a piecewise approximation approach to nonlinear systems stability and region of attraction this paper discusses piecewise bilinear models and recurrent fuzzy systems as particular classes of dynamic fuzzy systems which can be dealt with in the same framework due to their structural similarity as universal approximators they are capable of representing any continuous nonlinear system dynamics with arbitrary accuracy by means of a rule base first basic definitions of both systems are revisited here and unifying canonical forms are given then a stability criterion is derived for the system classes which is based on approximating piecewise quadratic lyapunov functions and formulated in terms of linear matrix inequalities the main contribution is the demonstration of the stability analysis and estimation of region of attraction by means of these models outperforming methods reported in the literature

frechet differentiability of the solutions of a semilinear abstract cauchy problem abstract sufficient conditions are found under which the solutions z t q of a semilinear abstract cauchy problem of the form d d t z t a q z t f q t z t are frechet differentiable with respect to the parameter q an explicit form is provided for the sensitivity equation satisfied by the frechet derivative d q z t q

boundary control of an unstable heat equation via measurement of domain averaged temperature in this note a feedback boundary controller for an unstable heat equation is designed the equation can be viewed as a model of a thin rod with not only the heat loss to a surrounding medium stabilizing but also the heat generation inside the rod destabilizing the heat generation adds a destabilizing linear term on the right hand side of the equation the boundary control law designed is in the form of an integral operator with a known continuous kernel function but can be interpreted as a backstepping control law this interpretation provides a lyapunov function for proving stability of the system the control is applied by insulating one end of the rod and applying either dirichlet or neumann boundary actuation on the other

boundary feedback stabilization of fisher s equation abstract the aim of this work is to design an explicit finite dimensional boundary feedback controller for locally exponentially stabilizing the equilibrium solutions to fisher s equation in both l 2 0 1 and h 1 0 1 the feedback controller is expressed in terms of the eigenfunctions corresponding to unstable eigenvalues of the linearized equation this stabilizing procedure is applicable for any level of instability which extends the result of barbu 2013 for nonlinear parabolic equations the effectiveness of the approach is illustrated by a numerical simulation

semidefinite programming relaxations for semialgebraic problems a hierarchy of convex relaxations for semialgebraic problems is introduced for questions reducible to a finite number of polynomial equalities and inequalities it is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility the main tools employed are a semidefinite programming formulation of the sum of squares decomposition for multivariate polynomials and some results from real algebraic geometry the techniques provide a constructive approach for finding bounded degree solutions to the positivstellensatz and are illustrated with examples from diverse application fields

convex computation of the region of attraction of polynomial control systems we address the long standing problem of computing the region of attraction roa of a target set e g a neighborhood of an equilibrium point of a controlled nonlinear system with polynomial dynamics and semialgebraic state and input constraints we show that the roa can be computed by solving an infinite dimensional convex linear programming lp problem over the space of measures in turn this problem can be solved approximately via a classical converging hierarchy of convex finite dimensional linear matrix inequalities lmis our approach is genuinely primal in the sense that convexity of the problem of computing the roa is an outcome of optimizing directly over system trajectories the dual infinite dimensional lp on nonnegative continuous functions approximated by polynomial sum of squares allows us to generate a hierarchy of semialgebraic outer approximations of the roa at the price of solving a sequence of lmi problems with asymptotically vanishing conservatism this sharply contrasts with the existing literature which follows an exclusively dual lyapunov approach yielding either nonconvex bilinear matrix inequalities or conservative lmi conditions the approach is simple and readily applicable as the outer approximations are the outcome of a single semidefinite program with no additional data required besides the problem description the approach is demonstrated on several numerical examples

the stability and exponential stabilization of the heat equation with memory abstract in this paper we investigate the stability of heat equation with a constant memory kernel we prove that the system does not decay for the positive kernel and is polynomially stable but not exponentially for the negative kernel in particular a more detailed description of the decay property for the negative kernel is obtained by dividing initial values into two categories moreover we show that the system can be exponentially stabilized by boundary control for any constant memory kernel

