Partial Integral Operators Research Articles

Uniform convergence of solutions to stochastic hybrid models of gene regulatory networks

In a recent paper by Kurasov et al. (Math Biosci 305:170–177, 2018), a hybrid gene regulatory network was proposed to model gene expression dynamics by using a stochastic system of coupled partial differential equations. In more recent work, the existence and strong convergence of the solutions to equilibrium were proven. In this article, we improve upon their result by showing that the convergence rate is independent of the initial state, therefore proving that the solutions converge not only strongly but even uniformly to equilibrium. To this end, we make use of a recent convergence theorem for stochastic, irreducible semigroups that contain partial integral operators.

Computing Optimal Upper Bounds on the H2-norm of ODE-PDE Systems using Linear Partial Inequalities

Recently, a broad class of linear delayed and ODE-PDEs systems was shown to have an equivalent representation using Partial Integral Equations (PIEs). In this paper, we use this PIE representation, combined with algorithms for convex optimization of Partial Integral (PI) operators to bound the H2-norm for input-output systems of this class. Specifically, the methods proposed here apply to delayed and ODE-PDE systems (including delayed PDE systems) in one or two spatial variables where the disturbance does not enter through the boundary.For such systems, we define a notion of H2-norm using an initial state-to-output framework and show that this notion reduces to more traditional concepts under the assumption of existence of a strongly continuous semigroup. Next, we consider input-output systems for which there exists a PIE representation and for such systems show that computing a minimal upper bound on the H2-norm of delayed and PDE systems can be equivalently formulated as a convex optimization problem subject to linear PI operator inequalities (LPIs). We convert, then, these optimization problems to Semi-Definite Programming (SDP) problems using the PIETOOLS toolbox. Finally, we apply the results to several numerical examples – focusing on time-delay systems (TDS) for which comparable H2 approximation results are available in the literature. The numerical results demonstrate the accuracy of the computed upper bound on the H2-norm.

Delay-Dependent Stability for Load Frequency Control System via Linear Operator Inequality

The stability analysis problem is considered for multiarea load frequency control (LFC) systems with electric vehicles (EVs) and time delays. The novel linear operator inequality approach is proposed and a less conservative delay-dependent stability condition is achieved. First, the model of the multiarea LFC system with EVs and delays is expressed by the partial integral equation (PIE) at the first time. Then, a complete quadratic Lyapunov-Krasovskii functional is built in the form of an inner product of the linear partial integral (PI) operator. The novel stability criteria with less conservatism are proposed in the form of linear operator inequality. Moreover, the relationships between the delay margins and the controller parameters are shown. Finally, the simulations are conducted on both one-area and two-area LFC systems to show the effectiveness of the proposed approach.

A Fuzzy-PIE Representation of T-S Fuzzy Systems with Delays and Stability Analysis via LPI method

Inspired by the recently proposed Partial Integral Equality(PIE) representation for linear delay systems, this paper proposes a fuzzy-PIE representation for T-S fuzzy systems with delays for the first time. Inspired by the free-weighting matrix technique, this paper introduces the free-weighting Partial Integral (PI) operators. Based on the novel representation and free-weighting PI operators, the stability issue is investigated for the T-S fuzzy systems with delays. The corresponding conditions are given as Linear Partial Inequality (LPI) and can be solved by the MATLAB toolbox PIETOOLS. Compared with the existing results, our method has no need of the bounding technique and a large amount of matrix operation. The numerical examples show the superiority of our method. This paper adds to the expanding field of LPI approach to fuzzy systems with delays.

The analytical analysis of fractional order Fokker-Planck equations

<abstract><p>In the current note, we broaden the utilization of a new and efficient analytical computational scheme, approximate analytical method for obtaining the solutions of fractional-order Fokker-Planck equations. The approximate solution is obtained by decomposition technique along with the property of Riemann-Liouuille fractional partial integral operator. The Caputo-Riemann operator property for fractional-order partial differential equations is calculated through the utilization of the provided initial source. This analytical scheme generates the series form solution which is fast convergent to the exact solutions. The obtained results have shown that the new technique for analytical solutions is simple to implement and very effective for analyzing the complex problems that arise in connected areas of science and technology.</p></abstract>

Uniform convergence of stochastic semigroups

For stochastic C0-semigroups on L1-spaces there is a wealth of results that show strong convergence to an equilibrium as t → ∞, given that the semigroup contains a partial integral operator. This has plenty of applications to transport equations and in mathematical biology. However, up to now partial integral operators do not play a prominent role in theorems which yield uniform convergence of the semigroup rather than only strong convergence.In this article we prove that, for irreducible stochastic semigroups, uniform convergence to an equilibrium is actually equivalent to being partially integral and uniformly mean ergodic. In addition to this Tauberian theorem, we also show that our semigroup is uniformly convergent if and only if it is partially integral and the dual semigroup satisfies a certain irreducibility condition. Our proof is based on a uniform version of a lower bound theorem of Lasota and Yorke, which we combine with various techniques from Banach lattice theory.

Boundedness of Mixed Norms of Partial Integral Operators

Boundedness of Mixed Norms of Partial Integral Operators

Partial integral operators of non-negative ordersin weighted Lebesgue spaces

Partial integral operators of non-negative ordersin weighted Lebesgue spaces

On the essential and the discrete spectra of a Fredholm type partial integral operator

We investigate the structure of the essential spectrum of a self-adjoint Fredholm type partial integral operator H. We obtain an explicit description of the essential spectrum of H and prove that an eigenvalue of H exists.

