The Markov power moment problem in problems of controllability and frequency extinguishing for the wave equation on a half-axis

In this paper necessary and sufficient conditions of null-controllability and approximate null-controllability are obtained for the wave equation on a half-axis. Controls solving these problems are found explicitly. Moreover, bang-bang controls solving the approximate null-controllability problem are constructed with the aid of solutions of a frequency extinguishing problem in the restricted band (−a,a) for this equation and the Markov power moment problem.     

Geometric Criterion for Controllability under Arbitrary Constraints on the Control

In this paper, necessary and sufficient conditions for null-controllability of a linear system under geometric constraints on the control are given, without the assumption that the origin is an equilibrium point of the system. The criterion for controllability uses the concept of a return condition on an interval which is introduced in the paper. This condition generalizes the existence of an equilibrium point.     

Stability and Controllability in Non-autonomous Time-discrete Dynamical Systems

In this paper stability and attractivity in non-autonomous time-discrete dynamical systems is investigated with the aid of Lyapunov functions. The results are applied to the problem of stabilization of controlled systems by feedback controls. In the final section of the paper we give sufficient conditions for norm-bounded null-controllability of linear systems.     

A Maximum Principle for SDEs of Mean-Field Type

We study the optimal control of a stochastic differential equation (SDE) of mean-field type, where the coefficients are allowed to depend on some functional of the law as well as the state of the process. Moreover the cost functional is also of mean-field type, which makes the control problem time inconsistent in the sense that the Bellman optimality principle does not hold. Under the assumption of a convex action space a maximum principle of local form is derived, specifying the necessary conditions for optimality. These are also shown to be sufficient under additional assumptions. This maximum principle differs from the classical one, where the adjoint equation is a linear backward SDE, since here the adjoint equation turns out to be a linear mean-field backward SDE. As an illustration, we apply the result to the mean-variance portfolio selection problem.     

Farkas-type theorems for interval linear systems

We describe some Farkas-type necessary and sufficient conditions for solvability and feasibility of interval linear equations and inequalities, in a unified form. For the convenient choice in practice of theory and calculations, some different but equivalent forms of the Farkas-type conditions for interval linear systems are also discussed.     

On asymptotic equivalence of perturbed linear systems of differential and difference equations

Standard results on asymptotic integration of systems of linear differential equations give sufficient conditions which imply that a system is strongly asymptotically equivalent to its principal diagonal part. These involve certain dichotomy conditions on the diagonal part as well as growth conditions on the off-diagonal perturbation terms. Here, we study perturbations with a triangularly-induced structure and see that growth conditions can be substantially weakened. In addition, we give results for not necessarily triangular perturbations which in some sense “interpolate” between the classical theorems of Levinson and Hartman-Wintner. Some analogous results for systems of linear difference equations are also given.     

Recurrency and Transiency of Markov Processes Governed by Coupled Systems of Second–Order Elliptic Operators

Markov processes corresponding to coupled systems of the form k=1,2,...,m, are considered. Here L_k are second–order linear elliptic operators and d_k,j are non–negative functions. We prove sufficient conditions for the recurrency and transiency of the Markov processes corresponding to such systems and alos provide some examples which show that it is possible to have a recurrent system even if some or all its components are transient; where by components, we mean the Markov processes engendered by the uncoupled equations. We also provide all example which shows that a system composed of recurrent processes may itself be transient. Finally, we consider necessary and sufficient conditions for the exterior Dirichlet problem of the coupled system to have a bounded solution as.     

Approximate controllability of second-order semilinear evolution systems with state-dependent infinite delay

In this article, we study the problem of approximate controllability for a class of semilinear second-order control systems with state-dependent delay. We establish some sufficient conditions for approximate controllability for this kind of systems by constructing fundamental solutions and using the resolvent condition and techniques on cosine family of linear operators. Particularly, theory of fractional power operators for cosine families is also applied to discuss the problem so that the obtained results can be applied to the systems involving derivatives of spatial variables.~To illustrate the applications of the obtained results, two examples are presented in the end.     

Average case tractability of non-homogeneous tensor product problems with the absolute error criterion

We study average case tractability of non-homogeneous tensor product problems with the absolute error criterion. We consider algorithms that use finitely many evaluations of arbitrary linear functionals. For general non-homogeneous tensor product problems, we obtain the matching necessary and sufficient conditions for strong polynomial tractability in terms of the one-dimensional eigenvalues. We give some examples to show that strong polynomial tractability is not equivalent to polynomial tractability, and polynomial tractability is not equivalent to quasi-polynomial tractability. But for non-homogeneous tensor product problems with decreasing eigenvalues, we prove that strong polynomial tractability is always equivalent to polynomial tractability, and strong polynomial tractability is even equivalent to quasi-polynomial tractability when the one-dimensional largest eigenvalues are less than one. In particular, we find an example that quasi-polynomial tractability with the absolute error criterion is not equivalent to that with the normalized error criterion even if all the one-dimensional largest eigenvalues are one. Finally we consider a special class of non-homogeneous tensor product problems with improved monotonicity condition of the eigenvalues.     

