A Generalized Comparison Theorem for BSDEs and Its?Applications

This paper establishes a generalized comparison theorem for one-dimensional backward stochastic differential equations (BSDEs) whose generators are uniformly continuous in z and satisfy a kind of weakly monotonic condition in y. As applications, two new existence and uniqueness theorems for solutions of BSDEs are obtained. In the one-dimensional setting, these results generalize some corresponding results in Pardoux and Peng (Syst. Control Lett. 14:55–61, 1990), Mao (Stoch. Process. Their Appl. 58:281–292, 1995), El Karoui et al. (Math. Finance 7:1–72, 1997), Pardoux (Nonlinear Analysis, Differential Equations and Control, Montreal, QC, 1998, Kluwer Academic, Dordrecht, 1999), Cao and Yan (Adv. Math. 28(4):304–308, 1999), Briand and Hu (Probab. Theory Relat. Fields 136(4):604–618, 2006), and Jia (C. R. Acad. Sci. Paris, Ser. I 346:439–444, 2008).     

A class of evolution equations: Existence of solutions with functional boundary conditions

We consider the evolution equation whose right-hand side is the sum of a linear unbounded operator generating a compact strongly continuous semigroup and a continuous operator acting in function spaces. We prove the existence of a solution that stays within a given closed convex set and moreover, satisfies a functional boundary condition, particular cases of which are the Cauchy initial condition, periodicity condition, mixed condition including continuous transformations of spatial variables, etc. The main result is illustrated by using an example of the boundary-value problem for a partial operator-differential equation. Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 48–60, January, 1999.     

Optimal control for 3D stochastic Navier–Stokes equations

Loeb space methods are used to prove existence of an optimal control for general 3D stochastic Navier–Stokes equations with multiplicative noise. The possible non-uniqueness of the solutions mean that it is necessary to utilize the notion of a non-standard approximate solution developed in the paper by N.J. Cutland and Keisler H.J. 2004, Global attractors for 3-dimensional stochastic Navier–Stokes equations, Journal of Dynamics and Differential Equations, pp. 16205–16266, for the study of attractors.     

Spectral Doubling of Normal Operators and Connections with Antiunitary Operators

Equations such as AB = B ^T A have been studied in the finite dimensional setting in (Linear Algebra Appl 369:279–294, 2003). These equations have implications for the spectrum of B, when A is normal. Our aim is to generalize these results to an infinite dimensional setting. In this case it is natural to use JB*J for some conjugation operator J in place of B ^T . Our main result is a spectral pairing theorem for a bounded normal operator B which is applied to the study of the equation KB = B*K for K an antiunitary operator. In particular, using conjugation operators, we generalize the notion of Hamiltonian operator and skew-Hamiltonian operator in a natural way, derive some of their properties, and give a characterization of certain operators B for which AB = (JB*J)A and BA = A(JB*J) and also those B with KB = B*K for certain antiunitary operators K.     

Remarks on two-parameter stochastic differential equations in Hilbert space

An existence theorem for the solution of a two-parameter stochastic Goursat problem in Hilbert space is proved. In doing this we assume that the equation contains a principal linear term with an operator that is in general unbounded and two-parameter white noise.Translated fromTeoriya Sluchaínykh Protsessov, Vol. 14, pp. 64–71, 1986.     

带跳线性随机微分方程的近似能控性

研究了Poisson随机测度驱动的线性随机微分方程的近似能控性,通过对偶方法,得到了近似能控性的一个代数判据:由方程系数决定的某种不变空间V是退化空间{0}.此外,还给出了有限步计算验证该判据的程序算法.     

EXISTENCE OF SOLUTION AND APPROXIMATE CONTROLLABILITY OF A SECOND-ORDER NEUTRAL STOCHASTIC DIFFERENTIAL EQUATION WITH STATE DEPENDENT DELAY

This paper has two sections which deals with a second order stochastic neutral partial differential equation with state dependent delay. In the first section the existence and uniqueness of mild solution is obtained by use of measure of non-compactness. In the second section the conditions for approximate controllability are investigated for the distributed second order neutral stochastic differential system with respect to the approximate controllability of the corresponding linear system in a Hilbert space. Our method is an extension of co-author N. Sukavanam's novel approach in [22]. Thereby, we remove the need to assume the invertibility of a controllability operator used by authors in [5], which fails to exist in infinite dimensional spaces if the associated semigroup is compact. Our approach also removes the need to check the invertibility of the controllability Gramian operator and associated limit condition used by the authors in [20], which are practically difficult to verify and apply. An example is provided to illustrate the presented theory.     