parameter differentiability of the solution of a nonlinear abstract cauchy problem in this article an abstract nonlinear evolution equation with a parameter appearing in the nonlinear part is considered by using the theory of analytic semigroups and a generalization of gronwall s lemma for singular kernels sufficient conditions to ensure differentiability of the solution with respect to the parameter are derived in terms of the smoothness of the nonlinear part of the equation the results are applied to a nonlinear system of partial differential equations modeling the dynamics of shape memory alloys

estimation of boundary parameters in general heterodirectional linear hyperbolic systems abstract in this paper we extend recent results on state and boundary parameter estimation in coupled systems of linear partial differential equations pdes of the hyperbolic type consisting of n rightward and one leftward convecting equations to the general case which involves an arbitrary number of pdes convecting in both directions two adaptive observers are derived based on swapping design where one introduces a set of filters that can be used to express the system states as linear static combinations of the filter states and the unknown parameters standard parameter identification laws can then be applied to estimate the unknown parameters one observer which requires sensing at both boundaries generates estimates of the boundary parameters and system states while the second observer estimates the parameters from sensing limited to the boundary anti collocated with the uncertain parameters proof of boundedness of the adaptive laws is offered and sufficient conditions ensuring exponential convergence are derived the theory is verified in simulations

robust region of attraction estimation we propose a method to compute invariant subsets of the region of attraction for asymptotically stable equilibrium points of polynomial dynamical systems with bounded parametric uncertainty parameter independent lyapunov functions are used to characterize invariant subsets of the robust region of attraction a branch and bound type refinement procedure reduces the conservatism we demonstrate the method on an example from the literature and uncertain controlled short period aircraft dynamics

local exponential stabilization of fisher s equation using the backstepping technique abstract this paper deals with the boundary stabilization problem for the 1d fisher s partial differential equation pde defined on a bounded interval which is a nonlinear unstable distributed parameter system the stabilizing controller derived using the backstepping technique for linear parabolic pdes associated with fisher s pde can be extended to the nonlinear case in the work we can prove that the boundary controller based on the linear backstepping synthesis approach can make the closed loop nonlinear fisher s system exponentially stable with any desired decay rate locally in both l 2 0 1 and h 1 0 1 respectively the effectiveness of the control synthesis approach is illustrated by a numerical simulation

stabilization and controllability of first order integro differential hyperbolic equations in the present article we study the stabilization of first order linear integro differential hyperbolic equations for such equations we prove that the stabilization in finite time is equivalent to the exact controllability property the proof relies on a fredholm transformation that maps the original system into a finite time stable target system the controllability assumption is used to prove the invertibility of such a transformation finally using the method of moments we show in a particular case that the controllability is reduced to the criterion of fattorini

on the estimation of asymptotic stability regions state of the art and new proposals this paper deals with the problem of the estimation of regions of asymptotic stability for continuous autonomous nonlinear systems the first part of the work provides a comprehensive survey of the existing methods and of their applications in engineering fields in the second part certain topological considerations are first developed and the trajectory reversing method is then presented together with a theorem on which it is based in the final part several examples of application are reported showing the efficiency of the proposed technique for low order second and third systems

a generalization of zubov s method to perturbed systems a generalization of zubov s theorem on representing the domain of attraction via the solution of a suitable partial differential equation is presented for the case of perturbed systems with a singular fixed point for the construction it is necessary to consider solutions in the viscosity sense as a consequence maximal robust lyapunov functions can be characterized as viscosity solutions

backstepping based boundary observer for a class of time varying linear hyperbolic pides in this paper a luenberger type boundary observer is presented for a class of distributed parameter systems described by time varying linear hyperbolic partial integro differential equations first known limitations due to the minimum observation time for simple transport equations are restated for the considered class of systems then the backstepping method is applied to determine the unknown observer gain term by avoiding the framework of gevrey functions which is typically used for the time varying case it is shown that the backstepping method can be employed without severe limitations on the regularity of the time varying terms a modification of the underlying volterra transformation ensures that the observer error dynamics is equivalent to the behavior of a predefined exponentially stable target system the magnitude of the observer gain term can be traded for lower decay rates of the observer error after the theoretic results have been proven the effectiveness of the proposed design is demonstrated by simulation examples

