Optimal Resource Consumption Control with Interval Restrictions

Some method is developed for calculating the optimal resource consumption control with interval restrictions on the components of the control vector. The approach is based on the sequential adjustment of the values of a quasioptimal control actions up to their limit values. The connection is found between the deviations of the initial conditions of the adjoint system and the deviations of the values of the quasioptimal control from the limit values. The rule for specifying the initial approximation is given, and the specific features of the rule are noted. An iterative algorithm is developed, and an example is given.     

Approximate solution to the resource consumption minimization problem. II. Estimates for the proximity of controls

The proximity is estimated of a resource quasioptimal control to the optimal control. We give a method for subdividing the bounded region of initial conditions into subregions and bringing the quasioptimal control closer to the resource optimal control.     

Resource-optimal control of linear systems

A numerical method for minimizing the resource consumption for linear dynamical systems is proposed. It is based on forming a finite-time control that steers the linear system from an arbitrary initial state to the desired terminal state in a given fixed time; this control gives an approximate solution of the problem. It is shown that the structure of the finite-time control makes it possible to determine the structure of the resource-optimal control. A method for determining an initial approximation is described, and an iterative algorithm for calculating the optimal control is proposed. A system of linear algebraic equations relating the deviations of the initial conditions in the adjoint system to the deviations of the phase coordinates from the prescribed terminal state at the terminal point in time is obtained. A computational algorithm is described. The radius of local convergence is found and the quadratic rate of convergence is established. It is proved that the computational procedure and the sequence of controls converge to the resource-optimal control.     

Boundary element methods for Maxwell's equations on non-smooth domains

Summary. Variational boundary integral equations for Maxwell's equations on Lipschitz surfaces in are derived and their well-posedness in the appropriate trace spaces is established. An equivalent, stable mixed reformulation of the system of integral equations is obtained which admits discretization by Galerkin boundary elements based on standard spaces. On polyhedral surfaces, quasioptimal asymptotic convergence of these Galerkin boundary element methods is proved. A sharp regularity result for the surface multipliers on polyhedral boundaries with plane faces is established. Received January 5, 2001 / Revised version received August 6, 2001 / Published online December 18, 2001 Correspondence to: C. Schwab     

Sequential synthesis of time-optimal control for linear systems with disturbances

A method of sequential synthesis of time-optimal control for a linear system with unknown disturbances is considered. A system of linear algebraic equations is obtained which relates the increments of phase coordinates to the increments of initial conditions of a normalized adjoint system and to the increment of control completion time. Evaluations consist in solving repeatedly a system of linear algebraic equations and integrating a matrix differential equation on the displacement intervals of control switching times and on the displacement interval of final control time. A procedure of correcting the switching times and the completion time in moving along the phase trajectory of a controllable object is examined. Simple and constructive conditions are specified for a discontinuous mode to occur, for a representation point to move along the switching manifolds, and for the optimal control structure to transform in moving along the phase trajectory of a system with uncontrollable disturbance. A computational algorithm is presented. It is proved that a sequence of controls converges locally at a quadratic rate and globally to a time-optimal control.     

Numerical methods for the linear Boltzmann transport equation in slab geometry

An iterative method for computing numerical solutions of a finite-difference system corresponding to the linear Boltzmann equation in slab geometry is presented. This iterative scheme gives a straightforward marching process starting from the given boundary and initial conditions. It is shown that with a suitable initial iteration the sequence of iterations converges monotonically to a unique solution of the finite-difference system. This monotone convergence leads to improved upper and lower bounds of the solution in each iteration, and to the well-posedness of the discrete system in the sense of Hadamard. It also leads to the convergence of the discrete system to the continuous system as the mesh size of the space–velocity–time variables approaches to zero. Under a mild restriction on the time-increment the discrete system is numerically stable, independent of the mesh-size of the space and velocity. An error estimate for the computed solution due to simultaneous initial and iteration error is obtained. Also given are some numerical results for the time-dependent and the steady-state solutions.     

The fast Fourier transform and the numerical solution of one-dimensional boundary integral equations

Summary Here we present a fully discretized projection method with Fourier series which is based on a modification of the fast Fourier transform. The method is applied to systems of integro-differential equations with the Cauchy kernel, boundary integral equations from the boundary element method and, more generally, to certain elliptic pseudodifferential equations on closed smooth curves. We use Gaussian quadratures on families of equidistant partitions combined with the fast Fourier transform. This yields an extremely accurate and fast numerical scheme. We present complete asymptotic error estimates including the quadrature errors. These are quasioptimal and of exponential order for analytic data. Numerical experiments for a scattering problem, the clamped plate and plane estatostatics confirm the theoretical convergence rates and show high accuracy.     

