Stability of invariant regions for an alternating direction method for systems of nonlinear reaction-diffusion equations

In this paper, we consider an alternating direction method for the numerical solution of systems of nonlinear reaction-diffusion equations, with homogeneous Dirichlet boundary conditions, and we consider some properties of the difference scheme. If the system admits a bounded invariant regionS of the phase space, we prove that, under certain conditions on the mesh,S is also invariant for the difference equations. Moreover we find an error bound which tends to diamS ast→+∞ and which is 0 (h) fort fixed. Finally, we derive a time-independent error bound for a special case of mild nonlinearity. Work performed under the auspicies of G.N.I.M.-C.N.R. in the context of Program of Preventive Medicine (Project MPP1), C.N.R.     

Robust stabilisation of nonlinear plants via left coprime factorizations

In this paper we take steps towards the development of a robust stabilization theory for nonlinear plants. An approach using the left coprime factorizations of the plant and controller under certain differential boundedness assumptions is used. We first focus attention on a characterization of the class of all stabilizing nonlinear controllers K_Q for a nonlinear plant G, parameterized in terms of an arbitrary stable (nonlinear) operator Q. Also, we consider the dual class of all plants G_S stabilized by a given nonlinear controller K and parameterized in terms of an arbitrary stable (nonlinear) operator S. We show that a necessary and sufficient condition for K_Q to stabilize G_S with Q, S not necessarily stable, is that S stabilizes Q. This robust stabilization result is of interest for the solution of problems in the areas of nonlinear adaptive control and simultaneous stabilization. It specializes to known results for linear operators.     

Affine invariant scale-space

A newaffine invariant scale-space for planar curves is presented in this work. The scale-space is obtained from the solution of a novel nonlinear curve evolution equation which admits affine invariant solutions. This flow was proved to be the affine analogue of the well knownEuclidean shortening flow. The evolution also satisfies properties such ascausality, which makes it useful in defining a scale-space. Using an efficient numerical algorithm for curve evolution, this continuous affine flow is implemented, and examples are presented. The affine-invariant progressive smoothing property of the evolution equation is demonstrated as well.     

Global -gain design for a class of nonlinear systems

It is known that the so-called control problem of a nonlinear system is locally solvable if the corresponding problem for the linearized system can be solved by linear feedback. In this paper we prove that this condition suffices to solve also a global control problem, for a fairly large class of nonlinear systems, if one is free to choose a state-dependent weight of the control input. Using a two-way (backward and forward) recursive induction argument, we simultaneously construct, starting from a solution of the Riccati algebraic equation, a global solution of the Hamilton–Jacobi–Isaacs partial differential equation arising in the nonlinear control, as well as a state feedback control law that achieves global disturbance attenuation with internal stability for the nonlinear systems.     

Application of facial reduction to H ∞ state feedback control problem

One often encounters numerical difficulties in solving linear matrix inequality (LMI) problems obtained from H _∞ control problems. For semidefinite programming (SDP) relaxations for combinatorial problems, it is known that when either an SDP relaxation problem or its dual is not strongly feasible, one may encounter such numerical difficulties. We discuss necessary and sufficient conditions to be not strongly feasible for an LMI problem obtained from H _∞ state feedback control problems and its dual. Moreover, we interpret the conditions in terms of control theory. In this analysis, facial reduction, which was proposed by Borwein and Wolkowicz, plays an important role. We show that the dual of the LMI problem is not strongly feasible if and only if there exist invariant zeros in the closed left-half plane in the system, and present a remedy to remove the numerical difficulty with the null vectors associated with invariant zeros in the closed left-half plane. Numerical results show that the numerical stability is improved by applying it.     

