Discrete Optimal Control: Second Order Optimality Conditions

In this paper, we present a survey and refinement of our recent results in the discrete optimal control theory. For a general nonlinear discrete optimal control problem (P) , second order necessary and sufficient optimality conditions are derived via the nonnegativity ( I S 0) and positivity ( I >0) of the discrete quadratic functional I corresponding to its second variation. Thus, we fill the gap in the discrete-time theory by connecting the discrete control problems with the theory of conjugate intervals, Hamiltonian systems, and Riccati equations. Necessary conditions for I S 0 are formulated in terms of the positivity of certain partial discrete quadratic functionals, the nonexistence of conjugate intervals, the existence of conjoined bases of the associated linear Hamiltonian system, and the existence of solutions to Riccati matrix equations. Natural strengthening of each of these conditions yields a characterization of the positivity of I and hence, sufficiency criteria for the original problem (P) . Finally, open problems and perspectives are also discussed.     

Necessary Conditions for Nonsmooth,Infinite-Horizon,Optimal Control Problems

Necessary conditions are proved for deterministic nonsmooth optimal control problems involving an infinite horizon and terminal conditions at infinity. The necessary conditions include a complete set of transversality conditions.     

Second Order Sufficient Conditions for Optimal Control Problems with Non-unique Minimizers: An Abstract Framework

Standard second order sufficient conditions in optimal control theory provide not only the information that an extremum is a weak local minimizer, but also tell us that the extremum is locally unique. It follows that such conditions will never cover problems in which the extremum is continuously embedded in a family of constant cost extrema. Such problems arise in periodic control, when the cost is invariant under time translations, in shape optimization, where the cost is invariant under Euclidean transformations (translations and rotations of the extremal shape), and other areas where the domain of the optimization problem does not really comprise elements in a linear space, but rather an equivalence class of such elements. We supply a set of sufficient conditions for minimizers that are not locally unique, tailored to problems of this nature. The sufficient conditions are in the spirit of earlier conditions for ‘non-isolated’ minima, in the context of general infinite dimensional nonlinear programming problems provided by Bonnans, Ioffe and Shapiro, and require coercivity of the second variation in directions orthogonal to the constant cost set. The emphasis in this paper is on the derivation of directly verifiable sufficient conditions for a narrower class of infinite dimensional optimization problems of special interest. The role of the conditions in providing easy-to-use tests of local optimality of a non-isolated minimum, obtained by numerical methods, is illustrated by an example in optimal control.     

First Order LSFEM for Second Order Hyperbolic Problems

In the last years the interest in least squares finite element methods has grown due to some interesting properties of these methods (cf. the monographs [1], [2]). While for many elliptic problems the theoretical background has been established, only a few articles analyse least squares methods for transient problems. In this article some new stability estimates for the least squares method for the wave equation in 1 and 2 dimensions are derived. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Second-Order Necessary Optimality Conditions in Optimal Impulsive Control Problems

Impulsive optimal control problems are studied. Under the Frobenius conditions, second-order necessary optimality conditions are proved without any a priori normality assumptions.     

Necessary Optimality Conditions for Optimal Control Problems with Nonsmooth Mixed State and Control Constraints

In this paper we study an optimal control problem with nonsmooth mixed state and control constraints. In most of the existing results, the necessary optimality condition for optimal control problems with mixed state and control constraints are derived under the Mangasarian-Fromovitz condition and under the assumption that the state and control constraint functions are smooth. In this paper we derive necessary optimality conditions for problems with nonsmooth mixed state and control constraints under constraint qualifications based on pseudo-Lipschitz continuity and calmness of certain set-valued maps. The necessary conditions are stratified, in the sense that they are asserted on precisely the domain upon which the hypotheses (and the optimality) are assumed to hold. Moreover necessary optimality conditions with an Euler inclusion taking an explicit multiplier form are derived for certain cases.     

First-Order Necessary Conditions in Optimal Control

Journal of Optimization Theory and Applications - In an earlier analysis of strong variation algorithms for optimal control problems with endpoint inequality constraints, Mayne and Polak provided...     

Necessary Conditions for Impulsive Nonlinear Optimal Control Problems without a priori Normality Assumptions

First-order and second-order necessary conditions of optimality for an impulsive control problem that remain informative for abnormal control processes are presented and derived. One of the main features of these conditions is that no a priori normality assumptions are required. This feature follows from the fact that these conditions rely on an extremal principle which is proved for an abstract minimization problem with equality constraints, inequality constraints, and constraints given by an inclusion in a convex cone. Two simple examples illustrate the power of the main result.The first author was partially supported by the Russian Foundation for Basic Research Grant 02-01-00334. The second author was partially supported by the Russian Foundation for Basic Research Grant 00-01-00869. The third author was partially supported by Fundacao para a Ciencia e Tecnologia and by INVOTAN Grant.     