Zur L∞-Konvergenz finiter Elemente bei parabolischen Differentialgleichungen

For a parabolic model problem it is shown that the convergence rate of higher order finite element approximation is quasioptimal in L∞. Moreover, the estimate does not depend on the interval [0, T]. The essential tool of the proof is a modified weighted norm technique.     

Un test d'auto-similarité pour les processus gaussiens à accroissements stationnaires

In this Note, we present a method for testing self-similarity of discretized observations of a Gaussian process with stationary increments. The test is based on the estimation of a distance between the process and a set of processes containing all the fractional Brownian motions. This distance is constructed from two estimations of multiscale generalized quadratic variations expectations. The second estimation requires to estimate by regression the self-similarity index H. Both these estimators of H present good robustness and computing time properties compared with maximum likelihood approach, with nearly similar convergence rate.     

Computation of Control for Linear Approximately Controllable System Using Tikhonov Regularization

Computation of control for a controlled partial differential equation is a di?cult task, especially when the control problem is ill posed. In this paper, we propose a method of computing the regularized control of a diffusion control system using Tikhonov regularization approach when the system is approximately controllable. The method proposed here for choosing regularization parameter guarantees the convergence of the proposed control.     

New Algorithm for Computing LQ Suboptimal Output Feedback Gains of Decentralized Control Systems

In this paper, the problem of computing the suboptimal output feedback gains of decentralized control systems is investigated. First, the problem is formulated. Then, the gradient matrices based on the index function are derived and a new algorithm is established based on some nice properties. This algorithm shows that a suboptimal gain can be computed by solving several ordinary differential equations (ODEs). In order to find an initial condition for the ODEs, an algorithm for finding a stabilizing output feedback gain is exploited, and the convergence of this algorithm is discussed. Finally, an example is given to illustrate the proposed algorithm.     

On the boundary element method for some nonlinear boundary value problems

Summary Here we analyse the boundary element Galerkin method for two-dimensional nonlinear boundary value problems governed by the Laplacian in an interior (or exterior) domain and by highly nonlinear boundary conditions. The underlying boundary integral operator here can be decomposed into the sum of a monotoneous Hammerstein operator and a compact mapping. We show stability and convergence by using Leray-Schauder fixed-point arguments due to Petryshyn and Neas.Using properties of the linearised equations, we can also prove quasioptimal convergence of the spline Galerkin approximations.This work was carried out while the first author was visiting the University of Stuttgart     

Analysis of a bilinear finite element for shallow shells I: Approximation of inextensional deformations

We consider a bilinear reduced-strain finite element formulation for a shallow shell model of Reissner-Naghdi type. The formulation is closely related to the facet models used in engineering practice. We estimate the error of this scheme when approximating an inextensional displacement field. We make the strong assumptions that the domain and the finite element mesh are rectangular and that the boundary conditions are periodic and the mesh uniform in one of the coordinate directions. We prove then that for sufficiently smooth fields, the convergence rate in the energy norm is of optimal order uniformly with respect to the shell thickness. In case of elliptic shell geometry the error bound is furthermore quasioptimal, whereas in parabolic and hyperbolic geometries slightly enhanced smoothness is required, except for the degenerate cases where the characteristic lines are parallel with the mesh lines. The error bound is shown to be sharp.

     

The boundary element spline collocation for nonuniform meshes on the torus

The spline collocation method for a class of biperiodic strongly elliptic pseudodifferential operators is considered. As trial functions tensor products of odd degree splines are used and the collocation is imposed at the nodal points of the tensor product mesh. It is shown that the collocation problem is uniquely solvable if the maximum mesh length is small enough. Moreover, the approximation is stable and quasioptimal with respect to a norm depending on the order of the operator and the degree of approximating splines. Some convergence results are given for general and quasiuniform meshes. The results cover for example the single layer and the hypersingular operators.     

