat restrictions are usually reduced to integral constraints. All these constraints are essential for robots controlled by electric actuators. In this chapter, the author extends the well-known Kalman's method originally developed for linear systems without control constraints to systems subject to mixed, state, and integral constraints. In Kalman's approach, the open-loop control is formed as a linear combination of the natural modes of the system. The author derives sufficient controllability conditions which ensure that the obtained control satisfies all imposed constraints and brings our system to the prescribed terminal state in finite time. The proposed technique is applied to a dynamical system of the fourth order which is a model for mechanical systems controlled by electric drives.
This paper is concerned with the design of a controller-observer scheme for the exponential stabilization of a class of singularly perturbed nonlinear systems. The controller design uses a sliding mode technique and is divided in two phases: slow feedback control and fast feedback control so that a final composite control is obtained. Assuming that only the fast state is available and the system's output is a function of the slow state, an observer design is presented. A stability analysis is also made to provide sufficient conditions for the ultimate boundedness of the full order closed-loop system when the slow state is estimated by means of the observer. An application to the model of a permanent magnet stepper motor is given to show the controller-observer methodology and stability analysis.
