Optimal control of infinite-dimensional uncertain systems

In this paper, we consider a minimax problem of optimal control for a class of strongly nonlinear uncertain evolution equations on a Banach space. We prove the existence of optimal controls. A nontrivial example of a class of systems governed by a nonlinear partial differential equation with uncertain spatial parameters is presented for illustration.This work was supported in part by the National Science and Engineering Research Council of Canada under Grant No. A7109 and The Engineering Faculty Development Fund, University of Ottawa.The authors would like to thank the anonymous reviewers for their valuable comments and suggestions.     

On the maximum principle for time-optimal controls for a class of distributed-boundary control problems

In this note, we present the necessary conditions of optimality for time-optimal controls for a class of distributed-boundary control problems in general Banach spaces using the semigroup theory. Theorem 3.1 is based on a recent general maximum principle due to Barbu (Ref. 1), which was proved for strictly convex reflexive Banach spaces. Theorem 3.2 generalizes this result (for time-optimal control problems) by lifting the assumption.This work was supported by the National Science and Engineering Council of Canada under Grant No. 7109.     

Finite-time null controllability for a class of linear evolution equations on a Banach space with control constraints

We present some necessary and sufficient conditions for null controllability for a class of general linear evolution equations on a Banach space with constraints on the control space. We also present a result on the existence of time-optimal controls and some partial results on the maximum principle. Some interesting insights that can be obtained from these results are discussed, and the paper is concluded with an application to a boundary control problem.This work was supported in part by the National Science and Engineering Council of Canada under Grant No. 7109.The author is thankful to Professor L. Cesari for many helpful suggestions and also for calling his attention to the recent papers of Professor K. Narukawa.     

Relaxation and existence of optimal controls for systems governed by evolution inclusions in separable Banach spaces

In this paper, we examine relaxed control systems governed by evolution inclusions in a separable Banach space. First, we establish the existence of admissible trajectories, correcting an earlier result of Ahmed. Then, we obtain a compactness result for the set of admissible trajectories. Using this compactness result, we prove the existence of optimal solutions for optimal control problems; furthermore, we show that the values of the original and relaxed problems are equal. Finally, we show that the original trajectories are dense in the set of relaxed trajectories. An example is worked out.This research was supported by NSF Grant No. DMS-86-02313.     

二阶微分包含的边值问题及其应用

在Banach中讨论了一类二阶微分包含的边值问题和一类反馈控制问题.结合不动点定理,对解的存在性给出了两个充分性条件,最后应用前面的结果建立了反馈控制问题解的存在性定理.     

On the optimal controls of a class of systems governed by second order parabolic partial delay-differential equations with first boundary conditions

Summary In this paper we consider the question of existence of optimal controls for a class of systems governed by second order parabolic partial delay-differential equations with first boundary conditions and with controls appearing in the coefficients. In Theorems2.2 and2.3 we present existence and uniqueness of solutions of the first boundary problems. In Theorems3.1 and3.2 we prove that whenever the coefficients of the system converge in the w*-topology (L¹ topology on L^∞) the corresponding solutions converge weakly in an appropriate Sobolev space. Using these basic results we present two theorems (Theorems4.1 and4.2) on the existence of optimal controls. Entrata in Redazione il 21 gennaio 1978.     

Optimal control of systems governed by time-delayed,second-order,linear, parabolic partial differential equations with a first boundary condition

In this paper, we consider a class of systems governed by time-delayed, second-order, linear, parabolic partial differential equations with first boundary conditions. The existence and uniqueness of solutions of this class of systems are established in Theorem 3.2. A necessary condition for optimality for the corresponding controlled system is presented in Theorem 5.1. For the proof of this theorem, we develop several preparatory results in Sections 2, 3, and 4.     

On solutions of differential inclusions in homogeneous spaces of functions

In this paper we study the solvability of some classes of differential inclusions with multivalued linear operators in homogeneous spaces of functions. These spaces include a large number of functional spaces like periodic functions and Bohr and Stepanov almost periodic functions. As an application, we consider some existence results for feedback control systems governed by degenerate differential equations of Sobolev type in a Banach space.     

On the optimal control of systems governed by quasilinear integro-partial differential equations of parabolic type

Recently, M. N. O?uztöreli presented certain results on the existence and uniqueness of solutions of systems governed by a linear integro-partial differential equation of parabolic type with delayed arguments. Since his results admit only smooth coefficients, they could not be used directly in the study of the optimal control problems with bounded measurable control variables appearing in the coefficients of the system equations. In this paper, we consider a class of systems described by second-order quasilinear parabolic integro-partial differential equations with all but the second-order coefficients assumed bounded measurable. Our principal results are: Theorem 3.5, which establishes the existence and uniqueness of solutions of this class of systems (with controls in the coefficients), and Theorem 4.4, which gives a necessary condition for optimality for the corresponding controlled system.     

Feedback control systems governed by evolution equations

ABSTRACT

The goal of this paper is to provide systematic approaches to study the feedback control systems governed by evolution equations in separable reflexive Banach spaces. We firstly give some existence results of mild solutions for the equations by applying the Banach's fixed point theorem and the Leray–Schauder alternative fixed point theorem with Lipschitz conditions and some types of boundedness conditions. Next, by using the Filippove theorem and the Cesari property, a new set of sufficient assumptions are formulated to guarantee the existence of feasible pairs for the feedback control systems. Some existence results for an optimal control problem are given. Finally, we apply our main result to obtain a controllability result for semilinear evolution equations and existence results for a class of differential variational inequalities and Clarke's subdifferential inclusions.     

