Interconnections and Initial Conditions of Linear Systems With First-Order Representations

This paper considers the initial value problem of an interconnection composed of linear systems described by the first-order differential/algebraic equations (DAEs). An initial condition of the system variable for which the DAE has a solution is called admissible. For the interconnected system, we formulate the invariance of the admissible initial condition sets (AICSs) of the sub-systems under interconnection. Namely, the AICSs are said to be invariant if they remain unchanged even when additional constraints due to interconnection are imposed on the system variables. It is shown that the feedback and regular feedback structures of the interconnection guarantee the invariance of the AICSs in the senses of impulsive-smooth distributions and smooth distributions, respectively. The results in this paper justify the use of a feedback controller in the control system design.     

First-Order Necessary Optimality Conditions for General Bilevel Programming Problems

In Ref. 1, bilevel programming problems have been investigated using an equivalent formulation by use of the optimal value function of the lower level problem. In this comment, it is shown that Ref. 1 contains two incorrect results: in Proposition 2.1, upper semicontinuity instead of lower semicontinuity has to be used for guaranteeing existence of optimal solutions; in Theorem 5.1, the assumption that the abnormal part of the directional derivative of the optimal value function reduces to zero has to be replaced by the demand that a nonzero abnormal Lagrange multiplier does not exist.     

On the Cauchy Problems of Fractional Evolution Equations with Nonlocal Initial Conditions

In this paper, new criterions which allow us to relax the compactness and Lipschitz continuity on nonlocal item, ensuring the existence and uniqueness of mild solutions for the Cauchy problems of fractional evolution equations with nonlocal initial conditions, are established. The results obtained in this paper essentially extend some existing results in this area. Finally, we present two applications to the abstract results.     

A Note on First-Order Sufficient Optimality Conditions for Pareto Problems

We take into consideration the first-order sufficient conditions, established by Jiménez and Novo (Numer. Funct. Anal. Optim. 2002; 23:303–322) for strict local Pareto minima. We give here a more operative condition for a strict local Pareto minimum of order 1.     

First-Order Necessary Conditions of Optimality for Strongly Monotone Nonlinear Control Problems

Necessary conditions for the optimality of a pair (y*, u*) with respect to the cost functional g(y) + h(u) subject to AyBu + f are given in terms of generalized gradients. Here, g is locally Lipschitz, h is convex, A is a maximal strongly monotone operator, and B is linear. Two examples of applications of our necessary conditions to nonlinear partial differential equations of elliptic type are presented.     

Further Gurtin-type Variational Principles for Linear Initial Value Problems

Following the variational principles for linear initial valueproblems associated with the wave and heat conduction equationsdiscussed by Gurtin and Leitmann and using time convolutionsit is shown that general variational principles exist for theseproblems with sources on the boundaries and within the regionunder consideration.     

On Rate-Type Viscoplastic Problems with Linear Boundary Conditions

In this paper we introduce the abstract concept of “linear boundary conditions” in the study of deformable bodies. We establish two existence and uniqueness results concerning respectively quasistatic and dynamic problems involving such type of boundary conditions. We also apply these existence results in the study of viscoplastic problems involving classical boundary conditions.     

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...     

非光滑半定规划的一阶最优性条件

首次考虑了非光滑半定规化问题．运用与非线性规划类似的技巧,把现存的理论扩展到约束是结构稀疏矩阵的情况,给出了其一阶最优性条件。考虑了严格互补条件不成立的情形．在约束矩阵为对角阵条件下,所用的正则条件与传统非线性优化意义下的是一致的．     

First-Order Optimality Conditions in Generalized Semi-Infinite Programming

In this paper, we consider a generalized semi-infinite optimization problem where the index set of the corresponding inequality constraints depends on the decision variables and the involved functions are assumed to be continuously differentiable. We derive first-order necessary optimality conditions for such problems by using bounds for the upper and lower directional derivatives of the corresponding optimal value function. In the case where the optimal value function is directly differentiable, we present first-order conditions based on the linearization of the given problem. Finally, we investigate necessary and sufficient first-order conditions by using the calculus of quasidifferentiable functions.     

Initial Value Problems of Fractional Order with Fractional Impulsive Conditions

In this paper we intend to accomplish two tasks firstly, we address some basic errors in several recent results involving impulsive fractional equations with the Caputo derivative, and, secondly, we study initial value problems for nonlinear differential equations with the Riemann–Liouville derivative of order 0 < α ≤ 1 and the Caputo derivatives of order 1 < δ < 2. In both cases, the corresponding fractional derivative of lower order is involved in the formulation of impulsive conditions.     

