Characterization of Lyapunov pairs in the nonlinear case and applications

The paper presents a characterization of the Lyapunov pairs for a general initial value problem with a possibly multivalued m$m$-accretive operator on an arbitrary Banach space by means of the contingent derivative related to the operator. The proof is based on tangency and flow-invariance arguments combined with a priori estimates and approximation. The abstract results are applied to obtain precise a priori estimates for the mild solutions. They readily lead to the existence of global solutions and various controllability properties. Important Lyapunov pairs are pointed out in connection with specific problems.     

The Nehari problem for infinite-dimensional linear systems of parabolic type

A complete solution is obtained to the suboptimal Nehari extension problem for transfer functions of parabolic systems with Dirichlet boundary control and smooth observations. The solutions are given in terms of the realization (–A, B, C), whereA is a uniformly strongly elliptic operator of order two with smooth coefficients defined on a bounded open domain ofR _d ,B=AB _D andB _D is the Dirichlet map associated with Dirichlet boundary conditions andC is a bounded observation map fromL ₂() to the output spaceY. The approach is to solve an equivalentJ-spectral factorization problem for this particular realization.     

The Nehari problem for the Pritchard-Salamon class of infinite-dimensional linear systems: a direct approach

A complete solution is obtained to the Nehari problem for symbols which have a realization as an exponentially stable Pritchard-Salamon system (A, B, C). This allows for the possibility thatB andC be unbounded and have infinite rank. The approach is to solve an equivalentJ-spectral factorization problem for this particular realization.     

Spectral stability of general non-negative self-adjoint operators with applications to Neumann-type operators

We prove a stability theorem for the eigenvalues of general non-negative self-adjoint linear operators with compact resolvents and by applying it we prove a sharp stability result for the dependence of the eigenvalues of second order uniformly elliptic linear operators with homogeneous Neumann boundary conditions upon domain perturbation.     

The finite section method for Moore-Penrose inversion of Toeplitz operators

It is shown that for certain classes of infinite block Toeplitz matricesT(a)=[a _j-k ] ₀ the Moore-Penrose inverses of the finite sectionT _n (a)=[a _j-k ] ₀ ^n–1 converge to the Moore-Penrose inverse ofT(a). Furthermore the convergence for modified finite section methods and the finite section method for Wiener-Hopf integral and related operators are studied.     

Carleman estimates for semi-discrete parabolic operators and application to the controllability of semi-linear semi-discrete parabolic equations

In arbitrary dimension, in the discrete setting of finite-differences we prove a Carleman estimate for a semi-discrete parabolic operator, in which the large parameter is connected to the mesh size. This estimate is applied for the derivation of a (relaxed) observability estimate, that yield some controlability results for semi-linear semi-discrete parabolic equations. Sub-linear and super-linear cases are considered.     

Carleman estimates for degenerate parabolic operators with applications to null controllability

We prove an estimate of Carleman type for the one dimensional heat equation $$ u_t - \left( {a\left( x \right)u_x } \right)_x + c\left( {t,x} \right)u = h\left( {t,x} \right),\quad \left( {t,x} \right) \in \left( {0,T} \right) \times \left( {0,1} \right), $$ where a(·) is degenerate at 0. Such an estimate is derived for a special pseudo-convex weight function related to the degeneracy rate of a(·). Then, we study the null controllability on [0, 1] of the semilinear degenerate parabolic equation $$ u_t - \left( {a\left( x \right)u_x } \right)_x + f\left( {t,x,u} \right) = h\left( {t,x} \right)\chi _\omega \left( x \right), $$ where (t, x) ∈(0, T) × (0, 1), ω=(α, β) ⊂⊂ [0, 1], and f is locally Lipschitz with respect to u. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

The forward problem for the electromagnetic Helmholtz equation with critical singularities

We study the forward problem of the magnetic Schrödinger operator with potentials that have a strong singularity at the origin. We obtain new resolvent estimates and give some applications on the spectral measure and on the solutions of the associated evolution problem.     

Circulant preconditioners for pricing options

We use the normalized preconditioned conjugate gradient method with Strang’s circulant preconditioner to solve a nonsymmetric Toeplitz system A_nx=b, which arises from the discretization of a partial integro-differential equation in option pricing. By using the definition of family of generating functions introduced in [16], we prove that Strang’s circulant preconditioner leads to a superlinear convergence rate under certain conditions. Numerical results exemplify our theoretical analysis.     

