On the optimal control of integral-functional equations

The problem of the optimal control of stochastic integral-functional equations of neutral type with an intergral quality functional is considered. For the case of a linear quadratic problem an explicit form of the optimal control is presented.

A class of equations which originated in the synthesis of Volterra equations, and stochastic differential equations with after-effects of neutral type are discussed. The problem of the optimal control of such systems is an essential development of the theory of controlled differential equations /1–8/. Examples of real objects whose mathematical models contain equations with an after-effect are discussed in /9/. A study of integral equations of neutral type is essential in controlling the motion of bodies in a continuous medium, /10/. Volterra equations first arose in the theory of creep and form the basis of this theory /11, 12/.     

The Fundamental Solution and Its Role in the Optimal Control of Infinite Dimensional Neutral Systems

In this work, we shall consider standard optimal control problems for a class of neutral functional differential equations in Banach spaces. As the basis of a systematic theory of neutral models, the fundamental solution is constructed and a variation of constants formula of mild solutions is established. We introduce a class of neutral resolvents and show that the Laplace transform of the fundamental solution is its neutral resolvent operator. Necessary conditions in terms of the solutions of neutral adjoint systems are established to deal with the fixed time integral convex cost problem of optimality. Based on optimality conditions, the maximum principle for time varying control domain is presented. Finally, the time optimal control problem to a target set is investigated.     

Runge–Kutta–collocation methods for systems of functional–differential and functional equations

Systems of functional–differential and functional equations occur in many biological, control and physics problems. They also include functional–differential equations of neutral type as special cases. Based on the continuous extension of the Runge–Kutta method for delay differential equations and the collocation method for functional equations, numerical methods for solving the initial value problems of systems of functional–differential and functional equations are formulated. Comprehensive analysis of the order of approximation and the numerical stability are presented.     

Maximum principle for optimal control of neutral stochastic functional differential systems

This paper is concerned with optimal control of neutral stochastic functional differential equations (NSFDEs). The Pontryagin maximum principle is proved for optimal control, where the adjoint equation is a linear neutral backward stochastic functional equation of Volterra type (VNBSFE). The existence and uniqueness of the solution are proved for the general nonlinear VNBSFEs. Under the convexity assumption of the Hamiltonian function, a sufficient condition for the optimality is addressed as well.     

Contractivity and exponential stability of solutions to nonlinear neutral functional differential equations in banach spaces

A series of contractivity and exponential stability results for the solutions to nonlinear neutral functional differential equations (NFDEs) in Banach spaces are obtained,which provide unified theoretical foundation for the contractivity analysis of solutions to nonlinear problems in functional differential equations (FDEs),neutral delay differential equations (NDDEs) and NFDEs of other types which appear in practice.     

On abstract degenerate neutral differential equations

We introduce a new abstract model of functional differential equations, which we call abstract degenerate neutral differential equations, and we study the existence of strict solutions. The class of problems and the technical approach introduced in this paper allow us to generalize and extend recent results on abstract neutral differential equations. Some examples on nonlinear partial neutral differential equations are presented.     

Controlling linear functional differential equation

Necessary conditions for the optimal control of a linear system of neutral type functional differential equations are obtained. We show that a success in reducing the initial problem to the problem of solvability of a certain boundary value problem is, in principle, a question of one's ability to construct adjoint operators to the operators appearing in the initial control problem.     

A unified theory of necessary conditions for nonlinear nonconvex control systems

We consider optimal problems for a general nonlinear nonconvex input-output relation for Banach space valued functions. A maximum principle is obtained using Ekeland's variational principle. The formulation applies to systems described by ordinary differential equations, functional differential equations, and partial differential equations (both for distributed and boundary control systems).This work was supported in part by the National Science Foundation under Grant No. DMS-8200645.     

Maximum principles for control systems described by measure functional differential equations

Three optimal control problems involving measure functional differential equations are considered. The necessary conditions, in the form of the Pontryagin maximum principle, for an optimal control are obtained. This is accomplished by the application of a theorem by Debovistskii-Milyutin. A simple example is also illustrated to show the applicability of the results obtained.     

Maximum principle for the optimal control of a hyperbolic equation in one space dimension,part 2: Application

The optimal open-loop control of a beam subject to initial disturbances is studied by means of a maximum principle developed for hyperbolic partial differential equations in one space dimension. The cost functional representing the dynamic response of the beam is taken as quadratic in the displacement and its space and time derivatives. The objective of the control is to minimize a performance index consisting of the cost functional and a penalty term involving the control function. Application of the maximum principle leads to boundary-value problems for hyperbolic partial differential equations subject to initial and terminal conditions. The explicit solution of this system is obtained yielding the expressions for the state and optimal control functions. The behavior of the controlled and uncontrolled beam is studied numerically, and the effectiveness of the proposed control is illustrated.     

