model structure simplification of nonlinear systems via immersion this paper concerns the simplification of model structures of nonlinear systems while preserving their input output maps the basic technique is an immersion of a system which is a mapping of the initial state from the original system to another system of an identical input output map the necessary and sufficient conditions are presented for the immersibility of a given system into a state space representation of such particular structures as rational functions or polynomials with respect to the state the conditions are so mild that many types of nonlinear systems can be represented by rational or polynomial model structures moreover it is also shown that a polynomial state space representation can always be constructed to be at most quadratic with respect to the state

observer design using a generalized time scaled block triangular observer form this paper presents an observer design for uncontrolled nonlinear multi output continuous time systems based on a time scaled block triangular observer form tbtof the tbtof generalizes an established block triangular observer form btof by including generalized time scaling functions tsfs which have both state and output dependence introducing these stability preserving tsfs broadens the class of systems which admits a btof based observer design necessary conditions for the tsfs are provided when a tbtof exists an observer can be constructed subsystem at a time sufficient conditions are given to prove estimate error stability a physical example illustrates the tbtof construction and observer design

observer design for nonlinear control systems abstract this paper deals with the observer design problem of a wide class of nonlinear systems subjected to bounded nonlinearities a sufficient liapunovlike condition is provided and the proposed dynamic observer is a direct extension of the one in linear case

exact finite dimensional filters via systems realization for a class of discrete time nonlinear systems abstract we obtain necessary and sufficient conditions for the existence of a finite dimensional filter for the discrete time nonlinear system x k 1 x k y k h x k x k k k 0 1 this system is distinguished by the absence of noise in the dynamic and by the correlation between the state and the intensity of noise in the observations the necessary and sufficient condition provides an explicit formula for the minimal filter and various system theoretic properties of and of the minimal filter

duality between the trigonometric bcn sutherland system and a completed rational ruijsenaars schneider van diejen system we present a new case of duality between integrable many body systems where two systems live on the action angle phase spaces of each other in such a way that the action variables of each system serve as the particle positions of the other one our investigation utilizes an idea that was exploited previously to provide group theoretic interpretation for several dualities discovered originally by ruijsenaars in the group theoretic framework one applies hamiltonian reduction to two abelian poisson algebras of invariants on a higher dimensional phase space and identifies their reductions as action and position variables of two integrable systems living on two different models of the single reduced phase space taking the cotangent bundle of u 2n as the upstairs space we demonstrate how this mechanism leads to a new dual pair involving the bcn trigonometric sutherland system thereby we generalize earlier results pertaining to the an trigonometric sutherland system as well as a recent work by pusztai on the hyperbolic bcn sutherland system

restricted dynamic observer error linearizability following the general dynamic observer error linearization problem state equivalence to a triangular nonlinear observer canonical form with index d and the restricted dynamic observer error linearization problem are defined in this paper the necessary and sufficient conditions for the above two problems are given since our proofs are constructive a desired state transformation can also be found in the theorem

numerical methods for a class of nonlocal diffusion problems with the use of backward sdes we propose a novel numerical approach for nonlocal diffusion equations du et al 2012 with integrable kernels based on the relationship between the backward kolmogorov equation and backward stochastic differential equations bsdes driven by levy jumps processes the nonlocal diffusion problem under consideration is converted to a bsde for which numerical schemes are developed as a stochastic approach the proposed method completely avoids the challenge of iteratively solving non sparse linear systems arising from the nature of nonlocality this allows for embarrassingly parallel implementation and also enables adaptive approximation techniques to be incorporated in a straightforward fashion moreover our method recovers the convergence rates of classic deterministic approaches e g finite element or collocation methods due to the use of high order temporal and spatial discretization schemes in addition our approach can handle a broad class of problems with general inhomogeneous forcing terms as long as they are globally lipschitz continuous rigorous error analysis of the new method is provided as several numerical examples that illustrate the effectiveness and efficiency of the proposed approach

lax representation of the hyperbolic van diejen dynamics with two coupling parameters in this paper we construct a lax pair for the classical hyperbolic van diejen system with two independent coupling parameters built upon this construction we show that the dynamics can be solved by a projection method which in turn allows us to initiate the study of the scattering properties as a consequence we prove the equivalence between the first integrals provided by the eigenvalues of the lax matrix and the family of van diejen s commuting hamiltonians also at the end of the paper we propose a candidate for the lax matrix of the hyperbolic van diejen system with three independent coupling constants

