Value function and optimality condition for semilinear control problems. II: Parabolic case

In this paper we continue to study properties of the value function and of optimal solutions of a semilinear Mayer problem in infinite dimensions. Applications concern systems governed by a state equation of parabolic type. In particular, the issues of the joint Lipschitz continuity and semiconcavity of the value function are treated in order to investigate the differentiability of the value function along optimal trajectories.The research of P. Cannarsa was partially supported by the Italian National Project MURST Equazioni di Evoluzione e Applicazioni Fisico-Matematiche.     

On Controllability of an Elastic Ring

We study the exact controllability problem for a ring under stretching tension that varies in time. We are looking for a couple of forces, which drive the state solution to rest. We show that applying two forces is necessary for controllability and the ring is controllable in the time interval greater than the optical length of the string. We also explain why one force would not be enough to control the ring. We use the method of moments to reduce the controllability problem to a moment problem for the controlling forces. The solution of that problem is based on an auxiliary basis property result. Both method of moments and proof of the basis property are developed for the model with time-dependent parameters. S.A.’s research was supported in part by the NSF, grant ARC–0724860. B.B.’s research was supported in part by University of Tennessee at Chattanooga Faculty Research Grant. S.I.’s research was supported in part by the Russian Foundation for Basic Research, grants 08-01-00595a and 08-01-00676a.     

On pole assignment of distributed parameter systems with unbounded input elements

In this paper, the pole assignment problem is considered for a class of distributed parameter systems with unbounded input element and with multiple spectral structure. A formula on the spectrum of the closed loop operator is proved and a formula of pole assignment is obtained. Finally, an example concerning a beam vibration is given. This work is supported by the National Natural Sciences Foundation of China and the National Key Project of China, and partly by the Post Doctoral Science Foundation of China and the Youth Science Foundation of Shanxi.     

Digital modeling and control of multiple time-delayed distributed power grid

Distributed power grid (DPG) control systems are so highly interconnected that the effects of local disturbances as well as transmission time delays can be amplified as they propagate through a complex network of transmission lines. These effects deteriorate control performance and could possibly destabilize the overall system. In this paper, a new approximated discretization method and digital design for DPG control systems with multiple state, input and output delays as well as a generalized bilinear transformation method are presented. Based on a procedure for the generation of impulse response data, the multiple fractional/integer time-delayed continuous-time system is transformed to a discrete-time model with multiple integer time delays. To implement the digital modeling, the singular value decomposition (SVD) of a Hankel matrix together with an energy loss level is employed to obtain an extended discrete-time state space model. Then, the extended discrete-time state space model of the DPG control system is reformulated as an integer time-delayed discrete-time system by computing its observable canonical form. The proposed method can closely approximate the step response of the original continuous time-delayed DPG control system by choosing various energy loss levels. For completeness, an optimal digital controller design for the DPG control system and a generalized bilinear transformation method with a tunable parameter are also provided, which can re-transform the integer time-delayed discrete-time model to its continuous-time model. Illustrative examples are given to demonstrate the effectiveness of the developed method.     

Asymptotic coefficients for Gaussian radial basis function interpolants

Gaussian radial basis functions (RBFs) on an infinite interval with uniform grid pacing h are defined by ?(x;α,h)≡exp(-[α²/h²]x²). The only significant numerical parameter is α, the inverse width of the RBF functions relative to h. In the limit α→0, we demonstrate that the coefficients of the interpolant of a typical function f(x) grow proportionally to exp(π²/[4α²]). However, we also show that the approximation to the constant f(x)≡1 is a Jacobian theta function whose coefficients do not blow up as α→0. The subtle interplay between the complex-plane singularities of f(x) (the function being approximated) and the RBF inverse width parameter α are analyzed. For α≈1/2, the size of the RBF coefficients and the condition number of the interpolation matrix are both no larger than O(10⁴) and the error saturation is smaller than machine epsilon, so this α is the center of a “safe operating range” for Gaussian RBFs.     

An analytic approximation to the cardinal functions of Gaussian radial basis functions on an infinite lattice

Gaussian radial basis functions (RBFs) have been very useful in computer graphics and for numerical solutions of partial differential equations where these RBFs are defined, on a grid with uniform spacing h, as translates of the “master” function (x;α,h)≡exp(-[α²/h²]x²) where α is a user-choosable constant. Unfortunately, computing the coefficients of (x-jh;α,h) requires solving a linear system with a dense matrix. It would be much more efficient to rearrange the basis functions into the equivalent “Lagrangian” or “cardinal” basis because the interpolation matrix in the new basis is the identity matrix; the cardinal basis C_j(x;α,h) is defined by the set of linear combinations of the Gaussians such that C_j(kh)=1 when k=j and C_j(kh)=0 for all integers . We show that the cardinal functions for the uniform grid are C_j(x;h,α)=C(x/h-j;α) where C(X;α)≈(α²/π)sin(πX)/sinh(α²X). The relative error is only about 4exp(-2π²/α²) as demonstrated by the explicit second order approximation. It has long been known that the error in a series of Gaussian RBFs does not converge to zero for fixed α as h→0, but only to an “error saturation” proportional to exp(-π²/α²). Because the error in our approximation to the master cardinal function C(X;α) is the square of the error saturation, there is no penalty for using our new approximations to obtain matrix-free interpolating RBF approximations to an arbitrary function f(x). The master cardinal function on a uniform grid in d dimensions is just the direct product of the one-dimensional cardinal functions. Thus in two dimensions . We show that the matrix-free interpolation can be extended to non-uniform grids by a smooth change of coordinates.     

