A slight improvement to Korenblum's constant

Let A²(D) be the Bergman space over the open unit disk D in the complex plane. Korenblum conjectured that there is an absolute constant c∈(0,1) such that whenever |f(z)|?|g(z)| in the annulus c<|z|<1, then ‖f(z)‖?‖g(z)‖. This conjecture had been solved by Hayman [W.K. Hayman, On a conjecture of Korenblum, Analysis (Munich) 19 (1999) 195-205. [1]], but the constant c in that paper is not optimal. Since then, there are many papers dealing with improving the upper and lower bounds for the best constant c. For example, in 2004 C. Wang gave an upper bound on c, that is, c<0.67795, and in 2006 A. Schuster gave a lower bound, c>0.21. In this paper we slightly improve the upper bound for c.     

最大模原理的推广

本文把最大模原理及 Phragmén—— Lindelof定理推广到具有保域性的一类连续函数上 ,得到若干结果 .     

On the maximum principle on complete Finsler manifolds

In this paper, we generalize Omori–Yau maximum principle to Finsler geometry. As an application, we obtain some Liouville-type theorems of subharmonic functions on forward complete Finsler manifolds.     

Nonsmooth maximum principle for infinite-horizon problems

In this paper, we consider a class of infinite-horizon discounted optimal control problems with nonsmooth problem data. A maximum principle in terms of differential inclusions with a Michel type transversality condition is given. It is shown that, when the discount rate is sufficiently large, the problem admits normal multipliers and a strong transversality condition holds. A relationship between dynamic programming and the maximum principle is also given.The author is indebted to Francis Clarke for helpful suggestions and discussions.     

On the strong maximum principle for degenerate parabolic equations

Variants of the strong maximum principle are established for subsolutions to degenerate parabolic equations for which the standard version of the strong maximum principle does not hold. The results are formulated for viscosity solutions.     

Lower inequalities of heat semigroups by using parabolic maximum principle

Using parabolic maximum principle, we apply the analytic method to obtain lower comparison inequalities for non-negative weak supersolutions of the heat equation associated with a regular strongly ρ-local Dirichle form on the abstract metric measure space. As an application, we obtain lower estimates for heat kernels on some Riemannian manifolds.     

Optimal control: Nonlocal conditions,computational methods,and the variational principle of maximum

This paper surveys theoretical results on the Pontryagin maximum principle (together with its conversion) and nonlocal optimality conditions based on the use of the Lyapunov-type functions (solutions to the Hamilton-Jacobi inequalities). We pay special attention to the conversion of the maximum principle to a sufficient condition for the global and strong minimum without assumptions of the linear convexity, normality, or controllability. We give the survey of computational methods for solving classical optimal control problems and describe nonstandard procedures of nonlocal improvement of admissible processes in linear and quadratic problems. Furthermore, we cite some recent results on the variational principle of maximum in hyperbolic control systems. This principle is the strongest first order necessary optimality condition; it implies the classical maximum principle as a consequence.     

An extension to Banach space of Pontryagin's maximum principle

An extension of Pontryagin's maximum principle to the case where the state space is infinite dimensional is given. The control process is governed by ordinary nonlinear differential equations. A property of control processes, which is analogous to well-known, nonlinear interior mapping theorems, makes up the basis for the proofs.     

A stochastic maximum principle for processes driven by fractional Brownian motion

We prove a stochastic maximum principle for controlled processes X(t)=X^(u)(t) of the form

dX(t)=b(t,X(t),u(t)) dt+σ(t,X(t),u(t)) dB^(H)(t),

where B^(H)(t) is m-dimensional fractional Brownian motion with Hurst parameter . As an application we solve a problem about minimal variance hedging in an incomplete market driven by fractional Brownian motion.     

On the sufficiency of the linear maximum principle for discrete-time control problems

This note presents a family of linear maximum principles for the discrete-time optimal control problem, derived from the saddle-point theorem of mathematical programming. Some simple examples illustrate the applicability of the main theoretical results.     

An application of Neustadt's abstract maximum principle to probability kinematics

An important class of problems in philosophy can be formulated as mathematical programming problems in an infinite-dimensional vector space. One such problem is that of probability kinematics: the study of how an individual ought to adjust his degree-of-belief function in response to new information. Much work has recently been done to establish maximum principles for these generalized programming problems (Refs. 3–4). Perhaps, the most general treatment of the problem presented to date is that by Neustadt (Ref. 1). In this paper, the problem of probability kinematics is formulated as a generalized mathematical programming problem and necessary conditions for the optimal revised degree-of-belief function are derived from an abstract maximum principle contained in Neustadt's paper.This work was supported by the National Research Council of Canada.The author is grateful to G. J. Lastman and J. A. Baker of the University of Waterloo for numerous suggestions made for improvement of this paper. The problem of probability kinematics was brought to the author's attention by W. L. Harper of the University of Western Ontario.     

Normality of the maximum principle for nonconvex constrained Bolza problems

We consider a Bolza optimal control problem with state constraints. It is well known that under some technical assumptions every strong local minimizer of this problem satisfies first order necessary optimality conditions in the form of a constrained maximum principle. In general, the maximum principle may be abnormal or even degenerate and so does not provide a sufficient information about optimal controls. In the recent literature some sufficient conditions were proposed to guarantee that at least one maximum principle is nondegenerate, cf. [A.V. Arutyanov, S.M. Aseev, Investigation of the degeneracy phenomenon of the maximum principle for optimal control problems with state constraints, SIAM J. Control Optim. 35 (1997) 930–952; F. Rampazzo, R.B. Vinter, A theorem on existence of neighbouring trajectories satisfying a state constraint, with applications to optimal control, IMA 16 (4) (1999) 335–351; F. Rampazzo, R.B. Vinter, Degenerate optimal control problems with state constraints, SIAM J. Control Optim. 39 (4) (2000) 989–1007]. Our aim is to show that actually conditions of a similar nature guarantee normality of every nondegenerate maximum principle. In particular we allow the initial condition to be fixed and the state constraints to be nonsmooth. To prove normality we use J. Yorke type linearization of control systems and show the existence of a solution to a linearized control system satisfying new state constraints defined, in turn, by linearization of the original set of constraints along an extremal trajectory.     

Pontryagin’s maximum principle for constrained impulsive control problems

Necessary conditions in the form of Pontryagin’s maximum principle are derived for impulsive control problems with mixed constraints. A new mathematical concept of impulsive control is introduced as a requirement for the consistency of the impulsive framework. Additionally, this control concept enables the incorporation of the engineering needs to consider conventional control action while the impulse develops. The regularity assumptions under which the maximum principle is proved are weaker than those in the known literature. Ekeland’s variational principle and Lebesgue’s discontinuous time variable change are used in the proof. The article also contains an example showing how such impulsive controls could be relevant in actual applications.     

A maximum principle for optimal control problems with mixed constraints

Necessary conditions in the form of maximum principles are derivedfor optimal control problems with mixed control and state constraints.Traditionally, necessary condtions for problems with mixed constraintshave been proved under hypothesis which include the requirementthat the Jacobian of the mixed constraint functional, with respectto the control variable, have full rank. We show that it canbe replaced by a weaker ‘interiority’ hypothesis.This refinement broadens the scope of the optimality conditions,to cover some optimal control problems involving differentialalgebraic constraints, with index greater than unity.