Global L2-gain design for a class of nonlinear systems

It is known that the so-called H_∞ control problem of a nonlinear system is locally solvable if the corresponding problem for the linearized system can be solved by linear feedback. In this paper we prove that this condition suffices to solve also a globalH_∞ control problem, for a fairly large class of nonlinear systems, if one is free to choose a state-dependent weight of the control input. Using a two-way (backward and forward) recursive induction argument, we simultaneously construct, starting from a solution of the Riccati algebraic equation, a global solution of the Hamilton–Jacobi–Isaacs partial differential equation arising in the nonlinear H_∞ control, as well as a state feedback control law that achieves global disturbance attenuation with internal stability for the nonlinear systems.     

Exponential stability and stabilization of extended linearizations via continuous updates of Riccati‐based feedback

Many recent works on the stabilization of nonlinear systems target the case of locally stabilizing an unstable steady‐state solution against small perturbations. In this work, we explicitly address the goal of driving a system into a nonattractive steady state starting from a well‐developed state for which the linearization‐based local approaches will not work. Considering extended linearizations or state‐dependent coefficient representations of nonlinear systems, we develop sufficient conditions for the stability of solution trajectories. We find that if the coefficient matrix is uniformly stable in a sufficiently large neighborhood of the current state, then the state will eventually decay. On the basis of these analytical results, we propose a scheme that is designed to maintain the stabilization property of a Riccati‐based feedback constant during a certain period of the state evolution. We illustrate the general applicability of the resulting algorithm for setpoint stabilization of nonlinear autonomous systems and its numerical efficiency in 2 examples.     

On l∞ and l2 robustness of spatially invariant systems

We consider spatiotemporal systems and study their l_∞ and l₂ robustness properties in the presence of spatiotemporal perturbations. In particular, we consider spatially invariant nominal models and provide necessary and sufficient conditions for system robustness for the cases when the underlying perturbations are linear spatiotemporal varying, and nonlinear spatiotemporal invariant, unstructured or structured. It turns out that these conditions are analogous to the scaled small gain condition (which is equivalent to a spectral radius condition and a linear matrix inequality for the l_∞ and l₂ cases, respectively) derived for standard linear time‐invariant models subject to time‐varying linear and time‐invariant nonlinear perturbations. Copyright © 2009 John Wiley & Sons, Ltd.     

Effects of population size on the performance of genetic algorithms and the role of crossover

One of the most important parameters in the application of genetic algorithms (GAs) is the population size N. In many cases, the choice of N determines the quality of the solutions obtained. The study of GAs with a finite population size requires a stochastic treatment of evolution. In this study, we examined the effects of genetic fluctuations on the performance of GA calculations. We considered the role of crossover by using the stochastic schema theory within the framework of the Wright-Fisher model of Markov chains. We also applied the diffusion approximation of the Wright-Fisher model. In numerical experiments, we studied effects of population size N and crossover rate pc on the success probability S. The success probability S is defined as the probability of obtaining the optimum solution within the limit of reaching the stationary state. We found that in a GA with pc, the diffusion equation can reproduce the success probability S. We also noted the role of crossover, which greatly increases S.     

Numerical solution of some initial optimal control problems using the reproducing kernel Hilbert space technique

ABSTRACT

In this paper, we present a general technique for solving a class of linear/nonlinear optimal control problems. In fact, an analytical solution of the state variable is represented in the form of a series in a reproducing kernel Hilbert space. Sometimes with the aid of this series form, we can also present the optimal control variable in a series form. An iterative method is given to obtain the approximate optimal control and state variables and the cost functional is numerically obtained. Convergence analysis of the method is also provided. Several numerical examples are tested to demonstrate the applicability and efficiency of the method.     

Optimal control for unstructured nonlinear differential-algebraic equations of arbitrary index

We study optimal control problems for general unstructured nonlinear differential-algebraic equations of arbitrary index. In particular, we derive necessary conditions in the case of linear-quadratic control problems and extend them to the general nonlinear case. We also present a Pontryagin maximum principle for general unstructured nonlinear DAEs in the case of restricted controls. Moreover, we discuss the numerical solution of the resulting two-point boundary value problems and present a numerical example. This research was supported through the Research-in-Pairs Program at Mathematisches Forschungsinstitut Oberwolfach. V. Mehrmann’s research was supported by Deutsche Forschungsgemeinschaft, through Matheon, the DFG Research Center “Mathematics for Key Technologies” in Berlin.     