high gain observers in the state and estimation of robots having elastic joints the paper deals with the state and parameter estimation of robots having elastic joints the estimation of the unknown parameters of the robot is reduced to the estimation of its state variables by a state space immersion the luenberger observer is used in the state estimation of the extended robot model the use of high gains is studied in the cancellation of nonlinearities in order to simplify the observer design the high gain induces a time scale separation between the robot model and the observer and therefore the singular perturbation theory can be used in the stability analysis of the estimate error dynamics in particular it is shown that the error dynamics reaches the stable equilibrium in a very fast transient ensuring that the slow dynamics of the observer is just that of the given robot

commutativity of immersion and linearization a given nonlinear system can be represented via an immersion as rational or polynomial functions thus leading to a simplified model structure an immersion is a mapping of the initial state from the original state space to another state space while exactly preserving the input output map in this note we show that the linearization of the system after immersion has an identical input output map to the linearization of the original system before immersion in other words immersion and linearization commute this is potentially useful for applications such as linear control design and sensitivity analysis after nonlinear identification and has important implications for system approximation by linearization

an algorithm for system immersion into nonlinear observer form siso case the class of nonlinear systems that can be put into nonlinear observer form linear system with output injection can be broadened if we employ the system immersion we provide an immersion algorithm for siso nonlinear systems with relative degree r the proposed algorithm is an extension of the previous results and does not require the relative degree 1 assumption in addition it is seen that the immersion can always be computed via algebraic computations and simple integrations except in a very special case in which the relative degree equals the system dimension in this case only one first order differential equation appears in the algorithm

free motion on the poisson su n group is the phase space of the poisson su n we explicitly calculate the symplectic structure of define an analogue of the hamiltonian of the free motion on su n and solve the corresponding equations of motion velocity is related to the momentum by a nonlinear legendre transformation

differential equations in the spectral parameter we determine all the potentialsv x for the schrodinger equation 2 v x k2 such that some family of eigenfunctions satisfies a differential equation in the spectral parameterk of the formb k k o x o for each suchv x we determine the algebra of all possible operatorsb and the corresponding functions x

a stable multistep scheme for solving backward stochastic differential equations in this paper we propose a stable multistep scheme on time space grids for solving backward stochastic differential equations in our scheme the integrands which are conditional mathematical expectations derived from the original equations are approximated by using lagrange interpolating polynomials with values of the integrands at multiple time levels they are then numerically evaluated using the gauss hermite quadrature rules and polynomial interpolations on the spatial grids error estimates are rigorously proved for the semidiscrete version of the proposed scheme for backward stochastic differential equations with certain types of simplified generator functions finally various numerical examples and comparisons with some other methods are presented to demonstrate high accuracy of the proposed multistep scheme

discrete time observer error linearizability via restricted dynamic systems in this technical note we define the observer error linearization problem of a discrete time autonomous nonlinear system via a restricted dynamic system this is the dual to restricted dynamic feedback linearization in a loose sense necessary and sufficient conditions for this problem are obtained in terms of the index

dynamic observer error linearization in this paper a new framework of observer error linearization problem is proposed the main idea of our approach is twofold the one is to introduce an auxiliary dynamics whose input is the system output and the other is to transform the augmented system into an observable linear system with an injection term which contains the system output as well as the state of the auxiliary dynamics it is a natural extension of the recently developed dynamic observer error linearization where the injection term contains only newly defined output it is also shown that whenever an n dimensional system is immersible into an n d dimensional linear system up to an output injection then it can be also dynamically observer error linearizable in our sense with a d dimensional auxiliary dynamics moreover we show that the converse is not true by providing a counterexample which implies that our approach is applicable to a strictly wider class of systems than that of the system immersion method

an immersion based observer design for rank observable nonlinear systems in this note an observer design for a wide class of nonlinear systems is proposed it is based on some high gain design for the particular case of state affine systems up to triangular nonlinearity on the one hand and immersion of nonlinear systems into such a form on the other hand conditions for the observer convergence are given in terms of appropriate excitation and a constructive procedure for the immersion is presented the methodology is illustrated by examples of application

an efficient method for observer design with approximately linear error dynamics exact error linearization is a well known full order observer design method which yields linear time invariant stable error dynamics in normal form coordinates the difficult procedure associated with exact error linearization has led to the simpler extended luenberger observer design however extended luenberger observers can only guarantee a first order approximation to the exact error linearization design this paper provides an explicit formula involving only derivative lie derivative and bracket and matrix inversion operations for computing an nth order approximation of the exact error linearization observer hence the proposed method is a computational simple tool for designing observers for systems admitting exact error linearization observers two examples demonstrate the application of the method