On the Numerical Solution of Differential Games for Neutral-Type Linear Systems

The paper deals with a zero-sum differential game, in which the dynamics of a conflict-controlled system is described by linear functional differential equations of neutral type and the quality index is the sum of two terms: the first term evaluates the history of motion of the system realized up to the terminal time, and the second term is an integral–quadratic evaluation of the corresponding control realizations of the players. To calculate the value and construct optimal control laws in this differential game, we propose an approach based on solving a suitable auxiliary differential game, in which the motion of a conflict-controlled system is described by ordinary differential equations and the quality index evaluates the motion at the terminal time only. To find the value and the saddle point in the auxiliary differential game, we apply the so-called method of upper convex hulls, which leads to an effective solution in the case under consideration due to the specific structure of the quality index and the geometric constraints on the control actions of the players. The efficiency of the approach is illustrated by an example, and the results of numerical simulations are presented. The constructed optimal control laws are compared with the optimal control procedures with finitedimensional approximating guides, which were developed by the authors earlier.     

中立型随机泛函微分方程的Khasminskii型定理

本文对中立型随机泛函微分方程建立了Khasminskii型定理,这个定理显示在局部Lipschitz条件但是不要求线性增长的条件下,中立型随机泛函微分方程存在一个全局解.本文的这个解存在性条件可以包含更广的一类非线性中立型随机泛函微分方程.最后,本文给出一个例子来阐述我们的思想.     

Study of optimal control problems on the half-line by the averaging method

We justify the application of the averaging method to optimal control problems for systems of differential equations on the half-line. For optimal control problems for systems of differential equations linear in the control, we prove the existence of optimal controls for the exact and averaged problems. We show that an optimal control in the averaged problem is ɛ-optimal in the exact problem.     

Optimal birth control of population dynamics

The authors studied optimal birth control policies for an age-structured population of McKendrick type which is a distributed parameter system involving 1st order partial differential equations with nonlocal bilinear boundary control. The functional analytic approach of Dubovitskii and Milyutin is adopted in the investigation. Maximum principles for problems with a free end condition and fixed final horizon are developed, and the time optimal control problems, the problem with target sets, and infinite planning horizon case are investigated.     

中立型泛函微分方程的周期解

对于中立型泛函微分方程,证明了解的毕竟有界性蕴含周期解的存在性,把常微分方程中著名的Yoshizawa周期解存在定理推广到中立型泛函微分方程,然后利用所得结果给出一类产生于电力系统的中立型时滞泛函微分方程周期解存在惟一与吸引的条件。     

The first variation and Pontryagin’s maximum principle in optimal control for partial differential equations

A modification of the classical needle variation, namely, the so-called two-parameter variation of controls is proposed. The first variation of a functional is understood as a repeated limit. It is shown that the modified needle variation can be effectively used to derive necessary optimality conditions for a rather wide class of optimal control problems involving partial differential equations with weak solutions. Specifically, the two-parameter variation is used to obtain necessary optimality conditions in the form of a maximum principle for the optimal control of divergent hyperbolic equations.     

On optimality conditions for the guaranteed result in control problems for time-delay systems

For control problems under disturbance of dynamical systems described by differential equations with discrete and distributed time delays and with initial data satisfying the Lipschitz property, the corresponding Lipschitzness of the optimal guaranteed result functional is established and inequalities for its directional derivatives are obtained.     

非线性中立型泛函微分方程Runge-Kutta 法的稳定性和收敛性

本文涉及Runge-Kutta 法变步长求解非线性中立型泛函微分方程(NFDEs) 的稳定性和收敛性.为此, 基于Volterra 泛函微分方程Runge-Kutta 方法的B- 理论, 引入了中立型泛函微分方程Runge-Kutta 方法的EB (expanded B-theory)-稳定性和EB-收敛性概念. 之后获得了Runge-Kutta 方法变步长求解此类方程的EB - 稳定性和EB- 收敛性. 这些结果对中立型延迟微分方程和中立型延迟积分微分方程也是新的.     

A generalization of a minimax theorem of Fan via a theorem of the alternative

A minimization problem for a functional on a convex subsetC of a normed linear space is considered. Under certain hypotheses, optimality in a certain subset ofC implies the validity of first-order necessary optimality conditions for the problem inC. The result is applied to a problem in optimal periodic control of neutral functional differential equations.This work was partially supported by a grant from Deutsche Forschungsgemeinschaft and by AFOSR under Grant No. AFOSR-84-0398.