Stability and Neimark–Sacker bifurcation in Runge–Kutta methods for a predator–prey system

We investigate the discretization of a predator–prey system with two delays under the general Runge–Kutta methods. It is shown that if the exact solution undergoes a Hopf bifurcation at τ=τ*, then the numerical solution undergoes a Neimark–Sacker bifurcation at τ(h)=τ*+O(h ^p ) for sufficiently small step size h, where p≥1 is the order of the Runge–Kutta method applied. The direction of Neimark–Sacker bifurcation and stability of bifurcating invariant curve are the same as that of delay differential equation.     

Coupling of optimal variation of parameters method with Adomian’s polynomials for nonlinear equations representing fluid flow in different geometries

In this study, we describe a modified analytical algorithm for the resolution of nonlinear differential equations by the variation of parameters method (VPM). Our approach, including auxiliary parameter and auxiliary linear differential operator, provides a computational advantage for the convergence of approximate solutions for nonlinear boundary value problems. We consume all of the boundary conditions to establish an integral equation before constructing an iterative algorithm to compute the solution components for an approximate solution. Thus, we establish a modified iterative algorithm for computing successive solution components that does not contain undetermined coefficients, whereas most previous iterative algorithm does incorporate undetermined coefficients. The present algorithm also avoid to compute the multiple roots of nonlinear algebraic equations for undetermined coefficients, whereas VPM required to complete calculation of solution by computing roots of undetermined coefficients. Furthermore, a simple way is considered for obtaining an optimal value of an auxiliary parameter via minimizing the residual error over the domain of problem. Graphical and numerical results reconfirm the accuracy and efficiency of developed algorithm.

     

Discretization methods with analytical solutions for a convection–reaction equation with higher-order discretizations

We introduce an improved second-order discretization method for the convection–reaction equation by combining analytical and numerical solutions. The method is derived from Godunov's scheme, see [S.K. Godunov, Difference methods for the numerical calculations of discontinuous solutions of the equations of fluid dynamics, Mat. Sb. 47 (1959), pp. 271–306] and [R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2002.], and uses analytical solutions to solve the one-dimensional convection-reaction equation. We can also generalize the second-order methods for discontinuous solutions, because of the analytical test functions. One-dimensional solutions are used in the higher-dimensional solution of the numerical method.

The method is based on the flux-based characteristic methods and is an attractive alternative to the classical higher-order total variation diminishing methods, see [A. Harten, High resolution schemes for hyperbolic conservation laws, J. Comput. Phys. 49 (1993), pp. 357–393.]. In this article, we will focus on the derivation of analytical solutions embedded into a finite volume method, for general and special solutions of the characteristic methods.

For the analytical solution, we use the Laplace transformation to reduce the equation to an ordinary differential equation. With general initial conditions, e.g. spline functions, the Laplace transformation is accomplished with the help of numerical methods. The proposed discretization method skips the classical error between the convection and reaction equation by using the operator-splitting method.

At the end of the article, we illustrate the higher-order method for different benchmark problems. Finally, the method is shown to produce realistic results.     

Stability and convergence of F iterative scheme with an application to the fractional differential equation

In this paper, we prove that F iterative scheme is almost stable for weak contractions. Further, we prove convergence results for weak contractions as well as for generalized non-expansive mappings due to Hardy and Rogers via F iterative scheme. We also prove that F iterative scheme converges faster than the some known iterative schemes for weak contractions. An illuminative numerical example is formulated to support our assertion. Finally, utilizing our main result the solution of nonlinear fractional differential equation is approximated.