A Fréchet derivative-free cubically convergent method for set-valued maps

We introduce a new iterative method in order to approximate a locally unique solution of variational inclusions in Banach spaces. The method uses only divided differences operators of order one. An existence–convergence theorem and a radius of convergence are given under some conditions on divided difference operator and Lipschitz-like continuity property of set-valued mappings. Our method extends the recent work related to the resolution of nonlinear equation in Argyros (J Math Anal Appl 332:97–108, 2007) and has the following advantages: faster convergence to the solution than all the previous known ones in Argyros and Hilout (Appl Math Comput, 2008 in press), Hilout (J Math Anal Appl 339:53–761, 2008, Positivity 10:673–700, 2006), and we do not need to evaluate any Fréchet derivative. We provide also an improvement of the ratio of our algorithm under some center-conditions and less computational cost. Numerical examples are also provided.      

Dual pairs in fluid dynamics

This article is a rigorous study of the dual pair structure of the ideal fluid (Phys D 7:305–323, 1983) and the dual pair structure for the n-dimensional Camassa–Holm (EPDiff) equation (The breadth of symplectic and poisson geometry: Festshrift in honor of Alan Weinstein, 2004), including the proofs of the necessary transitivity results. In the case of the ideal fluid, we show that a careful definition of the momentum maps leads naturally to central extensions of diffeomorphism groups such as the group of quantomorphisms and the Ismagilov central extension.     

On stationarity of stochastic retarded linear equations with unbounded drift operators

This article continues the study of Liu [Statist. Probab. Lett. 78(2008): 1775–1783; Stoch. Anal. Appl. 29(2011): 799–823] for stationary solutions of stochastic linear retarded functional differential equations with the emphasis on delays which appear in those terms including spatial partial derivatives. As a consequence, the associated stochastic equations have unbounded operators acting on the point or distributed delayed terms, while the operator acting on the instantaneous term generates a strongly continuous semigroup. We present conditions on the delay systems to obtain a unique stationary solution by combining spectrum analysis of unbounded operators and stochastic calculus. A few instructive cases are analyzed in detail to clarify the underlying complexity in the study of systems with unbounded delayed operators.     

A direct approach to the weighted admissibility of observation operators for bounded analytic semigroups

In this note we adopt the approach in Bonnit et al. (Czechoslov. Math. J. 60(2):527–539, 2010) to give a direct proof of some recent results in Haak and Le Merdy (Houst. J. Math., 2005) and Haak and Kunstmann (SIAM J. Control Optim. 45:2094–2118, 2007) which characterizes the L ^p -admissibility of type α depending on p of unbounded observation operators for bounded analytic semigroups.     

Implicit functions determined by equations with unbounded operators and the solvability of operator equations

For implicit functions determined by the equation T_x+F(λ,x)=0, where T is an unbounded operator, existence and differentiability conditions are established. As an application, the solvability conditions for equations of the form Tx+f(x)=0 are derived. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 248, 1998, pp. 216–224. Translated by L. Yu. Kolotilina.     

An analytic approach to stochastic Volterra equations with completely monotone kernels

We apply the semigroup setting of Desch and Miller to a class of stochastic integral equations of Volterra type with completely monotone kernels with a multiplicative noise term; the corresponding equation is an infinite dimensional stochastic equation with unbounded diffusion operator that we solve with the semigroup approach of Da Prato and Zabczyk. As a motivation of our results, we study an optimal control problem when the control enters the system together with the noise.      