Normal fields of functionals and optimal measurable sections

We study some classes of extended real-valued functions on measurable fields of metric spaces and give sufficient conditions for the existence of optimal measurable sections.The author is grateful to A. Kucia for Lemma 3.3, Theorem 3.1 (i), and helpful discussions.     

Convergence of a feasible directions algorithm for a distributed optimal control problem of parabolic type with terminal inequality constraints

In this paper, we consider a class of optimal control problems with control and terminal inequality constraints, where the system dynamics is governed by a linear second-order parabolic partial differential equation with first boundary condition. A feasible direction algorithm for solving this class of optimal control problems has already been obtained in the literature. The aim of this paper is to improve the convergence result by using a topology arising in the study of relaxed controls.     

Optimal control of systems governed by a class of nonlinear evolution equations in a reflexive Banach space

The paper presents a closure theorem for the attainable trajectories of a class of control systems governed by a large class of nonlinear evolution equations in reflexive Banach spaces. Several existence theorems for optimal controls are proven that include a terminal control problem, a time-optimal control problem, and a special Bolza problem. Some results of independent interest are also presented.This work was supported in part by the National Research Council of Canada under Grant No. 7109.The authors would like to thank Professor L. Cesari for pointing out that joint continuity off is required for the setsG andR to satisfy the upper semicontinuity property (Theorems 5.1 and 5.2).     

Accretive operators which are always single-valued in normed spaces

Set-valued accretive operators in Banach spaces have been extensively studied for several decades. Our main purpose in this paper is to establish a quite revealing result that says that every set-valued lower semi-continuous accretive mapping defined on a normed space is, indeed, single-valued on the interior of its domain. No reference to the well-known Michael’s Selection Theorem is needed. This result is used to extend known theorems concerning the existence of zeros for such operators, as well as showing existence of solutions for variational inclusions.     

Banach空间中一类广义混合非线性隐拟变分包含解的三步迭代

本文引入并研究了实Banach空间中一类新的广义混合非线性隐拟变分包含 ,通过对实Banach空间中的m 增生映象运用Nadler定理和Michael选择定理 ,构建了这类新的广义变分包含解的三步迭代算法 ,并证明了其解的存在性和由迭代算法生成的迭代序列的收敛性     

On the existence of optimal controls for quasilinear parabolic partial differential equations

A class of systems governed by quasilinear parabolic partial differential equations with first boundary conditions is considered. Existence of solutions for this class of systems and theira priori estimates are established. Further, a theorem on the existence of optimal controls for the corresponding control problem is obtained. Its proof is based on Filippov's implicit functions lemma. The control restraint setU is taken as a measurable multifunction.The authors wish to thank Professor L. Cesari for his most valuable comments and suggestions. In fact, a condition assumed in the original version of this paper was substantially relaxed by him. For details, see Remark 4.1.     

On the Solvability and Optimal Controls of Fractional Integrodifferential Evolution Systems with Infinite Delay

In this paper, we study the solvability and optimal controls of a class of fractional integrodifferential evolution systems with infinite delay in Banach spaces. Firstly, a more appropriate concept for mild solutions is introduced. Secondly, existence and continuous dependence of mild solutions are investigated by utilizing the techniques of a priori estimation and extension of step by steps. Finally, existence of optimal controls for system governed by fractional integrodifferential evolution systems with infinite delay is proved.     

Generalized controls: A necessary condition for optimality in a Banach space

A necessary condition is established for optimality in the case of problems where the constraints are simultaneously functions of the trajectory and the control (Problem 3.1, Theorem 5.7); this condition holds for generalized controls with values in a Hausdorff space and for a state space which is a Banach space. To demonstrate the result a new technique is used, based on the differentiability of the trajectory (Theorem 2.6) and the introduction of the notion of pseudosolution (Definition 2.8). These results are then applied to calculus of variations problems in a Banach space (Theorem 6.3).     

Banach空间中一类周期非线性受控系统最优控制的存在性

研究一类强非线性发展方程的周期解及相应的最优控制问题的存在性，首先，证明了Banach空间中一类包含非线性单调算子和非线性非单调扰动的强非线性发展方程周期解的存在性；其次，给出了保证相应的Lagrange最优控制的充分条件；最后，举例说明理论结果在拟线笥抛物方程周期问题及相应的最优控制问题中的应用。     

ON THE VITALI-HAHN-SAKS-NIKODYM THEOREM

Abstract

The classical Vitali-Hahn-Saks-Nikodym Theorem [5, Thm. I.4.8] gives a limit criterion for when a sequence of strongly additive vector measures on a σ-field of sets having their range in a Banach space can be expected to be uniformly strongly additive. In [16, Cor. 8], Saeki proved that the limit condition on the sequence of vector measures could be substantially weakened as long as the Banach space in play is “good enough”. Saeki's result was based upon his work on a class of set functions too large to have Rosenthal's Lemma at his disposal. In Section 2, we prove Saeki's result with Rosenthal's Lemma at the basis of our work and then augment our characterization of Banach spaces enjoying Saeki's result in [1] with another natural equivalent condition. In Section 3 we extend Saeki's result to Boolean algebras having the Subsequential Interpolation property.