     

Nonlinear least-absolute-values and minimax model fitting

Up to now, the least-absolute-values (l₁) and minimax (l_∞) criteria have seldom been used in model fitting, even if they are preferable for theoretical reasons: their non-differentiability causes numerical and analytical difficulties. Fortunately, a recent reformulation of l₁ and l_∞ as equivalent constrained differentiable problems enables one to numerically solve them using standard nonlinear programming software. This paper exploits this reformulation for analytical purposes. In particular, limits are derived concerning the ability of l₁ and l_∞ to resolve highly similar, nonlinearly parametric terms from error corrupted observations. Examples are exponentials of nearly equal decay in compartmental models and almost coinciding peaks in spectroscopy.     

The solution of a nonclassic problem for one-dimensional hyperbolic equation using the decomposition procedure

In this article, we propose a new approach for solving an initial–boundary value problem with a non-classic condition for the one-dimensional wave equation. Our approach depends mainly on Adomian's technique. We will deal here with new type of nonlocal boundary value problems that are the solution of hyperbolic partial differential equations with a non-standard boundary specification. The decomposition method of G. Adomian can be an effective scheme to obtain the analytical and approximate solutions. This new approach provides immediate and visible symbolic terms of analytic solution as well as numerical approximate solution to both linear and nonlinear problems without linearization. The Adomian's method establishes symbolic and approximate solutions by using the decomposition procedure. This technique is useful for obtaining both analytical and numerical approximations of linear and nonlinear differential equations and it is also quite straightforward to write computer code. In comparison to traditional procedures, the series-based technique of the Adomian decomposition technique is shown to evaluate solutions accurately and efficiently. The method is very reliable and effective that provides the solution in terms of rapid convergent series. Several examples are tested to support our study.     

Regularization Methods for Blind Deconvolution and Blind Source Separation Problems

This paper is devoted to blind deconvolution and blind separation problems. Blind deconvolution is the identification of a point spread function and an input signal from an observation of their convolution. Blind source separation is the recovery of a vector of input signals from a vector of observed signals, which are mixed by a linear (unknown) operator. We show that both problems are paradigms of nonlinear ill-posed problems. Consequently, regularization techniques have to be used for stable numerical reconstructions. In this paper we develop a rigorous convergence analysis for regularization techniques for the solution of blind deconvolution and blind separation problems. Convergence of regularized point spread functions and signals to a solution is established and a convergence rate result in dependence of the noise level is presented. Moreover, we prove convergence of the alternating minimization algorithm for the numerical solution of regularized blind deconvolution problems and present some numerical examples. Moreover, we show that many neural network approaches for blind inversion can be considered in the framework of regularization theory. Date received: August 17, 1999. Date revised: September 1, 2000.     

Development of a mixed control volume – Finite element method for the advection–diffusion equation with spectral convergence

In this paper we attack the problem of devising a finite volume method for computational fluid dynamics and related phenomena which can deal with complex geometries while attaining high-orders of accuracy and spectral convergence at a reasonable computational cost. As a first step towards this end, we propose a control volume finite element method for the solution of the advection–diffusion equation. The numerical method and its implementation are carefully tested in the paper where h- and p-convergence are checked by comparing numerical results against analytical solutions in several relevant test-cases. The numerical efficiency of a selected set of operations implemented is estimated by operation counts, ill-conditioning of coefficient matrices is avoided by using an appropriate distribution of interpolation points and control-volume edges.     

Sensitivity limitations in nonlinear feedback control

In this paper we define nonlinear sensitivity and complementary sensitivity operators of a feedback control loop and show that they satisfy a complemntarity constraint. We then consider the case of general nonlinear open-loop operators that give rise to nonlinear sensitivities that are Lipschitz operators on some Banach space. Under these conditions, we obtain lower bounds on the Lipschitz constants of both operators for open-loop nonminimum phase and unstable nonlinear systems. These results parallel those known in linear control theory on the H_∞ norms of S and T. We finally point to the relevance of the defined nonlinear sensitivities in robustness issues.