A UNIFIED FRAMEWORK FOR SOME INEXACT PROXIMAL POINT ALGORITHMS*

We present a unified framework for the design and convergence analysis of a class of algorithms based on approximate solution of proximal point subproblems. Our development further enhances the constructive approximation approach of the recently proposed hybrid projection–proximal and extragradient–proximal methods. Specifically, we introduce an even more flexible error tolerance criterion, as well as provide a unified view of these two algorithms. Our general method possesses global convergence and local (super)linear rate of convergence under standard assumptions, while using a constructive approximation criterion suitable for a number of specific implementations. For example, we show that close to a regular solution of a monotone system of semismooth equations, two Newton iterations are sufficient to solve the proximal subproblem within the required error tolerance. Such systems of equations arise naturally when reformulating the nonlinear complementarity problem.

     

Set-Valued Systems with Infinite-Dimensional Image and Applications

In infinite-dimensional spaces, we investigate a set-valued system from the image perspective. By exploiting the quasi-interior and the quasi-relative interior, we give some new equivalent characterizations of (proper, regular) linear separation and establish some new sufficient and necessary conditions for the impossibility of the system under new assumptions, which do not require the set to have nonempty interior. We also present under mild assumptions the equivalence between (proper, regular) linear separation and saddle points of Lagrangian functions for the system. These results are applied to obtain some new saddle point sufficient and necessary optimality conditions of vector optimization problems.     

Higher-order spectral analysis and weak asymptotic stability of convex processes

This paper deals with the asymptotic stability analysis of a discrete dynamical inclusion whose right-hand side is a convex process. We provide necessary and sufficient conditions for weak asymptotic stability, and obtain sharp estimates for the asymptotic null-controllability set. These estimates involve not only standard, but also higher-order spectral information on the convex process and its adjoint.     

Invariant convex bodies for strongly elliptic systems

We consider uniformly strongly elliptic systems of the second order with bounded coefficients. First, sufficient conditions for the invariance of convex bodies are obtained for linear systems without zero order term on bounded domains and quasilinear systems of special form on bounded domains and on a class of unbounded domains. These conditions are formulated in algebraic form. They describe relation between the geometry of the invariant convex body and the coefficients of the system. Next, necessary conditions, which are also sufficient, for the invariance of some convex bodies are found for elliptic homogeneous systems with constant coefficients in a half-space. The necessary conditions are derived by using a criterion on the invariance of convex bodies for normalized matrix-valued integral transforms also obtained in the paper. In contrast with the previous studies of invariant sets for elliptic systems, no a priori restrictions on the coefficient matrices are imposed.     

Invariants of multi-parameter approximate transformation groups

The problem of finding all functionally independent invariants for a given multi-parameter approximate transformation group is considered. The criterion for approximate function to be invariant under an approximate transformation group is similar to that of Lie's theory and is based on operators of the corresponding approximate Lie algebra. The calculation of invariants of an approximate transformation group reduces to solving the system of linear nonhomogeneous first-order PDEs. The sufficient conditions of compatibility and completeness for these systems are proved.     

关于线性方程组同解的条件

讨论线性方程组同解的条件,得到了两个线性方程组同解的充分必要条件.     

Control problem for the string equation with an external load

We consider a control problem for a string equation. The control is provided by an external load function, which is determined in the course of the solution of the problem in closed form. We obtain criteria for approximate null-controllability and null-controllability.     

Subdifferential estimate of the directional derivative,optimality criterion and separation principles

We provide an inequality relating the radial directional derivative and the subdifferential of proper lower semicontinuous functions, which extends the known formula for convex functions. We show that this property is equivalent to other subdifferential properties of Banach spaces, such as controlled dense subdifferentiability, optimality criterion, mean value inequality and separation principles. As an application, we obtain a first-order sufficient condition for optimality, which extends the known condition for differentiable functions in finite-dimensional spaces and which amounts to the maximal monotonicity of the subdifferential for convex lower semicontinuous functions. Finally, we establish a formula describing the subdifferential of the sum of a convex lower semicontinuous function with a convex inf-compact function in terms of the sum of their approximate ?-subdifferentials. Such a formula directly leads to the known formula relating the directional derivative of a convex lower semicontinuous function to its approximate ?-subdifferential.     

Boundary controllability of parabolic coupled equations

This paper is concerned with the boundary controllability of non-scalar linear parabolic systems. More precisely, two coupled one-dimensional parabolic equations are considered. We show that, in this framework, boundary controllability is not equivalent and is more complex than distributed controllability. In our main result, we provide necessary and sufficient conditions for the null controllability.     

On stability for switched linear positive systems

This paper addresses the stability properties of switched linear positive systems in continuous-time as well as in discrete-time. In the discrete-time case, some sufficient and necessary conditions for asymptotic stability are derived for pairs of second order systems. Similar conditions are also established for a finite number of second order systems. Furthermore, for higher order systems, some results on stability are provided in a similar manner. In particular, in this case, a common linear Lyapunov function guaranteeing the stability of the switched positive systems can be easily located by means of geometry properties. In the continuous-time case, a finite number of second order systems are considered. Some equivalent conditions for stability of such systems are developed.     

On the equivalence of two differential equations by means of reflecting functions coinciding

In this article, we discuss the equivalence of two differential systems by using the method of reflecting functions. We obtain some necessary and sufficient conditions under which certain differential equations are equivalent. Given these results, new types of differential systems equivalent to the given systems can be found. We also discussed the qualitative behavior of the periodic solutions of such differential systems. These results are new, in the sense that they generalize previous discussions on the equivalence of differential systems.