Asymmetric games for convolution systems with applications to feedback control of constrained parabolic equations

The paper is devoted to the study of some classes of feedback control problems for linear parabolic equations subject to hard/pointwise constraints on both Dirichlet boundary controls and state dynamic/output functions in the presence of uncertain perturbations within given regions. The underlying problem under consideration, originally motivated by automatic control of the groundwater regime in irrigation networks, is formalized as a minimax problem of optimal control, where the control strategy is sought as a feedback law. Problems of this type are among the most important in control theory and applications — while most challenging and difficult. Based on the Maximum Principle for parabolic equations and on the time convolution structure, we reformulate the problems under consideration as certain asymmetric games, which become the main object of our study in this paper. We establish some simple conditions for the existence of winning and losing strategies for the game players, which then allow us to clarify controllability issues in the feedback control problem for such constrained parabolic systems.     

Representation of feedback operators for parabolic control problems

In this paper we present results on existence and regularity of integral representations of feedback operators arising from parabolic control problems. The existence of such representations is important for the design of low order compensators and in the placement of sensors. This paper extends earlier results of J. A. Burns and B. B. King to problems with spatial dimensions.     

Homogeneous feedback design of differential inclusions based on control Lyapunov functions

This paper is concerned with the stabilization of differential inclusions. By using control Lyapunov functions, a design method of homogeneous controllers for differential equation systems is first addressed. Then, the design method is developed to two classes of differential inclusions without uncertainties: homogeneous differential inclusions and nonhomogeneous ones. By means of homogeneous domination theory, a homogeneous controller for differential inclusions with uncertainties is constructed. Comparing to the existing results in the literature, an improved formula of homogeneous controllers is proposed, and the difficulty of the controller design for uncertain differential inclusions is reduced. Finally, two simulation examples are given to verify the preset design.     

Stabilization of nonlinear switched systems using control Lyapunov functions

In this paper, we study the stabilization of general nonlinear switched systems by using control Lyapunov functions. The concept of control Lyapunov function for nonlinear control systems is generalized to switched control systems. The first part of our contribution provides a necessary and sufficient condition of stabilization. The main idea is to use a common control Lyapunov function; this is achieved with the converse Lyapunov theorem dedicated to switched systems. In the second part, an explicit construction of a common control Lyapunov function is addressed with respect to a finite family of switched systems. The approach uses a family of control Lyapunov functions attached to the subsystems.     

Conley barriers and their applications: Chain-recurrence and Lyapunov functions

Given a compact metric space X and a continuous map f from X to itself, we construct a barrier function for chain-recurrence. We use it to endow the space of chain-transitive components with a non-trivial ultrametric distance and to construct Lyapunov functions for f. Most of these constructions are then generalized on an arbitrary separable metric space to a continuous compactum-valued map.     

Lyapunov functions and attractors in arbitrary metric spaces

We prove two theorems concerning Lyapunov functions on metric spaces. The new element in these theorems is the lack of a hypothesis of compactness or local compactness. The first theorem applies to a discrete dynamical system on any metric space; the result is that if is an attractor for a continuous map of a metric space to itself, then there is a Lyapunov function for . The second theorem applies only to separable metric spaces; the theorem is that there is a complete Lyapunov function for any continuously-generated discrete dynamical system on a separable metric space. (A complete Lyapunov function is a real-valued function that is constant on orbits in the chain recurrent set, is strictly decreasing along all other orbits, and separates different components of the chain recurrent set.)

     

Chaos control and chaos synchronization for multi-scroll chaotic attractors generated using hyperbolic functions

In this paper, we design a series of chaotic systems that can generate one-directional, two-directional and three-directional multi-scroll chaotic attractors. Then, based upon the properties of these chaotic systems, we construct appropriate Lyapunov functions and design simple linear feedback controls to globally exponentially stabilize and synchronize these chaotic systems. Numerical simulation results are also presented to show the applicability of the proposed control laws.     

Non-coercive Lyapunov functions for infinite-dimensional systems

We show that the existence of a non-coercive Lyapunov function is sufficient for uniform global asymptotic stability (UGAS) of infinite-dimensional systems with external disturbances provided the speed of decay is measured in terms of the norm of the state and an additional mild assumption is satisfied. For evolution equations in Banach spaces with Lipschitz continuous nonlinearities these additional assumptions become especially simple. The results encompass some recent results on linear switched systems on Banach spaces. Finally, we derive new non-coercive converse Lyapunov theorems and give some examples showing the necessity of our assumptions.     

Lyapunov stability for impulsive control affine systems

This article studies several notions of Lyapunov stability for impulsive control affine systems in the setting of nonautonomous dynamical systems. It presents some relations between the stability of an impulsive control affine system and the stability of its adjacent control system. Stability of compact sets and their components are specially investigated. Lyapunov functionals are employed to characterize each type of stability of closed sets.     

Realistic feedback control of turbogenerators

In this paper, the optimal control of a turboalternator connected to an infinite bus is considered. The alternator is controlled through a linear feedback of the state variables. The feedback parameters are obtained by solving a two-point nonlinear boundary-value problem. The values obtained for these parameters depend on the strength and duration of the disturbance, since the model is nonlinear, contrary to the usual feedback control of a linear model. In contrast to the model used in Ref. 1, the model used here include the transfer functions of the governor, the turbine, and the voltage regulator.This work was supported in part by the National Research Council of Canada, Grant No. A-4146.     