无界单输入分布参数系统的一秩扰动和极点配置

本文考虑一类带有无界单输入元的分布参数系统的无条件基扰动和极点配置问题,推广了文［1－4］的结果,同时亦解决了文［5］中提出的一个猜想．     

Optimal quasi-static shape control for large aerospace antennae

In this paper, we propose an on-line control approach which will adjust the steady-state shape of a large antenna arbitrarily close to any achievable desired profile. The method makes use of distributed parameter system theory and allows refocusing using a limited number of control actuators and sensors.The controller gains are calculated by approximating the solution to an infinite-dimensional optimal quasi-static control problem. We prove a very general convergence result for such quasi-static controllers here and apply it to the antenna controller to show convergence using any Galerkin (finite-element) approximation method.This research was partially supported by the National Science Foundation, the National Aeronautics and Space Administration, and the Air Force Office of Scientific Research under Grants ECS-80-16173, NAG-1-171, and AFOSR-83-0124, respectively.     

Suboptimal shape control for quasi-static distributed-parameter systems

We propose an on-line control approach which will adjust the steady-state shape of a large antenna arbitrarily close to any achievable desired profile. The method makes use of distributed-parameter system theory and allows refocusing using a limited number of control actuators and sensors.The controller gains are calculated by approximating the solution to an infinite-dimensional optimal quasi-static control problem. The controller gain calculation is computationally simpler than that proposed in a companion paper. The Galerkin (finite element) approximation method is used for model reduction. We prove that both gain and state convergence can be achieved by using the proposed approximation scheme.This work was partially supported by the Air Force Office of Scientific Research, Grant No. AFOSR 83-0124, and by the National Aeronautics and Space Administration, Grant No. NAG-1-515.     

Procedure for asymptotic state and parameter estimation of nonlinear distributed parameter bioreactors

Due to measurement problem in biological variables, such as the biomass concentration and its specific growth rate, the on-line measurement of these variables is not ensured and this presents a great difficulty, especially when we consider the control problem. This paper presents and analyses an approach for estimating these biological variables through an example of a nonlinear distributed parameter bioreactor. An orthogonal collocation method is applied in the distributed system to obtain a system of ordinary differential equations. A simulation study shows the feasibility and the robustness of the estimator used for this nonlinear process.     

三角域上带两个形状参数的Bézier曲面的扩展

给出了三角域上带双参数λ1,λ2的类三次Bernstein基函数,它是三角域上三次Bernstein基函数的扩展.分析了该组基的性质并定义了三角域上带有两个形状参数λ1,λ2的类三次Bernstein-Bézier(B-B)参数曲面.该基函数及参数曲面分别具有与三次Bernstein基函数及三次B-B参数曲面类似的性质.当λ1,λ2取特殊的值时,可分别得到三次Bernstein基函数及三次B-B参数曲面以及参考文献中所定义的类三次Bernstein基函数及类三次B-B参数曲面.由实例可知,通过改变形状参数的取值,可以调整曲面的形状.     

双边截断型分布族参数函数的渐近最优经验Bayes估计

本文考虑一维双边截断型分布族参数函数在平方损失下的经验 Bayes估计问题 .给定θ,X的条件分布为f (x|θ) =ω(θ1,θ2 ) h(x) I[θ1,θ2 ] (x) dx其中θ =(θ1,θ2 )T(x) =(t1(x) ,t2 (x) ) =(min(x1,… ,xm) ,max(x1,… ,xm) )是充分统计量 ,其边缘密度为 f (t) ,本文通过 f (t)的核估计构造出θ的函数的经验 Bayes估计 ,并证明在一定的条件下是渐近最优的 (a.0 .)     

Spectral reduction for control systems modeling passive integrated circuits

Linear control systems modeling passive integrated circuits are examined. A new algebraic method of spectral reduction equipped with efficient tools for preserving passivity is proposed and justified. For RC networks (circuits), this method is similar to and can be regarded as an extension of the well-known PACT method, which is based on congruence transforms. Up to now, such an extension seemed to be impossible, and different techniques were used for the reduction of RCL and RCLM networks. Some numerical results are presented.