Duality and uniqueness of convex solutions to stationary Hamilton-Jacobi equations

Value functions for convex optimal control problems on infinite time intervals are studied in the framework of duality. Hamilton-Jacobi characterizations and the conjugacy of primal and dual value functions are of main interest. Close ties between the uniqueness of convex solutions to a Hamilton-Jacobi equation, the uniqueness of such solutions to a dual Hamilton-Jacobi equation, and the conjugacy of primal and dual value functions are displayed. Simultaneous approximation of primal and dual infinite horizon problems with a pair of dual problems on finite horizon, for which the value functions are conjugate, leads to sufficient conditions on the conjugacy of the infinite time horizon value functions. Consequently, uniqueness results for the Hamilton-Jacobi equation are established. Little regularity is assumed on the cost functions in the control problems, correspondingly, the Hamiltonians need not display any strict convexity and may have several saddle points.

     

On conservation laws and boundary conditions for “short waves” equation

In this paper we study local conservation laws for the equation of short waves in the form of a variational problem. We analyze an infinite symmetry group of the equation and generate a finite number of conservation laws corresponding to given infinite symmetries through appropriate boundary conditions.     

无穷时滞Liénard方程的周期解(英文)

讨论具有无穷时滞Liénard型方程x+(?²F(x))/(?x²)x+g(t,x_t)=p(t)的周期解问题, 利用重合度理论得到了周期解存在的充分条件.     

变系数Euler-Bernoulli梁振动发展系统的存在性

讨论变系数Euler-Bernoulli梁振动系统{uu(x,t) η(t)uxxxx(x,t)=0,0＜x＜1,0≤t≤T u(0,t)=ux(0,t)=0,0≤t≤t -uxxx(1,t) muu(1,t)=-αu1(1,t) βuxxx(1,t),0≤t≤T uxt(1,t) =-γuxx(1,t),0≤t≤t u(x,0)=u1(x),u1(x,0),0≤x≤1证明了该系统产生一个发展系统．     

On the Spectra of the Kink for Ferromagnetic XXZ Models

We consider the kink (infinite volume nontranslationally invariant ground states) of the ferromagnetic XXZ models on the multi-dimensional lattices. We obtained the following results: (i) The pure states satisfying the local zero energy condition are necessarily product states. (ii) The Hamiltonian in the GNS representation of the kink has no gap between the ground-state eigenvalue and the rest of the spectra.     

Double Structures and Soliton Double Solution Family for Einstein-Maxwell-Dilaton Theory

A doubleness symmetry property of the two-dimensional reduced Einstein-Maxwell-dilaton theory is found and, by using the so-called extended double complex method proposed previously, the motion equations of the theory are extended into a double form. Some new double symmetry transformations and double solution generating methods are given. As applications of these transformations and methods, an infinite chain of double solutions and a new soliton double solution family of the model are obtained. These results demonstrate that the theory under consideration possesses more and richer symmetry structures than previously expected. It seems that some of these results can not be obtained by using only the usual (non-double) scheme.     

Water transport on infinite graphs

If the nodes of a graph are considered to be identical barrels – featuring different water levels – and the edges to be (locked) water‐filled pipes in between the barrels, consider the optimization problem of how much the water level in a fixed barrel can be raised with no pumps available, that is, by opening and closing the locks in an elaborate succession. This model is related to an opinion formation process, the so‐called Deffuant model. We consider the initial water profile to be given by i.i.d. unif(0,1) random variables, investigate the supremum of achievable water levels at a given node – or to be more precise, the support of its distribution – and ask in which settings it becomes degenerate, that is, reduces to a single value. This turns out to be the case for all infinite connected quasi‐transitive graphs, with exactly one exception: the two‐sided infinite path.     

THE MITRA‐WAN FORESTRY MODEL: A DISCRETE‐TIME OPTIMAL CONTROL PROBLEM

In this paper, we present some new properties of the Mitra‐Wan forestry model written as a discrete‐time optimal control problem. For this problem, the set of stationary states is characterized. For the optimal long‐run management, we consider the following optimality criteria: average optimality, good control policies, bias optimality, and overtaking optimality. We establish relationships between these criteria and show that the value of average optimal policies is constant and equals the value in the optimal stationary state.     

On Yosida Approximations of McKean–Vlasov Type Stochastic Evolution Equations

This article is concerned with a semilinear McKean–Vlasov type stochastic evolution equation in a real Hilbert space. The main goal of the article is to study the existence and uniqueness of mild solutions, Yosida approximations of mild solutions of such equations, and to deduce the weak convergence of the corresponding induced probability measures. As an application, we also study the exponential stability of mild solutions of such equations.