Carleman Linearization Research Articles

A generalization of lyapunov's equation to nonlinear systems

In this paper a generalization of Lyapunov's equation for the stability of linear dynamical systems to globally asymptotically stable nonlinear systems is presented by embedding the system in a linear infinite-dimensional one on a tensor space by using Carleman linearization. This linear representation allows the definition and solution of a Lyapunov equation as in the usual linear case. The converse result is also discussed using the fact that globally asymptotically stable non linear systems are essentially linear.

SEMI-TENSOR PRODUCT OF MATRICES AND ITS SOME APPLICATIONS TO PHYSICS

In this paper we first give a general definition of a new kind of matrix products, called the semi-tensor product, which was firstly proposed in [4]. Certain new properties related to the later applications are proved. Using them, some problems in physics are investigated. First of all, the Carleman linearization of some dynamic physical systems is considered. It is used to investigate the invariants. A rigorous proof for the solvability is presented. Secondly, the problems of invariants of planar polynomial systems is converted to the solvability of a set of algebraic equations. Thirdly, we consider the contraction of a tensor field. A simple proof for general contraction is obtained.

Continuous time evolution from iterated maps and Carleman linearization

Using the Carleman linearization technique the continuous iteration of a mapping is studied. Based on the detailed analysis of the Carleman embedding matrix the precise mathematical meaning is given to such notion. The ordinary differential equations referring to continuous iterations are identified and the discussion of the relationship between them and the corresponding iterated maps is performed.

Symbolic and numerical analysis for studying complex nonlinear behavior

Carleman linearization and symbolic compution are used in order to derive explicit solutions in terms of exponential polynomials depending on the parameters and initial conditions. This new method is combined with a numerical algorithm in order to compute the Lyapunov exponents associated with the system. The aim of such an approach is to propose efficient tools in order to determine the intervals of the parameters where chaotic behavior exists.

Understanding of resonance phenomena on a catalyst under forced concentration and temperature oscillations

AbstractResonance is an interesting phenomenon that may be observed for reactions on catalytic surfaces during periodic forcing of operating variables. Forcing of the variables for non‐linear systems may result in substantially changed time averaged behaviour. These resonance phenomena have been observed experimentally by coincidence rather than by systematic analysis. It is not clear for what type of reaction kinetics such behaviour may be expected and predictions are therefore impossible. Clearly, this forms a serious obstacle for any practical application.In this work we set out to analyse the nature of resonance behaviour in heterogeneously catalysed reactions. A Langmuir Hinshelwood microkinetic model is analysed. It is demonstrated that for weakly non‐linear forcing variables — as inlet concentrations — forcing leads to resonance phenomena in terms of the reaction rate only in case high total surface occupancies exist in the steady state. In contrast, forcing of strongly non‐linear variables — like temperature — may give rise to resonance phenomena for both low and high surface occupancies. Necessary conditions for resonance to occur are derived. The analysis of resonance phenomena is greatly simplified by the availability of explicit analytical expressions as can be derived from Carleman linearization. We will demonstrate the merits of Carleman linearization as compared to numerical integration.

Discretization of nonlinear control systems via the Carleman linearization

A new explicit method for the discretization of nonlinear continuous-time systems is presented. A nonlinear discrete-time model is obtained by integrating a Carleman approximation of the original continuous-time system. In contrast to conventional explicit discretization methods, the proposed Carleman discretization does not deteriorate rapidly as the sampling period increases. Examples of open-loop and deadbeat closed-loop responses of an isothermal continuous stirred tank reactor reveal the advantages of the proposed discretization scheme.

A method for pulsed periodic optimization of chemical reaction systems

A new method for solving periodic optimization problems is introduced. The method is based on the Carleman linearization procedure of lumped-parameter systems and may be used to determine the optimal pulse periodic input. The method proves superior to the Pi criterion in several regards. The method is employed to study two periodic optimization problems, a system of consecutive and a system of parallel reactions taking place in an isothermal CSTR.

Infinite-dimensional Carleman linearization, the Lie series and optimal control of non-linear partial differential equations

The Carleman linearization and Lie series techniques are generalized to non-linear partial differential equations and applied to non-linear optimal control theory.

On the Stability Analysis of Nonlinear Systems

The stability of non-linear systems and non-linear delay systems is investigated in this paper via the methods of Carleman linearization and inequality analysis. Some new criteria for stability are obtained.