a posteriori error analysis of finite element method for linear nonlocal diffusion and peridynamic models abstract in this paper we present some results on a posteriori error analysis of finite element methods for solving linear nonlocal diffusion and bond based peridynamic models in particular we aim to propose a general abstract frame work for a posteriori error analysis of the peridynamic problems a posteriori error estimators are consequently prompted the reliability and efficiency of the estimators are proved connections between nonlocal a posteriori error estimation and classical local estimation are studied within continuous finite element space numerical experiments 1d are also given to test the theoretical conclusions

solution of the zakai equation by separation of variables an exact expression for the unnormalized conditional density is derived for certain nonlinear estimation problems the basic method is to solve the zakai equation using separation of variables the new filter is a generalization of the kalman bucy filter the benes filter and other nonlinear filters within a unified framework

constructive algorithm for system immersion into non linear observer form there exists a class of non linear systems which cannot be transformed into a non linear observer form an observable linear system up to output injection via diffeomorphism but can be immersed into a higher dimensional non linear observer form this class of systems can be characterized by a differential equation called characteristic equation if the system is an n dimensional system and it is immersible into n m dimensional observer form the characteristic equation involves n m 1 unknowns where n m unknowns are for the state immersion and one for the output transformation in general one should solve these unknowns simultaneously which makes the characterization difficult after investigating the algebraic structure of the characteristic equation we present an algorithm to check the immersibility under a constant rank assumption using the algorithm one can check the immersibility iteratively since only one unknown is involved at each step of the algorithm

nonlinear observers with linearizable error dynamics we present a new method for designing asymptotic observers for a class of nonlinear systems the error between the state of the system and the state of the observer in appropriate coordinates evolves linearly and can be made to decay arbitrarily exponentially fast

on the immersion of a discrete time polynomial analytic system into a polynomial affine one abstract the concept of immersion just recently introduced by fliess and kupka is related to the capability of reproducing the input output maps of a given system by means of another one in this paper we give necessary and sufficient conditions under which a polynomial analytic system i e a system whose dynamics are polynomial with respect to the input and nonlinear analytic with respect to the state is immersed into a polynomial affine system i e a system whose dynamics are polynomial in the input and linear in the state an application to the problem of the reproducibility of a linear analytic state feedback acting on a given system by means of a preprocessor suitably initialized is also performed

new classes of finite dimensional nonlinear filters abstract we derive sufficient conditions for the estimation algebras of two special classes of nonlinear estimation systems to be finite dimensional by carrying out the brockett wei norman program new classes of finite dimensional filters are obtained these systems can be regarded respectively as extensions of the kalman bucy filters and the benes filters

nonlinear observer design by dynamic observer error linearization in this note we address the problem of transforming a nonlinear system into nonlinear observer canonical form in the extended state space with the aid of dynamic system extension and introduction of virtual outputs as an intermediate step for the general problem we consider a restricted structure of dynamic extension which is obtained roughly speaking by adding the chains of integrators to the outputs of original system we propose sufficient conditions which can be verified using the system dynamics expressed in their original coordinates an illustrative example is included that demonstrates the advantage of the proposed method over the conventional method

further results on the observer design problem abstract in this paper the observer design problem for nonlinear systems is considered sufficient lyapunov like conditions are presented for the existence of a nonlinear observer the theory we develop considerably improves and extends the results of our recent work 14

linearization by output injection and nonlinear observers abstract observers can easily be constructed for those nonlinear systems which can be transformed into a linear system by change of state variables and output injection necessary and sufficient conditions for the existence of such a transformation are given

a state observer for nonlinear dynamical systems the problem of state observation for nonlinear systems is of main importance in automatic control in recent years many contributions have been presented in literature that solve the design problem for classes of nonlinear systems most works can be roughly classi ed into two categories in one the geometrical properties of vector elds de ning the system are exploited and therefore mathematical tools of the di erential geometry are extensively employed in the other one the functional properties of the input state and state output maps which de ne the system are mainly used while the geometrical methods often provide a general solution of the problem for systems that belong to restricted classes characterized by geometrical properties 1 2 3 4 5 6 7 13 14 17 18 19 the second approach gives solutions for classes of nonlinear systems whose geometrical nature is not investigated and only macroscopic parameters norms gains lipschitz constants and so on are involved in the design procedure 8 9 11 12 21 one main problem in state observation for nonlinear systems is that the property of driftobservability does not imply observability for any input function as shown in 16 19 thus it is interesting to investigate when the drift observability property allows the solution of the state observation problem nding conditions that restricts the class of input functions due to the mathematical tools employed this work may be considered as belonging to both the categories above mentioned in this paper nonlinear systems of the form