on the performance of nonlinear dynamical systems under parameter perturbation we present a method for analysing the deviation in transient behaviour between two parameterised families of nonlinear odes as initial conditions and parameters are varied within compact sets over which stability is guaranteed this deviation is taken to be the integral over time of a user specified positive definite function of the difference between the trajectories for instance the l 2 norm we use sum of squares programming to obtain two polynomials which take as inputs the possibly differing initial conditions and parameters of the two families of odes and output upper and lower bounds to this transient deviation equality can be achieved using symbolic methods in a special case involving linear time invariant parameter dependent systems we demonstrate the utility of the proposed methods in the problems of model discrimination and location of worst case parameter perturbation for a single parameterised family of ode models

disturbance rejection in general heterodirectional 1 d linear hyperbolic systems using collocated sensing and control abstract we solve a disturbance rejection problem for a general class of heterodirectional 1 d linear hyperbolic partial differential equations pdes the disturbance enters at one boundary while sensing and actuation is limited to the opposite boundary we do this by combining a separately designed controller and observer into an output feedback controller and proving stability of the overall closed loop system and exponential regulation of the control objective to zero the theory is demonstrated in a simulation

estimating the domain of attraction for uncertain polynomial systems estimating the domain of attraction da of equilibrium points is a problem of fundamental importance in systems engineering several approaches have been proposed for the case of known polynomial systems allowing one to find the largest estimate of the da leda for a given lyapunov function lf however the problem of estimating the robust da rda that is the da guaranteed for all possible uncertainties in an uncertain system it is still an unsolved problem in this paper lmi methods are proposed for estimating the rda in the case of systems depending polynomially in the state and in the uncertainty which is supposed to belong to a polytope specifically the issue of computing the robust leda rleda that is the intersection of all ledas is considered for common and parameter dependent lfs providing constant and parameter dependent lower bounds the computation of approximations with simple shape of the rleda in the case of parameter dependent lfs is also discussed

output feedback stabilization and estimation of the region of attraction for nonlinear systems a vector control lyapunov function perspective this technical note deals with local asymptotic stabilization of uncertain nonlinear systems by output feedback control our study provides a new design technique from a vector control lyapunov function perspective a comparison principle is developed and as a major consequence an effective estimation method of the region of attraction roa is established application to a class of minimum phase strict feedback nonlinear systems gives rise to a constructive stabilization design as well as an estimate of the resulting roa

region of attraction analysis of nonlinear stochastic systems using polynomial chaos expansion abstract a method is presented to estimate the region of attraction roa of stochastic systems with finite second moment and uncertainty dependent equilibria the approach employs polynomial chaos pc expansions to represent the stochastic system by a higher dimensional set of deterministic equations we first show how the equilibrium point of the deterministic formulation provides the stochastic moments of an uncertainty dependent equilibrium point of the stochastic system a connection between the boundedness of the moments of the stochastic system and the lyapunov stability of its pc expansion is then derived defining corresponding notions of a roa for both system representations we show how this connection can be leveraged to recover an estimate of the roa of the stochastic system from the roa of the pc expanded system two optimization programs obtained from sum of squares programming techniques are provided to compute inner estimates of the roa the first optimization program uses the lyapunov stability arguments to return an estimate of the roa of the pc expansion based on this result and user specifications on the moments for the initial conditions the second one employs the shown connection to provide the corresponding roa of the stochastic system the method is demonstrated by two examples

pontryagin s principle for state constrained control problems governed by parabolic equations with unbounded controls this paper deals with optimal control problems governed by semilinear parabolic equations with pointwise state constraints and unbounded controls under some strong stability assumption we obtain necessary optimality conditions in the form of a pontryagin s minimum principle in qualified form a pontryagin s principle in nonqualified form is also proved without any stability condition