On the number of negative eigenvalues of a partial integral operator

We find the lower boundary for the essential spectrum of a Fredholm type partial integral operator H. We also obtain an estimate for the number of eigenvalues below this boundary.

On the discrete spectrum of partial integral operators

We prove a theorem on the discrete spectrum of a partial integral selfadjoint operator with a continuous kernel.

Bounds and positivity conditions for operator valued functions in a Hilbert lattice

In this paper we investigate regular functions of a bounded operator A acting in a Hilbert lattice and having the form A=D + T, where T is a positive operator and D is a selfadjoint operator whose resolution of the identity P(t)\((a\le s \le b)\) has the property \(P(s_2)-P(s_1)\;\;(s_1<s_2)\) are non-negative in the sense of the order. Upper and lower bounds and positivity conditions for the considered operator valued functions are derived. Applications of the obtained estimates to functions of integral operators, partial integral operators, infinite matrices and differential equations are also discussed.

Partial radiation conditions and hypersingular integral operators

The structure of the partial radiation conditions is analyzed. It is found that the principal part of the operator defining these conditions defines a hypersingular operator. The series corresponding to the operator associated with the boundary conditions is transformed to a rapidly converging series.

Spectrum and resolvent of a partial integral operator

We consider the partial integral operator where w(.,. ), K(.,. ,. ), φ are given functions. Norm estimates for the resolvent and bounds for the spectrum are established. Our main tool is the direct integral of Hilbert spaces and recent estimates for norms of the resolvents of compact operators and operators with the compact Hermitian components.

Spectra of partial integral operators with a kernel of three variables

Abstract Let Ω= [a, b] × [c, d] and T 1, T 2 be partial integral operators in $$ C $$(Ω): (T 1 f)(x, y) = $$ \mathop \smallint \limits_a^b $$ k 1(x, s, y)f(s, y)ds, (T 2 f)(x, y) = $$ \mathop \smallint \limits_c^d $$ k 2(x, ts, y)f(t, y)dt where k 1 and k 2 are continuous functions on [a, b] × Ω and Ω × [c, d], respectively. In this paper, concepts of determinants and minors of operators E−τT 1, τ ∈ ℂ and E−τT 2, τ ∈ ℂ are introduced as continuous functions on [a, b] and [c, d], respectively. Here E is the identical operator in C(Ω). In addition, Theorems on the spectra of bounded operators T 1, T 2, and T = T 1 + T 2 are proved.

On Volterra partial integral equations in the space of continuous functions of three variables

It is well known that any Volterra integral equation of the second kind with compact operator is uniquely solvable. Partial integral operators are not compact, even in the general case of continuous kernels. Unique solvability conditions for Volterra partial integral equations of the second kind in the space of continuous functions of three variables are considered. Conditions for a Volterra partial integral equation to be equivalent to a three-dimensional Volterra integral equation with compact operator are obtained. Continuum analogues of matrix equations for some problems of scattering theory are reduced to the Volterra partial integral equations under examination.

A discrete “three-particle” Schrödinger operator in the Hubbard model

In the space L 2(T ν ×T ν ), where T ν is a ν-dimensional torus, we study the spectral properties of the “three-particle” discrete Schrodinger operator Ĥ = H0 + H1 + H2, where H0 is the operator of multiplication by a function and H1 and H2 are partial integral operators. We prove several theorems concerning the essential spectrum of Ĥ. We study the discrete and essential spectra of the Hamiltonians Ht and h arising in the Hubbard model on the three-dimensional lattice.

A Note on the Fredholm Property of Partial Integral Equations of Romanovskij Type

Some conditions are given, both necessary and sufficient, under which a partial integral equation of Romanovskij type defines a Fredholm operator of index zero in the space of continuous functions. In 1932, Romanovskij [6] has described a problem in the theory of Markov chains with two-sided link which leads to an equation of the form (1) x(t, s) = Rx(t, s) + f(t, s), (t, s) ∈ D := [a, b]× [a, b], where R is the linear operator defined by (2) Rx(t, s) = ∫ b a m(t, s, σ)x(σ, t) dσ which contains some continuous or measurable kernel function m : D× [a, b] → R. A particular feature of the operator (2) is that first the two variables in the unknown integrand x are inverted, and afterwards the integration is carried out with respect to the first variable. Equation (1) has been studied for continuous kernel functions in [6] by means of Fredholm determinants. In this connection it turned out that many results on the problem (1) are quite different from classical results on Fredholm integral equations, mainly due to the fact that the operator (2) is not compact and not even an integral operator. It is natural (and now common sense) to call operators of the form (2) partial integral operators, inasmuch as the integration in (2) is carried out only with respect to one variable while the other variables are “frozen.” AMS Mathematics Subject Classification. 45A05, 45B05, 45P05, 46E15, 47A53, 47B38, 47G10.

INVERTIBILITY AND SPECTRA OF OPERATORS ON TENSOR PRODUCTS OF HILBERT SPACES This research was supported by the Kamea Fund.

A class of operators on a tensor product of separable Hilbert spaces is considered. It contains various traditional operators. Invertibility, positive invertibility conditions and estimates for the norm of the resolvents are established. In addition, bounds for the spectrum are suggested. Applications to partial integral and integro-differential operators are discussed.

Spectrum perturbations of operators on tensor products of Hilbert spaces

We investigate the spectrum perturbations and spectrum localization of a class of operators on a tensor product of separable Hilbert spaces. In particular, estimates for the spectral radius and norm of the resolvent are derived. Applications to partial integral and integrodifferential operators are also discussed.