A formula for iterated derivatives along trajectories of nonlinear systems

The paper presents a formula for the n-th order total derivative of the output function of a nonlinear systems, in terms of derivatives of the inputs. The formula is given in terms of noncommuting variables and the shuffle product and has potential applications to input-output decoupling and inverses of nonlinear systems.     

An observability criterion for dynamic systems governed by Riccatidifferential equations

In this paper, we present an observability criterion for systems whose state is governed by a matrix Riccati differential equation and whose output is given by a affine transformation     

A capillary free boundary problem governed by the Navier-Stokes equations

In this paper an unsteady capillary free boundary in a viscous imcompressible fluid is studied from the numerical point of view. The linearized problem is formulated, and existence and uniqueness of a solution is proved. The linearized problem is discretized by a finite element method in space and a finite difference method in time. The numerical scheme is proved to be convergent and stable. The method is applied to the computation of free boundaries, eigenfrequencies, damping factors and vibration modes. The influence of the initial conditions, the boundary conditions, the contact angle, the Ohnesorge number and the Bond number is studied.     

Stability of linear systems governed by second order vector differential equations

We consider linear non-autonomous systems governed by second order vector ordinary differential equations. Explicit stability conditions are derived.     

Proximal point method for optimal control processes governed by ordinary differential equations

This paper is concerned with the proximal‐based approach to linear and finite‐difference approximations of constrained convex optimal control problems. We consider control systems governed by ordinary differential equations in the presence of additional terminal/state inequalities and propose a numerical method derived from the proximal point algorithm. The aim of the paper is to study the convergence properties of the obtained conceptual algorithm and to show that it can be used to compute approximate optimal controls. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

State-constrained optimal control problems governed by coupled nonlinear wave equations with memory

This work considers the optimal control problems governed by coupled nonlinear wave equations with memory, where the state constraint is a type of integral. First, we investigate the existence of the optimal solution for the optimal control problem, and then we deduce the maximum principle for the optimal solution with state constraint. Moreover, we give a concrete example which is covered by the main result.     

Real-time optimization and nonlinear model predictive control of processes governed by differential-algebraic equations

Optimization problems in chemical engineering often involve complex systems of nonlinear DAE as the model equations. The direct multiple shooting method has been known for a while as a fast off-line method for optimization problems in ODE and later in DAE. Some factors crucial for its fast performance are briefly reviewed. The direct multiple shooting approach has been successfully adapted to the specific requirements of real-time optimization. Special strategies have been developed to effectively minimize the on-line computational effort, in which the progress of the optimization iterations is nested with the progress of the process. They use precalculated information as far as possible (e.g. Hessians, gradients and QP presolves for iterated reference trajectories) to minimize response time in case of perturbations. In typical real-time problems they have proven much faster than fast off-line strategies. Compared with an optimal feedback control computable upper bounds for the loss of optimality can be established that are small in practice. Numerical results for the Nonlinear Model Predictive Control (NMPC) of a high-purity distillation column subject to parameter disturbances are presented.     

Numerical solution of optimal control problems governed by integro‐differential equations

In this paper, an efficient hybrid approximation scheme for solving optimal control problems governed by integro‐differential equations is proposed. The current approach is based on a generalization of the hybrid of block‐pulse functions and Legendre's polynomials. An upper bound for the generalized hybrid functions with respect to the maximum norm is acquired and its convergence is demonstrated. The optimal control problem under study is transcribed to a mathematical programming one. Two illustrative examples are considered to verify the capability and reliability of the proposed procedure.     

Dynamic programming approach to the optimal control of systems governed by Goursat-Darboux equations

The method of dynamic programming is extended to a class of ‘two-dimensional time’ problems, the Goursat-Darboux systems (hyperbolic systems with initial-boundary data on characteristics). A formal derivation of a maximum principle of Pontryagin's type is obtained by using the ‘semi-dynamic programming’ equations. Controlled Goursat-Darboux systems have applications to the probiems of gas absorption and drying processes.     