bilinearization up to output injection abstract this paper deals with nonlinear systems that can be steered to bilinear systems by output injection and nonlinear change of coordinates we give an abstract non constructive theorem and an algorithm which allows us to recognize such systems

adapted solution of a backward stochastic differential equation abstract let wt t 0 1 be a standard k dimensional weiner process defined on a probability space f p and let ft denote its natural filtration given a f1 measurable d dimensional random vector x we look for an adapted pair of processes x t y t t 0 1 with values in rd and rd k respectively which solves an equation of the form x t t 1 f s x s y s ds t 1 g s x s y s dw s x a linearized version of that equation appears in stochastic control theory as the equation satisfied by the adjoint process we also generalize our results to the following equation x t t 1 f s x s y s ds t 1 g s x s y s dw s x under rather restrictive assumptions on g

observer design based on triangular form generated by injective map a procedure is provided for constructing an injective map as a generalized coordinate change for triangular form transformation based on the triangular form a method is developed for designing nonlinear state observers

action angle maps and scattering theory for some finite dimensional integrable systems i the pure soliton case we construct an action angle transformation for the calogero moser systems with repulsive potentials and for relativistic generalizations thereof this map is shown to be closely related to the wave transformations for a large classl of hamiltonians and is shown to have remarkable duality properties all dynamics inl lead to the same scattering transformation which is obtained explicitly and exhibits a soliton structure an auxiliary result concerns the spectral asymptotics of matrices of the formm exp td ast it pertains to diagonal matricesd whose diagonal elements have pairwise different real parts and to matricesm for which certain principal minors are non zero

symplectic structures associated to lie poisson groups the lie poisson analogues of the cotangent bundle and coadjoint orbits of a lie group are considered for the natural poisson brackets the symplectic leaves in these manifolds are classified and the corresponding symplectic forms are described thus the construction of the kirillov symplectic form is generalized for lie poisson groups

on the duality between the hyperbolic sutherland and the rational ruijsenaars schneider models we consider two families of commuting hamiltonians on the cotangent bundle of the group and show that upon an appropriate single symplectic reduction they descend to the spectral invariants of the hyperbolic sutherland and of the rational ruijsenaars schneider lax matrices respectively the duality symplectomorphism between these two integrable models that was constructed by ruijsenaars using direct methods can then be interpreted geometrically simply as a gauge transformation connecting two cross sections of the orbits of the reduction group

brief paper full order observer design for a class of port hamiltonian systems we consider a special class of port hamiltonian systems for which we propose a design methodology for constructing globally exponentially stable full order observers using a passivity based approach the essential idea is to make the augmented system consisting of the plant and the observer dynamics to become strictly passive with respect to an invariant manifold defined on the extended state space on which the state estimation error is zero we first introduce the concept of passivity of a system with respect to a manifold by defining a new input and output on the extended state space and then perform a partial state feedback passivation which leads to the construction of the observer we then illustrate this observer design procedure on two physical examples the magnetic levitation system and the inverted pendulum on the cart system

canonical form observer design for non linear time variable systems an observer of canonical phase variable form for non linear time variable systems is introduced the development of this non linear time variable form requires regularity of the non linear time variable observability matrix of the system from the relationships derived during the development it follows that a non linear time variable observer can be dimensioned by an eigenvalue assignment with respect to the canonical state coordinates if a linearization of system non linearities about the reconstructed state trajectories is permissible this is an assumption similar to that of the extended kalman filter based on a linearization about the current estimate

poisson lie interpretation of trigonometric ruijsenaars duality a geometric interpretation of the duality between two real forms of the complex trigonometric ruijsenaars schneider system is presented the phase spaces of the systems in duality are viewed as two different models of the same reduced phase space arising from a suitable symplectic reduction of the standard heisenberg double of u n the collections of commuting hamiltonians of the systems in duality are shown to descend from two families of free hamiltonians on the double which are dual to each other in a poisson lie sense our results give rise to a major simplification of ruijsenaars proof of the crucial symplectomorphism property of the duality map

