Optimal Control of Non-well-posed Heat Equations

This work is concerned with Pontryagin＇s maximum principle of optimal control problems governed by some non-well-posed semilinear heat equations. A type of approach to the non-well-posed optimal control problem is given.     

TIME AND NORM OPTIMAL CONTROLS:A SURVEY OF RECENT RESULTS AND OPEN PROBLEMS

We present in this paper a survey of recent results on the relation between time and norm optimality for linear systems and the infinite dimensional version of Pontryagin’s maximum principle. In particular, we discuss optimality (or nonoptimality) of singular controls satisfying the maximum principle and smoothness of the costate in function of smoothness of the target.     

切比雪夫不等式的加权推广

This paper presents the popularization of the weighting of Chebyshev＇s inequality and discusses the relation between this popularization and some famous inequalities.     

关于Hill方程稳定性准则的控制方法证明

Optimal estimates of the stability of Hill‘s equation are obtained. The discussion is based on Pontryagin‘s maximum principle.     

ON THE GENERALIZATIONS OF KALANDIA＇S LEMMA

Based on Bernstein＇s Theorem, Kalandia＇s Lemma describes the error estimate and the smoothness of the remainder under the second part of Hoelder norm when a HSlder function is approximated by its best polynomial approximation. In this paper, Kalandia＇s Lemma is generalized to the cases that the best polynomial is replaced by one of its four kinds of Chebyshev polynomial expansions, the error estimates of the remainder are given out under Hoeder norm or the weighted HSlder norms.     

A generalized vector-valued variational principle in Fréchet spaces

In the framework of Frechet spaces, we give a generalized vector-valued Ekeland＇s variational principle, where the perturbation involves the subadditive functions of countable generating semi-norms. By modifying and developing the method of Cammaroto and Chinni, we obtain a density theorem on extremal points of the vector-valued variational principle, which extends and improves the related known results.     

HEAT KERNELS AND HARDY＇S UNCERTAINTY PRINCIPLE ON H-TYPE GROUPS

This article obtains an explicit expression of the heat kernels on H-type groups and then follow the estimate of heat kernels to deduce the Hardy＇s uncertainty principle on the nilpotent Lie groups.     

Large deviation principle of stochastic differential equations with non-Lipschitzian coefficients

We study the large deviation principle of stochastic differential equations with non-Lipschitzian and non-homogeneous coefficients. We consider at first the large deviation principle when the coefficients σ and b are bounded, then we generalize the conclusion to unbounded case by using bounded approximation program. Our results are generalization of S. Fang-T. Zhang＇s results.     

Proper Submanifolds in Product Manifolds

The authors obtain various versions of the Omori-Yau's maximum principle on complete properly immersed submanifolds with controlled mean curvature in certain product manifolds,in complete Riemannian manifolds whose k-Ricci curvature has strong quadratic decay,and also obtain a maximum principle for mean curvature flow of complete manifolds with bounded mean curvature.Using the generalized maximum principle,an estimate on the mean curvature of properly immersed submanifolds with bounded projection in N1 in the product manifold N1 ×N2 is given.Other applications of the generalized maximum principle are also given.     

An Simple Elementary Proof for the Inequality dn〈3^n

In this paper, by using a variation of the Chebyshev＇s method, we present a very simple, elementary proof of an inequality which has applications in number theory.     

A RISK-SENSITIVE STOCHASTIC MAXIMUM PRINCIPLE FOR OPTIMAL CONTROL OF JUMP DIFFUSIONS AND ITS APPLICATIONS

A stochastic maximum principle for the risk-sensitive optimal control problem of jump diffusion processes with an exponential-of-integral cost functional is derived assuming that the value function is smooth, where the diffusion and jump term may both depend on the control. The form of the maximum principle is similar to its risk-neutral counterpart. But the adjoint equations and the maximum condition heavily depend on the risk-sensitive parameter. As applications, a linear-quadratic risk-sensitive control problem is solved by using the maximum principle derived and explicit optimal control is obtained.     

Parameter Estimations in BurrⅫ Model Using Masked Data

We consider a series system of two independent and non-identical components which have different BurrⅫ distributed lifetime.The maximum likelihood and Bayes estimators of the parameters of the system＇s components are obtained based on masked system life test data.The conclusion is that the Bayes estimates are better than the maximum likelihood estimates in the sense of having smaller mean squared errors.     

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.     

AN ESTIMATE FOR THE MEAN CURVATURE OF SUBMANIFOLDS CONTAINED IN A HOROBALL

We obtain the Omori-Yau maximum principle on complete properly immersed submanifolds with the mean curvature satisfying certain condition in complete Riemannian manifolds whose radial sectional curvature satisfies some decay condition, which generalizes our previous results in [17]. Using this generalized maximum principle, we give an estimate on the mean curvature of properly immersed submanifolds in H^n × R^e with the image under the projection on H^n contained in a horoball and the corresponding situation in hyperbolic space. We also give other applications of the generalized maximum principle.     

ALEKSANDROV MAXIMUM PRINCIPLE AND BONY MAXIMUM PRINCIPLE FOR PARABOLIC EQUATIONS

The author simplifies the proof of Aleksandrov maximum principle for parabolic equations givenby Krylov and obtains finer results.He further proves Bony maximum principle for parabolicequations by using the above results.     

Relative Superlinearity Implies Infinitely Many Solutions to Sturm-Liouville BVPs with Laplacian Type Operators

By the use of continuation theorems and the duality principle it＇s proved that for both homogeneous and nonhomogeneous Sturm-Liouville boundary value problems with Laplacian type operators, relative superlinearity implies the existence of infinitely many solutions under a few weakly added conditions.     

DENSITY OF MARKOV SYSTEMS AND ZEROS OF CHEBYSHEV POLYNOMIALS

We raise and partly answer the question: whether there exists a Markov system with respectto which the zeros of the Chebyshev polynomials are dense, but the maximum length of a zerofree interval of the nth Chebyshev polynomial does not tends to zero. We also draw the conclu-tion that a Markov system, under an additional assumption, is dense if and only if the maxi-mum length of a zero free interval of the nth associated Chebyshev polynomial tends to zero.     

A Note on a Theorem of J. Szabados

In this note, we establish a companion result to the theorem of J. Szabados on the maximum of fundamental functions of Lagrange interpolation based on Chebyshev nodes.     

UNIQUENESS OF VISCOSITY SOLUTIONS OF FULLY NONLINEAR SECOND ORDER PARABOLIC PDE''''S

This paper investigates the maximum principle for viscosity solutions of fully nonlinear, second order parabolic, possibly degenerate partial differential equations. Using this maximum principle the anthor prove that viscosity solutions of initial and bo unoary value problem for parabolic equations are unique.     

Sto chastic Maximum Principle for Optimal Control of Forward-backward Sto chastic Pantograph Systems with Regime Switching

In this paper, we derive the stochastic maximum principle for optimal control problems of the forward-backward Markovian regime-switching system. The control system is described by an anticipated forward-backward stochastic pantograph equation and modulated by a continuous-time finite-state Markov chain. By virtue of classical variational approach, duality method, and convex analysis, we obtain a stochastic maximum principle for the optimal control.