Optimal control of parameter distributed systems with impulses

This paper concerns optimal control problems with impulses. The optimal magnitude of impulses and the spatial position of impulses are studied. We obtain maximum principles for these problems.     

Mobile control of distributed parameter systems

Book Review

Mobile control of distributed parameter systemsA. G. Butkovskiy and L. M. Pustyl'nikov: translated by L. W. Longdon, Ellis Horwood, Halsted Press, New York, Chichester, Brisbane, Toronto, 1987, 310 pp. US$45.00. ISBN 0-85312-507-4 (Ellis Horwood Limited), ISBN 0-470-20817-1 (Halsted Press).     

Experimental design for distributed parameter vector systems

We formulate an optimal design problem for the selection of best states to observe and optimal sampling times and locations for parameter estimation or inverse problems involving complex nonlinear partial differential systems. An iterative algorithm for implementation of the resulting methodology is proposed.     

Vibrations of systems with distributed parameters under the influence of random disturbances

Ukrainian Mathematical Journal -     

Stability of distributed parameter systems with finite-dimensional controller-compensators using singular perturbations

Using a singular perturbation formulation of the linear time-invariant distributed parameter system, we develop a method to design finite-dimensional feedback compensators of any fixed order which will stabilize the infinite-dimensional distributed parameter system. The synthesis conditions are given entirely in terms of a finite-dimensional reduced-order model; the stability results depend on an infinite-dimensional version of the Klimushchev-Krasovskii lemma presented here.     

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.     

Periodic measurement strategies for distributed parameter systems filtering

This paper deals with the optimization of the output matrix for a discrete time linear stochastic system. The output matrix varies as a periodic function of time, and its values are constrained to belong to a finite prescribed set. The aim is to minimize the average variance of the Kalman filter estimation error in the periodic steady state. The application regards the optimization both of the measurement points and of the scanning sequence for a distributed parameter system (DPS) of parabolic type. A modal approximation is used to reduce the DPS to finite dimension. The proposed solution algorithm makes use of heuristic rules that enable to overcome the difficulties arising from the cardinality of the admissible set, the possible slow convergence of the relevant Riccati equation and the high dimensionality of the lumped approximate model of the DPS. The numerical applications show that the periodic scanning policies, found by the optimization algorithm, cause a great improvement of the filter performance, with respect to the case where a single fixed sensor is used.     

Dynamic programming in problems of identifying distributed parameter systems

The problem of identifying the input of a system governed by a “semi-linear” evolution equation of parabolic type, based on the results of observations subject to undefined disturbances, is investigated. Estimates of the input, optimal in the sense of the so-called H_∞-criterion, are obtained. The information function of the system—the value function in an appropriate optimization problem—is evaluated. The relations between the information function and information sets are indicated. Optimality principles adequate to the proposed formulations of the problem are formulated and the corresponding dynamic programming equations are derived. Procedures for regularizing the problem, based on evolution equations of the input estimates, are proposed for the heat-conduction equation.     

Dynamical level set method for parameter identification of nonlinear parabolic distributed parameter systems

This article considers a dynamical level set method for the identification problem of the nonlinear parabolic distributed parameter system, which is based on the solvability and stability of the direct PDE (partial differential equation) in Sobolev space. The dynamical level set algorithms have been developed for ill-posed problems in Hilbert space. This method can be regarded as a asymptotical regularization method as long as a certain stopping rule is satisfied. Hence, the convergence analysis of the method is established similar to the proof of convergence of asymptotical regularization. The level set converges to a solution as the artificial time evolves to infinity. Furthermore, the proposed level set method is proved to be stable by using Lyapunov stability theorem, which is constructed in my previous article.Numerical tests are discussed to demonstrate the efficacy of the dynamical level set method, which consequently confirm the level set method to be a powerful tool for the identification of the parameter.     

Normal form observers for hyperbolic distributed parameter systems with spatially varying coefficients

Observer design for 1D linear distributed parameter systems of hyperbolic type with boundary measurement is discussed. The approach is based on the observer canonical form introduced on the basis of a functional-differential equation (f.d.e.) equivalent to the original posed boundary value problem. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The spectral factorization problem for multivariable distributed parameter systems

This paper studies the solution of the spectral factorization problem for multivariable distributed parameter systems with an impulse response having an infinite number of delayed impulses. A coercivity criterion for the existence of an invertible spectral factor is given for the cases that the delays are a) arbitrary (not necessarily commensurate) and b) equally spaced (commensurate); for the latter case the criterion is applied to a system consisting of two parallel transmission lines without distortion. In all cases, it is essentially shown that, under the given criterion, the spectral density matrix has a spectral factor whenever this is true for its singular atomic part, i.e. its series of delayed impulses (with almost periodic symbol). Finally, a small-gain type sufficient condition is studied for the existence of spectral factors with arbitrary delays. The latter condition is meaningful from the system theoretic point of view, since it guarantees feedback stability robustness with respect to small delays in the feedback loop. Moreover its proof contains constructive elements.     

Dissipativity for Lur’e distributed parameter control systems

In this work, dissipativity of Lur’e distributed parameter control systems has been addressed. Delay-dependent sufficient conditions for the dissipativity with respect to the infinite-dimensional version of energy supply rate (Q₁,S₁,R₁) characterized exclusively by unbounded operator Q₁ are established in terms of linear operator inequalities (LOIs). Finally, the heat equation illustrates our result.     

Stability of a class of stochastic distributed parameter systems with random boundary conditions

In this article we consider the question of stability of a class of stochastic systems governed by elliptic and parabolic second order partial differential equations with Neumann boundary conditions. Results on the “stability in the mean” are given in Theorems 1 and 2, and those on “almost sure stability” are presented in Theorems 3 and 4. These results are proved under the assumption that the perturbing forces are measurable stochastic processes defined on $\text{I × Ω}$. In Theorem 5 it is shown that the proofs require only minor modification to admit progressively measurable (predictable or optional) processes.     

Abstract theory of the optimal control of distributed parameter systems

Leningrad. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 29, No. 1, pp. 94–107, January–February, 1988.     

广义算子半群与广义分布参数系统的适定性

首先,针对广义分布参数系统的求解问题,提出了由Hilbert空间中有界线性算子所引导的广义算子半群和广义积分半群;其次,讨论了广义预解算子的性质、广义算子半群与广义积分半群的性质;最后,研究了广义分布参数系统的适定性问题.     

非线性时变广义分布参数系统的适定性问题

本文的主要目的是讨论非线性时变广义分布参数系统的适定性问题.首先,在Banach空间中引入由连续(可能是非线性的)算子引导的非线性广义发展算子,该非线性广义发展算子是广义发展算子的推广,并研究非线性广义发展算子的性质;然后,应用非线性广义发展算子讨论非线性时变广义分布参数系统的适定性问题,并给出解的构造性表达式.     

About the concept of measure-valued solutions to distributed parameter systems

The measure-valued (mv) solutions are commonly defined in the literature simply by putting Young measures into the respective partial differential equations. Here a few examples of mv-solutions to evolution problems (conservation laws, fluid dynamics and a so-called backward-forward heat problem) are re-investigated to show that such definitions are not satisfactorily selective, i.e. they admit (sometimes surprisingly) many mv-solutions apparently without any physical sense. Then an attempt for a selective definition in the case of the backward-forward heat problem is made, resulting to a certain evolution variational inequality.