immersion of non linear systems into linear systems up to output injection characteristic equation approach there exists a class of non linear systems which cannot be transformed into non linear observer form via diffeomorphism but can be immersed into higher dimensional non linear observer form this paper investigates when n dimensional non linear systems can be immersed into n m dimensional non linear systems in non linear observer form and provides a necessary and sufficient condition we also present an algorithm to check the immersibility in general at the first step of the algorithm one should solve a differential equation with m 1 unknowns rather than solving one with n m unknowns simultaneously when a solution to the differential equation can be obtained the immersibility can be checked by a recursive algorithm moreover when 2m n a straightforward algorithm is provided to check the immersibility

verifiable conditions for multioutput observer error linearizability the verifiable necessary and sufficient conditions for the multioutput observer error linearization problem have not been found over the last three decades in this technical note we give the unverifiable necessary and sufficient conditions which are more specific than the ones in the literature using these conditions a verifiable sufficient condition also necessary with additional restriction on the problem is derived since our proofs are constructive a desired state transformation can also be found in the theorem also our algorithmic conditions can be easily implemented by a matlab programming

adaptive observers for single output nonlinear systems an adaptive version of the nonlinear observer obtained by a j krener et al 1983 is presented this version involves the cancellation of nonlinear terms by output injection as an intermediate step necessary and sufficient conditions are given for transforming a nonlinear system by state space change of coordinates into the special adaptive observer form that was used by y bastin et al 1988 to design adaptive observers

theorems on the structure of finite dimensional estimation algebras abstract in this paper we proved several theorems concerning the structure of finite dimensional estimation algebras in particular under proper technical assumptions we proved the following 1 the observation of a filtering system must be linear if the estimation algebra is finite dimensional 2 all elements of a finite dimensional estimation algebra belong to a special class of polynomial differential operators 3 all finite dimensional estimation algebras are solvable

nonlinear system immersion observers and finite dimensional filters abstract nonlinear deterministic and stochastic systems which can be immersed into linear observable ones are identified by necessary and sufficient conditions those systems are shown to admit observers in the deterministic case and exact finite dimensional filters in the stochastic case the sufficient conditions obtained generalize earlier results in both cases

analysis and approximation of nonlocal diffusion problems with volume constraints a recently developed nonlocal vector calculus is exploited to provide a variational analysis for a general class of nonlocal diffusion problems described by a linear integral equation on bounded domains in mbrn the nonlocal vector calculus also enables striking analogies to be drawn between the nonlocal model and classical models for diffusion including a notion of nonlocal flux the ubiquity of the nonlocal operator in applications is illustrated by a number of examples ranging from continuum mechanics to graph theory in particular it is shown that fractional laplacian and fractional derivative models for anomalous diffusion are special cases of the nonlocal model for diffusion that we consider the numerous applications elucidate different interpretations of the operator and the associated governing equations for example a probabilistic perspective explains that the nonlocal spatial operator appearing in our model corresponds to the infinitesimal generator for a symmetric jump process sufficie

design of state observers from a drift observability property in this paper the problem of state observation with exponential error decay for nonlinear systems affine in the input is considered and an observer is proposed for such an observer the drift observability property of the system i e observability for zero input together with an assumption on the input amplitude or on the observation relative degree are sufficient conditions for exponential convergence of the observation error the existence of an exponential observer is correlated to the existence of a solution for an h sub spl infin riccati like inequality global and semiglobal convergence results are presented

continuous time random walks on bounded domains a useful perspective to take when studying anomalous diffusion processes is that of a continuous time random walk and its associated generalized master equation we derive the generalized master equations for continuous time random walks that are restricted to a bounded domain and compare numerical solutions with kernel density estimates of the probability density function computed from simulations the numerical solution of the generalized master equation represents a powerful tool in the study of continuous time random walks on bounded domains

macdonald polynomials and algebraic integrability we construct explicitly nonpolynomial eigenfunctions of the difference operators by macdonald in the case t qk k z this leads to a new more elementary proof of several macdonald conjectures proved first by cherednik we also establish the algebraic integrability of macdonald operators at t qk k z generalizing the result of etingof and styrkas our approach works uniformly for all root systems including the bcn case and related koornwinder polynomials moreover we apply it for a certain deformation of the an root system where the previously known methods do not work