convex computation of the maximum controlled invariant set for polynomial control systems we characterize the maximum controlled invariant mci set for discrete as well as continuous time nonlinear dynamical systems as the solution of an infinite dimensional linear programming problem for systems with polynomial dynamics and compact semialgebraic state and control constraints we describe a hierarchy of finite dimensional linear matrix inequality lmi relaxations whose optimal values converge to the volume of the mci set dual to these lmi relaxations are sum of squares sos problems providing a converging sequence of outer approximations to the mci set the approach is simple and readily applicable in the sense that the approximations are the outcome of a single semidefinite program with no additional input apart from the problem description a number of numerical examples illustrate the approach

control of 1d parabolic pdes with volterra nonlinearities part ii analysis for a class of stabilizing boundary controllers for nonlinear 1d parabolic pdes introduced in a companion paper we derive bounds for the gain kernels of our nonlinear volterra controllers prove the convergence of the series in the feedback laws and establish the stability properties of the closed loop system we show that the state transformation is at least locally invertible and include an explicit construction for computing the inverse of the transformation using the inverse we show l 2 and h 1 exponential stability and explicitly construct the exponentially decaying closed loop solutions we then illustrate the theoretical results on an analytically tractable example

boundary stabilisation to non stationary solutions for deterministic and stochastic parabolic type equations abstractin this work we design explicit finite dimensional boundary feedback laws for stabilisation to trajectories for parabolic type equations the simple form of the feedback allows to write the solution of the corresponding closed loop equation in a mild formulation via a kernel then taking advantage of this the stability is shown as an application null stabilisation for stochastic parabolic type equations is deduced as well as far as we know the present work is the first result on boundary feedback stabilisation to trajectories and for stochastic parabolic type equations with stability guaranteed independent of how large the level of the noise is

paper maximal lyapunov functions and domains of attraction for autonomous nonlinear systems in this paper we present various theoretical and computational methods for estimating the domain of attraction of an autonomous nonlinear system these methods are based on the concept of a maximal lyapunov function which is introduced in this paper a partial differential equation characterizing a maximal lyapunov function is derived and the relationships of this equation as compared to zubov s partial differential equation are discussed an iterative procedure is given for solving the new partial differential equation this procedure yields lyapunov function candidates that are rational functions rather than polynomials the method is applied to four two dimensional examples and one three dimensional example and it is shown that the estimates obtained using this method are in many cases substantially better than those obtained using known methods

control of 1 d parabolic pdes with volterra nonlinearities part i design boundary control of nonlinear parabolic pdes is an open problem with applications that include fluids thermal chemically reacting and plasma systems in this paper we present stabilizing control designs for a broad class of nonlinear parabolic pdes in 1 d our approach is a direct infinite dimensional extension of the finite dimensional feedback linearization backstepping approaches and employs spatial volterra series nonlinear operators both in the transformation to a stable linear pde and in the feedback law the control law design consists of solving a recursive sequence of linear hyperbolic pdes for the gain kernels of the spatial volterra nonlinear control operator these pdes evolve on domains t n of increasing dimensions n 1 and with a domain shape in the form of a hyper pyramid 0 x n x n 1 x 1 x 1 we illustrate our design method with several examples one of the examples is analytical while in the remaining two examples the controller is numerically approximated for all the examples we include simulations showing blow up in open loop and stabilization for large initial conditions in closed loop in a companion paper we give a theoretical study of the properties of the transformation showing global convergence of the transformation and of the control law nonlinear volterra operators and explicitly constructing the inverse of the feedback linearizing volterra transformation this in turn allows us to prove l 2 and h 1 local exponential stability with an estimate of the region of attraction where possible and explicitly construct the exponentially decaying closed loop solutions