Wiener-Hermite functional representation of nonlinear stochastic systems

In this paper functionals of the vector Wiener process W(t) are defined, from the perspective of representing the response of nonlinear stochastic systems described by Itô stochastic differential equations. Wiener kernels are found in closed form for the class of bilinear systems. The general case of nonlinear, analytic systems is studied through the use of Carleman linearization process, whereby the original system is converted into a bilinear one of infinite dimension. Wiener kernels and transfer functions are found by the application of a perturbation method. The case of the Duffing oscillator is studied, and the results obtained by the present analytical technique are compared with those obtained by Gaussian closure and digital simulations.

A note on Carleman linearization

Finite dimensional nonlinear ordinary differential equations du j dt = V j(u) (j = 1, …, n) can be embedded into a associated infinite dimensional linear systems with the help of the Carleman linearization. We show that the linear infinite systems can admit solutions which are not a solution of the associated nonlinear finite system.

Quadratic regulatory theory for analytic non-linear systems with additive controls

Quadratic regulator problems over finite and infinite time intervals are solved for non-linear systems which are analytic in the state and linear in the control. The solution approach is based on using a formal power series expansion method for solving a non-linear Hamiltonian system. The technique of Carleman linearization facilitates the approach. The unique solution to the non-linear optimal regulator problem is obtained in terms of a convergent power series which satisfies the Hamilton-Jacobi-Bellman equation of dynamic programming. The realization of the optimal regulator is investigated and sufficient conditions are derived from the optimal regulator using the technique of Carleman linearization. Several examples are given to illustrate the theory.

Normal forms, resonance and bifurcation analysis via the Carleman linearization

Equivalence between the normal form (NF) and the Carleman linearization of a nonlinear dynamic system is established. The near-identity nonlinear coordinate transformation which brings a system to its NF is shown to be the similarity transformation bringing a Carleman system to a Jordan canonical form. It is shown that the steady-state multiplicity is given by the nullity of the first Carleman matrix with noncomplete nullspace. The stability of the limit cycles at Hopf bifurcation is determined by the direction of the i ω-generalized eigenvector of the first odd order Carleman matrix which has a noncomplete i ω-eigenspace. The coefficients of the resonant terms that are retained in the NF are explicitly determined.

Periodic impulse-forcing of non-linear systems: a new method

A new method is developed that allows one to determine the performance of a nonlinear system undergoing impulse-forcing analytically. The method is based on the Carleman linearization procedure, which approximates a non-linear system of equations and a non-linear performance-measure by a linear system of higher order. A general explicit formula is derived which allows the determination of the state equations and of the performance measure as a function of the forcing period. An example illustrates the new method.

Steady state bifurcations and exact multiplicity conditions via Carleman linearization

The problem of steady-state bifurcations of vector fields under parameter perturbation is resolved by a linear algebraic method. Exact multiplicity conditions for any steady state are obtained in terms of the system parameters. No reduction of the steady-state system to one equation is required. Instead the one-dimensional case is included as a subspace in this generalized framework. The key point that this paper highlights is that the order of the steady multiplicity at bifurcation can be determined by examining the dimension of the kernel of the successive Carleman linear operators for all cases of practical interest. In particular, the dimension of the kernel of any Carleman linear operator of order l, equals l if l is less than the multiplicity, μ. However, the μth order Carleman operator retains a (μ − 1)-dimensional kernel.

A linear algebraic approach to steady-state bifurcation of chemical reaction systems

A new linear algebraic method that makes use of the Carleman linearization procedure is given for determining the local multiplicity features of chemical reaction systems. The method requires no reduction of the original steady-state equations and gives exact multiplicity conditions in terms of the nullities of the successive Carleman matrices. The steady-state bifurcation problem of a general class of catalytic reaction systems is resolved using this method.

Carleman linearization and moment equations of nonlinear stochastic equations†

Carleman linearization and moment equations of nonlinear stochastic equations^†

Carleman linearization techniques in nonlinear stochastic systems

Carleman linearization techniques in nonlinear stochastic systems

Non-linear autonomous systems of differential equations and Carleman linearization procedure

The non-linear autonomous of differential equations x ̇ i= ∑ j a ijx j+ ∑ j,k b ijkx jx k( x ̇ i=dx i/dt, i, j, k= 1,2,…n) which plays an important role in chemical kinetics and other fields of physics (turbulence and plasma physics) is investigated using the Carleman linearization procedure.

A note of Pearlman's state-space equivalence theorem

Using Carleman linearization and known results on state-affine systems, a simplified proof of Pearlman's [3] state-space equivalence theorem is given.