向量值测度与线性系统最优控制问题的必要条件

The vector-valued measure defined by the well-posed linear boundary value problems is discussed. The maximum principle of the optimal control problem with non-convex constraintis proved by using the vector-valued measure. Especially, the necessary conditions of the optimal control of elliptic systems is derived without the convexity of the control domain and the cost function. optimal control, maximum principle, distributed parameter system, linear system,vector-valued measure.     

Boundary Value Problems for First Order Stochastic Differential Equations

In this paper, we present a new technique to study nonlinear stochastic differential equations with periodic boundary value condition （in the sense of expectation）. Our main idea is to decompose the stochastic process into a deterministic term and a new stochastic term with zero mean value. Then by using the contraction mapping principle and Leray-Schauder fixed point theorem, we obtain the existence theorem. Finally, we explain our main results by an elementary example.     

一阶随机微分方程的边值问题

In this paper,we present a new technique to study nonlinear stochastic differential equations with periodic boundary value condition (in the sense of expectation).Our main idea is to decompose the stochastic process into a deterministic term and a new stochastic term with zero mean value.Then by using the contraction mapping principle and Leray-Schauder fixed point theorem,we obtain the existence theorem.Finally,we explain our main results by an elementary example.     

一阶中立型差分方程振动的充分必要条件

本文研究了一阶常系数中立型时滞差分方程△[x(n)-px(n-(τ))]+qx(n-σ)=0的振动性.当p＞1时,通过构造若干适当的函数,我们分别得到了在Τ-σ＜1,(τ)-σ=1和(τ)-σ＞1三种情形下该方程振动的充分必要条件.     

一阶中立型时滞差分方程振动的充分必要条件

研究了一阶常系数中立型时滞差分方程A[x(n)-px(n-τ)]+qx(n-σ)=0的振动性.通过构造若干适当的函数,分别得到了在01两种情况下该方程的一切解振动的充分必要条件.     

随机递归最优控制和混合最优控制问题

对随机递归最优控制问题即代价函数由特定倒向随机微分方程解来描述和递归混合最优控制问题即控制者还需 决定最优停止时刻, 得到了最优控制的存在性结果. 在一类等价概率测度集中,还给出了递归最优值函数的最小和最大数学期望.     

Necessary Conditions for Optimal Terminal Time Control Problems Governed by a Volterra Integral Equation

We prove the maximum principle for optimal terminal time control problems with the state governed by a Volterra integral equation and constraints depending on the terminal time and the state. We use Pontryagin-type perturbations to reduce the problem to a well-known result of optimizati n theory.Communicated by D. A. Carlson     

Meshfree Approximation for Stochastic Optimal Control Problems

In this work,we study the gradient projection method for solving a class of stochastic control problems by using a mesh free approximation ap-proach to implement spatial dimension approximation.Our main contribu-tion is to extend the existing gradient projection method to moderate high-dimensional space.The moving least square method and the general radial basis function interpolation method are introduced as showcase methods to demonstrate our computational framework,and rigorous numerical analysis is provided to prove the convergence of our meshfree approximation approach.We also present several numerical experiments to validate the theoretical re-sults of our approach and demonstrate the performance meshfree approxima-tion in solving stochastic optimal control problems.     

First Order Optimality Conditions for Generalized Semi-Infinite Programming Problems

We study first-order optimality conditions for the class of generalized semi-infinite programming problems (GSIPs). We extend various well-known constraint qualifications for finite programming problems to GSIPs and analyze the extent to which a corresponding Karush-Kuhn-Tucker (KKT) condition depends on these extensions. It is shown that in general the KKT condition for GSIPs takes a weaker form unless a certain constraint qualification is satisfied. In the completely convex case where the objective of the lower-level problem is concave and the constraint functions are quasiconvex, we show that the KKT condition takes a sharper form. The authors thank the anonymous referees for careful reading of the paper and helpful suggestions. The research of the first author was partially supported by NSERC.     

Necessary Conditions for Free End-Time, Measurably Time Dependent Optimal Control Problems with State Constraints

Recently, necessary conditions have been derived for fixed-time optimal control problems with state constraints, formulated in terms of a differential inclusion, under very weak hypotheses on the data. These allow the multifunction describing admissible velocities to be unbounded and possibly nonconvex valued. This paper extends the earlier necessary conditions, to allow for free end-times. A notable feature of the new free end-time necessary conditions is that they cover problems with measurably time dependent data. For such problems, standard analytical techniques for deriving free-time necessary conditions, which depend on a transformation of the time variable, no longer work. Instead, we use variational methods based on the calculus of 'essential values".