stabilization and structural assignment of dirichlet boundary feedback parabolic equations a parabolic equation defined on a bounded domain is considered with input acting on the boundary through the dirichlet b c expressed as a specified finite dimensional feedback of the solution the free system zero b c is assumed throughout to be unstable two main results are established first we provide in 2 a novel proof that fully solves the corresponding boundary feedback stabilization problem thereby removing an annoying assumption required by the technique of appl math optim 6 1980 pp 201 220 under certain natural algebraic conditions based on the finitely many unstable eigenvalues we establish the existence of general boundary vectors for which the corresponding feedback semigroup decays exponentially to zero in the l 2 omega operator norm or more generally in the h s omega operator norm 0 leqq s frac 1 2 however most of the paper is devoted to the second problem structural or spectral assignment which is a natural question relevant to the selfadj

global steady state controllability of one dimensional semilinear heat equations we investigate the problem of exact boundary controllability of semilinear one dimensional heat equations we prove that it is possible to move from any steady state to any other by means of a boundary control provided that both are in the same connected component of the set of steady states the proof is based on an effective feedback stabilization procedure which is implemented

exponential stabilization for a class of nonlinear parabolic pde systems via fuzzy control approach this paper deals with the exponential stabilization problem for a class of nonlinear spatially distributed processes that are modeled by semilinear parabolic partial differential equations pdes for which a finite number of actuators are used a fuzzy control design methodology is developed for these systems by combining the pde theory and the takagi sugeno t s fuzzy model based control technique initially a t s fuzzy parabolic pde model is proposed to accurately represent a semilinear parabolic pde system then based on the t s fuzzy model a lyapunov technique is used to design a continuous fuzzy state feedback controller such that the closed loop pde system is exponentially stable with a given decay rate the stabilization condition is presented in terms of a set of spatial differential linear matrix inequalities sdlmis furthermore a recursive algorithm is presented to solve the sdlmis via the existing linear matrix inequality optimization techniques finally numerical simulations on the temperature profile control of a catalytic rod are given to verify the effectiveness of the proposed design method

brief paper sliding mode boundary control of a parabolic pde system with parameter variations and boundary uncertainties this paper considers the stabilization problem of a one dimensional unstable heat conduction system rod modeled by a parabolic partial differential equation pde powered with a dirichlet type actuator from one of the boundaries by applying the volterra integral transformation a stabilizing boundary control law is obtained to achieve exponential stability in the ideal situation when there are no system uncertainties the associated lyapunov function is used for designing an infinite dimensional sliding manifold on which the system exhibits the same type of stability and robustness against certain types of parameter variations and boundary disturbances it is observed that the relative degree of the chosen sliding function with respect to the boundary control input is zero a continuous control law satisfying the reaching condition is obtained by passing a discontinuous signum signal through an integrator

boundary feedback stabilization of the schlogl system the schlogl system is governed by a nonlinear reaction diffusion partial differential equation with a cubic nonlinearity that determines three constant equilibrium states it is a classical example of a chemical reaction system that is bistable the constant equilibrium that is enclosed by the other two constant equilibrium points is unstable in this paper robin boundary feedback laws are presented that stabilize the system in a given stationary state or more generally in a given time dependent desired system orbit the exponential stability of the closed loop system with respect to the l 2 norm is proved in particular it is shown that with the boundary feedback law the unstable constant equilibrium point can be stabilized

a state space approach to a one dimensional mathematical model for the dynamics of phase transitions in pseudoelastic materials abstract an abstract formulation of a general mathematical model for the dynamics of shape memory alloys is presented using a state space approach the model is written as a semilinear cauchy problem in an appropriate hilbert space it is shown that the differential operator corresponding to the linear part of the system generates an exponentially stable analytic semigroup for this semigroup explicit decay rates are given in terms of the physical constants of the system and its continuity with respect to the model parameters is proved a proof of local existence of solutions is given and open problems are discussed

pontryagin s principle for state constrained boundary control problems of semilinear parabolic equations this paper deals with state constrained optimal control problems governed by semilinear parabolic equations we establish a minimum principle of pontryagin s type to deal with the state constraints we introduce a penalty problem by using ekeland s principle the key tool for the proof is the use of a special kind of spike perturbations distributed in the domain where the controls are defined conditions for normality of optimality conditions are given