ON/OFF strategy based minimum-time control of continuous Petri nets

This paper focuses on the target marking control problem of timed continuous Petri nets (TCPN), aiming to drive the system from an initial state to a desired final one. This problem is similar to the set-point control problem in a general continuous-state system. In a previous work, a simple and efficient ON/OFF controller was proposed for Choice-Free nets, and it was proved to be minimum-time (Wang, 2010). However, for general TCPN the ON/OFF controller may bring the system to “blocking” situations due to its “greedy” firing strategy, and the convergence to the final state is not ensured. In this work the ON/OFF controller is extended to general TCPN by adding more “fair” strategies to solve conflicts in the system: the ON/OFF+ controller is obtained by forcing proportional firings of conflicting transitions. Nevertheless, such kind of controller might highly slow down the system when transitions have flows of different orders of magnitude, therefore a balancing process is introduced, leading to the B-ON/OFF controller. A third approach introduced here is the MPC-ON/OFF controller, a combination of Model Predictive Control (MPC) and the ON/OFF strategy; it may achieve a smaller number of time steps for reaching the final states, but usually requires more CPU time for computing the control laws. All the proposed extensions are heuristic methods for the minimum-time control and their convergences are proved. Finally, an application example of a manufacturing cell is considered to illustrate the methods. It is shown that by using the proposed controllers, reasonable numbers of time steps for reaching the final state can be obtained with low computational complexity.     

有限时间迭代学习控制

针对任意初态情形, 借助于初始修正吸引子的概念,讨论不确定时变系统能够达到实际完全跟踪性能的迭代学习控制方法.闭环系统中含有限时间控制作用, 在预先指定的区间上实现零误差跟踪,且起始段的系统输出轨迹也可预先规划.分别讨论部分限幅学习与完全限幅学习, 证明闭环系统中各变量的一致有界性以及误差序列的一致收敛性. 变量有界性证明得益于提出的限幅学习算法,特别是完全限幅学习算法可确保参数估值的变化范围.     

A Generalization of the Arnold-Wendland Lemma to a Modified Collocation Method for Boundary Integral Equations in ℝ3

We prove quasioptimal and optimal order estimates in various Sobolev norms for the approximation of linear strongly elliptic periodic pseudodifferential equations in two independent variables by a modified method of nodal collocation by odd degree polynomial splines. In the one-dimensional case, our method coincides with the method of nodal collocation when odd degree polynomial splines are employed for the trial functions. The convergence analysis is based on an equivalence which we establish between our method and a nonstandard Galerkin method for an operator closely related to the given operator. This equivalence is realized through a crucial intermediate result (which we now term the Arnold-Wendland lemma) to connect the solution of central finite difference equations and that of certain nonstandard Galerkin equations. The results of this paper are genuine two-dimensional generalizations of the results obtained by ARNOLD and WENDLAND in [2] for the one-dimensional equations.     

Global controllability and computation of controls in nonlinear systems

The authors consider the problem of constructing an admissible open-loop control of bounded energy steering a nonlinear system from a given initial state to a given final state under the condition that the first-approximation system is completely controllable. The convergent iterative procedure for computing an admissible control is verified. It is shown that a nonlinear system locally controllable with respect to the first approximation becomes globally completely controllable for any boundary conditions from the stability region if the initial nonlinear system is stabilizable up to the asymptotic stability in the large and in the whole and the nonlinear terms either satisfy the global Cauchy-Lipschitz condition or are polynomials of a certain degree in state coordinates with arbitrary coefficients. The nonlinear system of algebraic equations to computation of whose solutions the problem of constructing the admissible control reduces is indicated. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 26, Nonlinear Dynamics, 2005.     

Convergence of Algorithms in Optimization and Solutions of Nonlinear Equations

It is desirable that an algorithm in unconstrained optimization converges when the guessed initial position is anywhere in a large region containing a minimum point. Furthermore, it is useful to have a measure of the rate of convergence which can easily be computed at every point along a trajectory to a minimum point. The Lyapunov function method provides a powerful tool to study convergence of iterative equations for computing a minimum point of a nonlinear unconstrained function or a solution of a system of nonlinear equations. It is surprising that this popular and powerful tool in the study of dynamical systems is not used directly to analyze the convergence properties of algorithms in optimization. We describe the Lyapunov function method and demonstrate how it can be used to study convergence of algorithms in optimization and in solutions of nonlinear equations. We develop an index which can measure the rate of convergence at all points along a trajectory to a minimum point and not just at points in a small neighborhood of a minimum point. Furthermore this index can be computed when the calculations are being carried out.     

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In this paper the local functional limit theorem for increments of a Brownian motion is derived with large and small deviations, and the local functional convergence rate for increments of Brownian motion in Holder norm with respect to (r,p)capacity is estimated.