Nonlinear stability in reaction-diffusion systems via optimal Lyapunov functions

We define optimal Lyapunov functions to study nonlinear stability of constant solutions to reaction-diffusion systems. A computable and finite radius of attraction for the initial data is obtained. Applications are given to the well-known Brusselator model and a three-species model for the spatial spread of rabies among foxes.     

Switched safe tracking control design for unmanned autonomous helicopter with disturbances

In this paper, the switched safe tracking control scheme is investigated for the attitude and altitude system of a medium-scale unmanned autonomous helicopter with output constraints and unknown external disturbances. To keep the attitude angles and altitude within the desired constrained range, an output boundary protection approach is adopted to generate an output constrained trajectory which is piecewise differentiable. The disturbance observer-based control method is employed to handle the unknown external disturbances of the system. Because of the piecewise differentiability of the output constrained trajectory, the closed-loop error system with the safe tracking controller can be seen as a switched system with jump dynamics. The multiple Lyapunov function method is adopted to guarantee the tracking performance with designed average dwell time. Simulation results of an example are provided to illustrate the effectiveness of the proposed control scheme for the unmanned autonomous helicopter system.     

Asymptotically linear solutions of differential equations via Lyapunov functions

We discuss the existence of solutions with oblique asymptotes to a class of second order nonlinear ordinary differential equations by means of Lyapunov functions. The approach is new in this field and allows for simpler proofs of general results regarding Emden-Fowler like equations.     

Existence of piecewise affine Lyapunov functions in two dimensions

In Marinosson (2002) [10], a method to compute Lyapunov functions for systems with asymptotically stable equilibria was presented. The method uses finite differences on finite elements to generate a linear programming problem for the system in question, of which every feasible solution parameterises a piecewise affine Lyapunov function. In Hafstein (2004) [2] it was proved that the method always succeeds in generating a Lyapunov function for systems with an exponentially stable equilibrium. However, the proof could not guarantee that the generated function has negative orbital derivative locally in a small neighbourhood of the equilibrium. In this article we give an example of a system, where no piecewise affine Lyapunov function with the proposed triangulation scheme exists. This failure is due to the triangulation of the method being too coarse at the equilibrium, and we suggest a fan-like triangulation around the equilibrium. We show that for any two-dimensional system with an exponentially stable equilibrium there is a local triangulation scheme such that the system possesses a piecewise affine Lyapunov function. Hence, the method might eventually be equipped with an improved triangulation scheme that does not have deficits locally at the equilibrium.     

Revised CPA method to compute Lyapunov functions for nonlinear systems

The CPA method uses linear programming to compute Continuous and Piecewise Affine Lyapunov functions for nonlinear systems with asymptotically stable equilibria. In [14] it was shown that the method always succeeds in computing a CPA Lyapunov function for such a system. The size of the domain of the computed CPA Lyapunov function is only limited by the equilibrium?s basin of attraction. However, for some systems, an arbitrary small neighborhood of the equilibrium had to be excluded from the domain a priori. This is necessary, if the equilibrium is not exponentially stable, because the existence of a CPA Lyapunov function in a neighborhood of the equilibrium is equivalent to its exponential stability as shown in [11]. However, if the equilibrium is exponentially stable, then this was an artifact of the method. In this paper we overcome this artifact by developing a revised CPA method. We show that this revised method is always able to compute a CPA Lyapunov function for a system with an exponentially stable equilibrium. The only conditions on the system are that it is C²$C^2$ and autonomous. The domain of the CPA Lyapunov function can be any a priori given compact neighborhood of the equilibrium which is contained in its basin of attraction. Whereas in a previous paper [10] we have shown these results for planar systems, in this paper we cover general n-dimensional systems.     

Optimal feedback control policies for stochastic,distributed-parameter systems

This paper considers optimal feedback control policies for a class of discrete stochastic distributed-parameter systems. The class under consideration has the property that the random variable in the dynamic systems depends only on the time and possesses the Markovian property with stationary transition probabilities. A necessary condition for optimality of a feedback control policy, which has form similar to the Hamiltonian form in the deterministic case, is derived via a dynamic programming approach.     

Optimal control of sources on some classes of functions

In this work, we investigate the problems of the control of sources’ motion and power, which influence the state of the objects described by partial differential equations. The functions defining the sources’ operation are taken from different classes of functions that are easy to implement from the technical point of view. For numerical solution to the problems considered, we obtain and validate formulae for the gradient of a target functional, which allow using first-order optimization methods, e.g. gradient projection method.     

Global asymptotic stability in n-species non-autonomous Lotka–Volterra competitive systems with infinite delays and feedback control

A non-autonomous Lotka–Volterra competition system with infinite delays and feedback control and without dominating instantaneous negative feedback is investigated. By means of a suitable Lyapunov functional, sufficient conditions are derived for the global asymptotic stability of the system. Some new results are obtained.