brief paper local stability analysis using simulations and sum of squares programming the problem of computing bounds on the region of attraction for systems with polynomial vector fields is considered invariant subsets of the region of attraction are characterized as sublevel sets of lyapunov functions finite dimensional polynomial parametrizations for lyapunov functions are used a methodology utilizing information from simulations to generate lyapunov function candidates satisfying necessary conditions for bilinear constraints is proposed the suitability of lyapunov function candidates is assessed solving linear sum of squares optimization problems qualified candidates are used to compute invariant subsets of the region of attraction and to initialize various bilinear search strategies for further optimization we illustrate the method on small examples from the literature and several control oriented systems

finite time stabilization for stochastic reaction diffusion systems with markovian switching via boundary control abstract this paper investigates the finite time stability fts for a class of stochastic markovian reaction diffusion systems smrdss first a boundary control strategy is put forward under the designed boundary controller a sufficient condition of fts for smrdss is provided based on the method of lyapunov krasovskii functional combined with inequality techniques when there exists the incompleteness of transition rate information we further study the problem of smrdss with partially unknown transition rates trs by adding a set of symmetric free matrices an fts criterion is also obtained for this case theoretical results show that the case of completely known trs is the special case of partially unknown trs finally simulation examples are presented to verify the validity of our derived results

polynomial approximation of high dimensional hamilton jacobi bellman equations and applications to feedback control of semilinear parabolic pdes a procedure for the numerical approximation of high dimensional hamilton jacobi bellman hjb equations associated to optimal feedback control problems for semilinear parabolic equations is proposed its main ingredients are a pseudospectral collocation approximation of the pde dynamics and an iterative method for the nonlinear hjb equation associated to the feedback synthesis the latter is known as the successive galerkin approximation it can also be interpreted as newton iteration for the hjb equation at every step the associated linear generalized hjb equation is approximated via a separable polynomial approximation ansatz stabilizing feedback controls are obtained from solutions to the hjb equations for systems of dimension up to fourteen

local stability analysis for uncertain nonlinear systems we propose a method to compute provably invariant subsets of the region of attraction for the asymptotically stable equilibrium points of uncertain nonlinear dynamical systems we consider polynomial dynamics with perturbations that either obey local polynomial bounds or are described by uncertain parameters multiplying polynomial terms in the vector field this uncertainty description is motivated by both incapabilities in modeling as well as bilinearity and dimension of the sum of squares programming problems whose solutions provide invariant subsets of the region of attraction we demonstrate the method on three examples from the literature and a controlled short period aircraft dynamics example

region of attraction analysis with integral quadratic constraints abstract a general framework is presented to estimate the region of attraction of attracting equilibrium points the system is described by a feedback connection of a nonlinear polynomial system and a bounded operator the input output behavior of the operator is characterized using an integral quadratic constraint this allows to analyze generic problems including for example hard nonlinearities and different classes of uncertainties adding to the state of practice in the field which is typically limited to polynomial vector fields the iqc description is also nonrestrictive with the main result given for both hard and soft factorizations optimization algorithms based on sum of squares techniques are then proposed with the aim to enlarge the inner estimates of the roa numerical examples are provided to show the applicability of the approaches these include a saturated plant where bounds on the states are exploited to refine the sector description and a case study with parametric uncertainties for which the conservativeness of the results is reduced by using soft iqcs

computation of an optimal communication schedule in a nonlinear networked control system using sum of squares abstract in this paper we consider the design of a control optimal communication schedule in nonlinear networked control with nonlinear affine plant systems we focus here on periodic communication schedules in control systems with time triggered control communication and assume the existence of predesigned controllers which need to be optimally integrated via the control communication system for this we formulate a quadratic cost function for the nonlinear affine plant the system and the cost are first discretized using an exact mixed discretization approach which allows to lift the problem from the continuous discrete domain into a purely discrete time representation periodicity due to the fixed communication schedule is dealt with the discrete lifting technique to allow for the computation of the cost nonlinearities are approximated in terms of a taylor series which allows the use of the sum of squares framework once a cost function of the networked control system for a given fixed communication schedule is defined it is then possible to compare and choose those sequences that minimize this cost for optimal controller integration