self duality and scattering map for the hyperbolic van diejen systems with two coupling parameters with an appendix by s ruijsenaars in this paper we construct global action angle variables for a certain two parameter family of hyperbolic van diejen systems following ruijsenaars ideas on the translation invariant models the proposed action angle variables come from a thorough analysis of the commutation relation obeyed by the lax matrix whereas the proof of their canonicity is based on the study of the scattering theory as a consequence we show that the van diejen system of our interest is self dual with a factorized scattering map also in an appendix by s ruijsenaars a novel proof of the spectral asymptotics of certain exponential type matrix flows is presented this result is of crucial importance in our scattering theoretical analysis

a new model in the calogero ruijsenaars family hamiltonian reduction is used to project a trivially integrable system on the heisenberg double of su n n to obtain a system of ruijsenaars type on a suitable quotient space this system possesses bcn symmetry and is shown to be equivalent to the standard three parameter bcn hyperbolic sutherland model in the cotangent bundle limit

brockett s problem of classification of finite dimensional estimation algebras for nonlinear filtering systems in his talk at the international congress of mathematicians in 1982 brockett proposed to classify all finite dimensional estimation algebras arising in nonlinear filters recently chen yau and leung siam j control optim 35 1997 pp 1132 1141 claimed that they had classified all finite dimensional estimation algebras of maximal rank when the dimension of the state space is less than or equal to four in this paper we introduce a series of completely new computations about the estimation algebra and we find two sets of new equations about the omega matrix as a consequence we can prove without any dimension assumption on the state space that the omega matrix is a constant matrix and thus we succeed in classifying all finite dimensional estimation algebras of maximal rank with the state space dimension being arbitrary

brief paper linearization through state immersion of nonlinear systems admitting lie symmetries in this paper it is shown that if a nonlinear system admits a lie symmetry that can be transformed into its poincare dulac normal form by a state diffeomorphism then under some technical conditions such a nonlinear system can be immersed into a linear one this allows us to compute in closed form the flow all algebraic invariant curves through semi invariants of the nonlinear system and lyapunov functions to study stability properties

continuous and discontinuous finite element methods for a peridynamics model of mechanics in contrast to classical partial differential equation models the recently developed peridynamic nonlocal continuum model for solid mechanics is an integro differential equation that does not involve spatial derivatives of the displacement field as a result the peridynamic model admits solutions having jump discontinuities so that it has been successfully applied to fracture problems the peridynamic model features a horizon which is a length scale that determines the extent of the nonlocal interactions based on a variational formulation continuous and discontinuous galerkin finite element methods are developed for the peridynamic model discontinuous discretizations are conforming for the model without the need to account for fluxes across element edges through a series of simple one dimensional computational experiments we investigate the convergence behavior of the finite element approximations and compare the results with theoretical estimates one issue addressed is the effect of the relative sizes of the horizon and the grid for problems with smooth solutions we find that continuous and discontinuous piecewise linear approximations result in the same accuracy as that obtained by continuous piecewise linear approximations for classical models piecewise constant approximations are less robust and require the grid size to be small with respect to the horizon we then study problems having solutions containing jump discontinuities for which we find that continuous approximations are not appropriate whereas discontinuous approximations can result in the same convergence behavior as that seen for smooth solutions in case a grid point is placed at the locations of the jump discontinuities such results are directly obtained in the general case we show that such results can be obtained through a simple automated abrupt local refinement of elements containing the discontinuity in order to reduce the number of degrees of freedom while preserving accuracy we also briefly consider a hybrid discretization which combines continuous discretizations in regions where the solution is smooth with discontinuous discretizations in small regions surrounding the jump discontinuities

exploiting isochrony in self triggered control event triggered control and self triggered control have been recently proposed as new implementation paradigms that reduce resource usage for control systems in self triggered control the controller is augmented with the computation of the next time instant at which the feedback control law is to be recomputed since these execution instants are obtained as a function of the plant state we effectively close the loop only when it is required to maintain the desired performance thereby greatly reducing the resources required for control in this paper we present a new technique for the computation of the execution instants by exploiting the concept of isochronous manifolds also introduced in this paper while our previous results showed how homogeneity can be used to compute the execution instants along some directions in the state space the concept of isochrony allows us to compute the executions instants along every direction in the state space moreover we also show in this paper how to homogenize smooth control systems thus making our results applicable to any smooth control system the benefits of the proposed approach with respect to existing techniques are analyzed in two examples

a new kind of accurate numerical method for backward stochastic differential equations in this paper we propose a new kind of numerical simulation method for backward stochastic differential equations bsdes we discretize the continuous bsdes on time space discrete grids use the monte carlo method to approximate mathematical expectations and use space interpolations to compute values at non grid points to demonstrate the accuracy and the effectiveness of our method several numerical examples are given