Pathwise Solutions of the 2-D Stochastic Primitive Equations

In this work we consider a stochastic version of the Primitive Equations (PEs) of the ocean and the atmosphere and establish the existence and uniqueness of pathwise, strong solutions. The analysis employs novel techniques in contrast to previous works (Ewald et al. in Anal. Appl. (Singap.) 5(2):183–198, 2007; Glatt-Holtz and Ziane in Discrete Contin. Dyn. Syst. Ser. B 10(4):801–822, 2008) in order to handle a general class of nonlinear noise structures and to allow for physically relevant boundary conditions. The proof relies on Cauchy estimates, stopping time arguments and anisotropic estimates.     

Generalized Solutions of Nonlocal Elliptic Problems

An elliptic equation of order 2m with general nonlocal boundary-value conditions, in a plane bounded domain G with piecewise smooth boundary, is considered. Generalized solutions belonging to the Sobolev space W ₂ ^m (G) are studied. The Fredholm property of the unbounded operator (corresponding to the elliptic equation) acting on L ₂(G), and defined for functions from the space W ₂ ^m (G) that satisfy homogeneous nonlocal conditions, is established.__________Translated from Matematicheskie Zametki, vol. 77, no. 5, 2005, pp. 665–682.Original Russian Text Copyright ©2005 by P. L. Gurevich.     

Approximate boundary controllability of Sobolev-type stochastic differential systems

The objective of this paper is to investigate the approximate boundary controllability of Sobolev-type stochastic differential systems in Hilbert spaces. The control function for this system is suitably constructed by using the infinite dimensional controllability operator. Sufficient conditions for approximate boundary controllability of the proposed problem in Hilbert space is established by using contraction mapping principle and stochastic analysis techniques. The obtained results are extended to stochastic differential systems with Poisson jumps. Finally, an example is provided which illustrates the main results.     

Construction of Approximate Saddle-Point Strategies for Differential Games in a Hilbert Space

Two-person zero-sum infinite-dimensional differential games with strategies and payoff as in Berkovitz (SIAM J. Control Optim. 23: 173–196, 1985) are studied. Using Yosida type approximations of the infinitesimal generator (of the unbounded dynamics) by bounded linear operators, we prove convergence theorems for the approximate value functions. This is used to construct approximate saddle-point strategies in feedback form. A.J. Shaiju: NBHM Postdoctoral Fellow and the financial support from NBHM is gratefully acknowledged.     

Carleman Estimates and Controllability of Linear Stochastic Heat Equations

   Abstract. This work is concerned with Carleman inequalities and controllability properties for the following stochastic linear heat equation (with Dirichlet boundary conditions in the bounded domain D ⊂ R ^d and multiplicative noise):

Singular Sturmian separation theorems for nonoscillatory symplectic difference systems

ABSTRACT

In this paper, we derive new singular Sturmian separation theorems for nonoscillatory symplectic difference systems on unbounded intervals. The novelty of the presented theory resides in two aspects. We introduce the multiplicity of a focal point at infinity for conjoined bases, which we incorporate into our new singular Sturmian separation theorems. At the same time we do not impose any controllability assumption on the symplectic system. The presented results naturally extend and complete the known Sturmian separation theorems on bounded intervals by J. V. Elyseeva [Comparative index for solutions of symplectic difference systems, Differential Equations 45(3) (2009), pp. 445–459, translated from Differencial'nyje Uravnenija 45 (2009), no. 3, 431–444], as well as the singular Sturmian separation theorems for eventually controllable symplectic systems on unbounded intervals by O. Do?lý and J. Elyseeva [Singular comparison theorems for discrete symplectic systems, J. Difference Equ. Appl. 20(8) (2014), pp. 1268–1288]. Our approach is based on developing the theory of comparative index on unbounded intervals and on the recent theory of recessive and dominant solutions at infinity for possibly uncontrollable symplectic systems by the authors [P. ?epitka and R. ?imon Hilscher, Recessive solutions for nonoscillatory discrete symplectic systems, Linear Algebra Appl. 469 (2015), pp. 243–275; P. ?epitka and R. ?imon Hilscher, Dominant and recessive solutions at infinity and genera of conjoined bases for discrete symplectic systems, J. Difference Equ. Appl. 23(4) (2017), pp. 657–698]. Some of our results, including the notion of the multiplicity of a focal point at infinity, are new even for an eventually controllable symplectic difference system.