stability region analysis using polynomial and composite polynomial lyapunov functions and sum of squares programming we propose using bilinear sum of squares programming for obtaining inner bounds of roas for dynamical systems with polynomial vector fields we search for polynomial as well as composite lyapunov functions comprised of pointwise maximums of polynomial functions results for several examples from the literature are presented using the proposed methods and the penbmi solver

boundary control of three dimensional inextensible marine risers this paper present a design of boundary controllers actuated by hydraulic actuators at the top end for global stabilization of a three dimensional riser system first a set of partial and ordinary differential equations describing motion of the riser and hydraulic systems is developed second several important properties of the riser system are derived based on these properties we show that the conventional formula to calculate the riser effective tension is oversimplified and a new formula is provided next boundary controllers are designed based on lyapunov s direct method the backstepping technique the derived properties of the riser system dynamics and poincare s inequalities finally the galerkin approximation method is used to prove existence and uniqueness of the solutions of the closed loop control system

output feedback boundary control of a ginzburg landau model of vortex shedding an exponentially convergent observer is designed for a linearized ginzburg landau model of vortex shedding in viscous flow past a bluff body measurements are restricted to be taken collocated with the actuation which is applied on the cylinder surface the observer is used in conjuction with a state feedback boundary controller designed in previous work to attenuate vortex shedding while the theoretical results apply to the linearized system under sufficiently smooth initial data that satisfy the boundary conditions simulations demonstrate the performance of the linear output feedback scheme on the nonlinear plant model

boundary feedback stabilization of an unstable heat equation in this paper we study the problem of boundary feedback stabilization for the unstable heat equation ut x t uxx x t a x u x t this equation can be viewed as a model of a heat conducting rod in which not only is the heat being diffused mathematically due to the diffusive term uxx but also the destabilizing heat is generating mathematically due to the term a u with a 0 we show that for any given continuously differentiable function a and any given positive constant l we can explicitly construct a boundary feedback control law such that the solution of the equation with the control law converges to zero exponentially at the rate of l this is a continuation of the recent work of boskovic krstic and liu ieee trans automat control 46 2001 pp 2022 2028 and balogh and krstic european j control 8 2002 pp 165 176

backstepping observers for a class of parabolic pdes in this paper we design exponentially convergent observers for a class of parabolic partial integro differential equations p i des with only boundary sensing available the problem is posed as a problem of designing an invertible coordinate transformation of the observer error system into an exponentially stable target system observer gain output injection function is shown to satisfy a well posed hyperbolic pde that is closely related to the hyperbolic pde governing backstepping control gain for the state feedback problem for several physically relevant problems the observer gains are obtained in closed form the observer gains are then used for an output feedback design in both collocated and anti collocated setting of sensor and actuator the order of the resulting compensator can be substantially lowered without affecting stability explicit solutions of a closed loop system are found in particular cases

boundary control of transverse motion of marine risers with actuator dynamics this paper presents a boundary controller to reduce transverse motion of flexible marine risers driven by a hydraulic system at the top end of the risers under environmental disturbances induced by waves wind and ocean currents the boundary controller is designed based on lyapunov s direct method and the backstepping technique proof of existence and uniqueness of the solutions of the closed loop control system is carried out by using the galerkin approximation method simulation results illustrate the effectiveness of the proposed boundary controller

region of attraction estimation using invariant sets and rational lyapunov functions this work addresses the problem of estimating the region of attraction ra of equilibrium points of nonlinear dynamical systems the estimates we provide are given by positively invariant sets which are not necessarily defined by level sets of a lyapunov function moreover we present conditions for the existence of lyapunov functions linked to the positively invariant set formulation we propose connections to fundamental results on estimates of the ra are presented and support the search of lyapunov functions of a rational nature we then restrict our attention to systems governed by polynomial vector fields and provide an algorithm that is guaranteed to enlarge the estimate of the ra at each iteration