On the existence of optimal controls of systems governed by quasilinear parabolic partial delay-differential equations

The optimal-control problem for a distributed-parameter system is considered. The state variables satisfy a parabolic partial delay-differential equation with first boundary conditions. An existence theorem for the solutions of such systems is established. This and an extension of known techniques for systems without delay are used to establish existence theorems for the optimal-control problem under mild conditions involving the weak Cesari property.     

An improved stability test and stabilisation of linear time-varying systems governed by second-order vector differential equations

In this article, the problems of exponential stability analysis and stabilisation of linear time-varying systems described by a class of second-order vector differential equations are considered. Using bounding techniques on the trajectories of a linear time-varying system, the stability problem of the time-varying system is transformed to that of a time-invariant system and a new sufficient condition for the exponential stability is obtained. Moreover, the new criterion is proven to be superior to a test presented in the recent literature. Finally, the proposed criterion is applied to the exponential stabilisation problem via state feedback. The results are illustrated by several numerical examples.     

The application of an approximate non-linear filter to systems governed by coupled ordinary and partial differential equations

An approximate non-linear filter is applied to systems described by coupled ordinary and partial differential equations with both volume and boundary disturbances and measurement errors. Measurements are assumed to be carried out at u discrete number of spatial positions in the distributed portion of the system. The filter is applicable to cases in which the data are available continuously in time and at discrete sampling instants. Both versions of the filter are applied to an ingot heating problem of some interest in steel-making. The numerical results show that the filter is rapidly convergent and quite robust.     

Optimal control of large space structures governed by a coupled system of ordinary and partial differential equations

In this paper we consider the problem of optimal regulation of large space structures in the presence of flexible appendages. For simplicity of presentation, we consider a spacecraft consisting of a rigid bus and a flexible beam. The complete dynamics of the system is given by a coupled set of ordinary and partial differential equations. We show that the solution of this hybrid system is defined in a product space of appropriate finite- and infinite-dimensional spaces. We develop necessary conditions for determining the control torque and forces for optimal regulation of attitude maneuvers of the satellite along with simultaneous suppression of elastic vibrations of the flexible beam.     

Multidimensional gravity,flux and black brane solutions governed by polynomials

Two families of composite black brane solutions are overviewed, fluxbrane and black brane ones, in a model with scalar fields and fields of forms. The metric of any solution is defined on a manifold which contains a product of several Ricci-flat “internal” spaces. The solutions are governed by moduli functions \(\mathcal{H}_s \) (for fluxbranes) and H _s (for black branes), obeying nonlinear differential equations with certain boundary conditions. Themaster equations for \(\mathcal{H}_s \) and H _s are equivalent to Toda-like equations and depend on a nondegenerate matrix A related to brane intersection rules. The functions H _s and \(\mathcal{H}_s \) , as was conjectured and confirmed (at least partly) earlier, should be polynomials in proper variables if A is a Cartan matrix of some semisimple finite-dimensional Lie algebra. The fluxbrane polynomials \(\mathcal{H}_s \) were shown to be used for the construction of black brane polynomials H _s . This approach is illustrated by examples of nonextremal electric black p-brane solutions related to Lie algebras A ₂, C ₂, and G ₂.     

Numercial solution of boundary-valtue problems for second-order elliptic equations with mixed derivatives by effective iteration methods

The paper considers the numerical solution of boundary-value problems for second-order elliptic equations with mixed derivatives occurring, in particular, in the mathematical simulation of physical processes in the anisotropic medium. The spectral properties of matrices obtained from a center-difference approximation of the problem are investigated. For an approximated solution of the system of linear algebraic equations, a new two-step skew-symmetric iteration method is used.     

Accelerating Krawczyk-like interval algorithms for the solution of nonlinear systems of equations by using second derivatives

Recently the properties of Krawczyk-like iterative interval methods for the solution of systems of nonlinear equations have been discussed in several papers (e.g. [2], [3], [5], [6]). These methods converge to a solution under relatively weak conditions provided an initial inclusion vector is known. In the present paper we describe a method that improves the convergence speed for an important class of problems by using second partial derivatives. This method is particularly interesting for large systems with a Jacobi matrix whose off-diagonal coefficients are all constant.