The invertibility of convolution type operators on a union of intervals and the corona theorem

The invertibility of convolution type operators on unions of intervals is studied. Sufficient conditions of invertibility for some classes of these operators are established. Solvability results forn-term corona problems are obtained using two different approaches: one involving reduction ton–1 Riemann-Hilbert problems in two variables and another involving reduction to two-term corona problems. The invertibility of the convolution operators on a union of intervals is also related to the invertibility of associated convolution operators on single intervals. Formulas for the inverse operators are given.Work partially supported by F. C. T. (Portugal) Project Praxis XXI/2/2.1/MAT/441/94 and by CONACYT (México) grant, Cátedra Patrimonial, No. 990017-EX.     

Continuity properties for modulation spaces, with applications to pseudo-differential calculus—I

Let M^p,q denote the modulation space with parameters p,q∈[1,∞]. If 1/p₁+1/p₂=1+1/p₀ and 1/q₁+1/q₂=1/q₀, then it is proved that . The result is used to get inclusions between modulation spaces, Besov spaces and Schatten classes in calculus of Ψdo (pseudo-differential operators), and to extend the definition of Toeplitz operators. We also discuss continuity of ambiguity functions and Ψdo in the framework of modulation spaces.     

Regularity of solutions for the Navier-Stokes system of incompressible flows on a polygon

We are concerned with singularities and regularities of solutions for the Navier-Stokes system of incompressible flows on a polygonal domain with a concave vertex. We subtract the corner singularities by the Stokes operator from the solution velocity and pressure functions of the system. It is shown that the stress intensity factors are functions of time variable, belong to a fractional Sobolev space on the time interval and can be expressed in terms of given data. An increased regularity for the remainder is obtained.     

An a priori error estimate for a monotone mixed finite-element discretization of a convection–diffusion problem

We present a local exponential fitting hybridized mixed finite-element method for convection–diffusion problem on a bounded domain with mixed Dirichlet Neuman boundary conditions. With a new technique that interpretes the algebraic system after static condensation as a bilinear form acting on certain lifting operators we prove an a priori error estimate on the Lagrange multipliers that requires minimal regularity. While an extension of more classical arguments provide an estimate for the other solution components.     

Dynamics of Properties of Toeplitz Operators with Radial Symbols

In the case of radial symbols we study the behavior of different properties (boundedness, compactness, spectral properties, etc.) of Toeplitz operators T_a⁽⁾ acting on weighted Bergman spaces over the unit disk , in dependence on , and compare their limit behavior under with corresponding properties of the initial symbol a.     

A regularity result of solution to the compressible Stokes equations on a convex polygon

A compressible Stokes problem is analyzed in a convex polygon D. The goal of this paper is to sort out a singularity of the pressure function at the corner and to establish the corresponding regularity result of the resulted remainder part of the solution. For this a solution formula is derived and the singular function of the Stokes problem is considered. It is seen that the lowest order of the regularity of the system is the same as that of the (incompressible) Stokes one.     

On the nonnegative self-adjoint solutions of the operator Riccati equation for infinite dimensional systems

The nonnegative self-adjoint solutions of the operator Riccati equation (ORE) are studied for stabilizable semigroup Hilbert state space systems with bounded sensing and control. Basic properties of the maximal solution of the ORE are investigated: stability of the corresponding closed loop system, structure of the kernel, Hilbert-Schmidt property. Similar properties are obtained for the nonnegative self-adjoint solutions of the ORE. The analysis leads to a complete classification of all nonnegative self-adjoint solutions, which is based on a bijection between these solutions and finite dimensional semigroup invariant subspaces contained in the antistable unobservable subspace.     

Superoptimal singular values and indices of matrix functions

LetG be a matrix-valued function on the unit circle which is the sum of a continuous function and anH function. We establish an inequality between corresponding terms of the sequence of singular values of the Hankel operator {s _j (H _G )} and a sequence formed from the superoptimal singular values ofG with repetitions. The number of times each superoptimal singular value is repeated is a positive integer index which is the winding number of a related scalar function on the circle and gives information about the superoptimal error functionG-Q withQH . In the second part of the paper we establish a property of invariance of the sum of the indices corresponding to a particular superoptimal singular value. This establishes the truth of two conjectures made in [PeY].The research was supported by an NSF grant in modern analysis.     

On the transport-diffusion algorithm and its applications to the Navier-Stokes equations

Summary This paper deals with an algorithm for the solution of diffusion and/or convection equations where we mixed the method of characteristics and the finite element method. Globally it looks like one does one step of transport plus one step of diffusion (or projection) but the mathematics show that it is also an implicit time discretization of thePDE in Lagrangian form. We give an error bound (h+t+h×h/t in the interesting case) that holds also for the Navier-Stokes equations even when the Reynolds number is infinite (Euler equation).