Global Solutions for Nonlinear Delay Evolution Inclusions with Nonlocal Initial Conditions

We prove a sufficient condition for the existence of global C ⁰-solutions for a class of nonlinear functional differential evolution equation of the form $ \left\{{ll} \displaystyle u'(t)\in Au(t)+f(t),&t\in\mathbb{R}_+, \\[2mm] f(t)\in F(t,u(t),u_t),&t\in\mathbb{R}_+, \\[2mm] u(t)=g(u)(t),& t\in [\,-\tau,0\,], \right. $ \left\{\begin{array}{ll} \displaystyle u'(t)\in Au(t)+f(t),&t\in\mathbb{R}_+, \\[2mm] f(t)\in F(t,u(t),u_t),&t\in\mathbb{R}_+, \\[2mm] u(t)=g(u)(t),& t\in [\,-\tau,0\,], \end{array}\right.     

First-Order Algorithms for Generalized Semi-Infinite Min-Max Problems

We present a first-order algorithm for solving semi-infinite generalized min-max problems which consist of minimizing a function f⁰(x) = F(¹(x), .... , ^m (x)), where F is a smooth function and each ⁱ is the maximum of an infinite number of smooth functions.In Section 3.3 of [17] Polak finds a methodology for solving infinite dimensional problems by expanding them into an infinite sequence of consistent finite dimensional approximating problems, and then using a master algorithm that selects an appropriate subsequence of these problems and applies a number of iterations of a finite dimensional optimization algorithm to each of these problems, sequentially. Our algorithm was constructed within this framework; it calls an algorithm by Kiwiel as a subroutine. The number of iterations of the Kiwiel algorithm to be applied to the approximating problems is determined by a test that ensures that the overall scheme retains the rate of convergence of the Kiwiel algorithm.Under reasonable assumptions we show that all the accumulation points of sequences constructed by our algorithm are stationary, and, under an additional strong convexity assumption, that the Kiwiel algorithm converges at least linearly, and that our algorithm also converges at least linearly, with the same rate constant bounds as Kiwiel's.     

Nonlocal and Multipoint Boundary Value Problems for Linear Evolution Equations

We derive the solution representation for a large class of nonlocal boundary value problems for linear evolution partial differential equations (PDE) with constant coefficients in one space variable. The prototypical example of such PDE is the heat equation, for which problems of this form model physical phenomena in chemistry and for which we formulate and prove a full result. We also consider the third‐order case, which is much less studied and has been shown by the authors to have very different structural properties in general. The nonlocal conditions we consider can be reformulated as multipoint conditions , and then an explicit representation for the solution of the problem is obtained by an application of the Fokas transform method. The analysis is carried out under the assumption that the problem being solved is well posed, i.e., it admits a unique solution. For the second‐order case, we also give criteria that guarantee well posedness.     

一阶离散周期问题的多解性

Let T 1 be an integer, T = {0, 1, 2,..., T- 1}. This paper is concerned with the existence of periodic solutions of the discrete first-order periodic boundary value problems△u(t)- a(t)u(t) = λu(t) + f(u(t- τ(t)))- h(t), t ∈ T,u(0) = u(T),where △u(t) = u(t + 1)- u(t), a : T → R and satisfies∏T-1t=0(1 + a(t)) = 1, τ : T → Z t- τ(t) ∈ T for t ∈ T, f : R → R is continuous and satisfies Landesman-Lazer type condition and h : T → R. The proofs of our main results are based on the Rabinowitz's global bifurcation theorem and Leray-Schauder degree.     

脉冲发展方程初值问题的正解

This paper deals with the existence of e-positive mild solutions（see Definition 1）for the initial value problem of impulsive evolution equation with noncompact semigroup u（t）＋ Au（t）= f（t,u（t））,t ∈ [0,＋∞）,t = tk,u-t=tk = Ik（u（tk））,k = 1,2,...,u（0）= x0 in an ordered Banach space E.By using operator semigroup theory and monotonic iterative technique,without any hypothesis on the impulsive functions,an existence result of e-positive mild solutions is obtained under weaker measure of noncompactness condition on nonlinearity of f.Particularly,an existence result without using measure of noncompaceness condition is presented in ordered and weakly sequentially complete Banach spaces,which is very convenient for application.An example is given to illustrate that our results are more valuable.