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This work addresses the idea of optimal stabilization, namely robustness of optimal stabilization with nonlinearity Lipschitz of distributed semilinear systems using bounded control. This problem is treated under the condition of the unbounded operator, we show that the system is stable once the exact observability assumption is executed together with a Lipschitz property of the nonlinear operator. The concept of bounded control is also investigated in realistic domain. The stabilizing feedback is characterized by the minimization of the problem of cost. We also give different applications to parabolic and hyperbolic equations.

In this work the ring of finite adeles $${\mathbb {A}}_f$$ of the rational numbers $${\mathbb {Q}}$$ is obtained as a completion of $${\mathbb {Q}}$$ with respect to a certain non-Archimedean metric related to the second Chebyshev function, which allows us to represent any finite adele as a convergent series, generalizing m-adic analysis. This polyadic analysis allows us to introduce a novel pseudodifferential operator $$D^{\alpha }$$ on $$L^2({\mathbb {A}}_f)$$ of fractional differentiation, similar to the Vladimirov operator on the p-adic numbers. The operator $$D^{\alpha }$$ is a positive selfadjoint unbounded operator whose spectrum $$\sigma (D^{\alpha })$$ is essential and it consists of a countable number of eigenvalues, which converges to zero, and zero. Moreover, a sort of multiresolution analysis on $${\mathbb {A}}_f$$ provides us with a wavelet basis which is an orthonormal basis of eigenfunctions of $$D^{\alpha }$$ as well. The Cauchy problem of a wave-type pseudodifferential equation $$\begin{aligned} u_{tt}(x,t)+D^{\alpha }_x u(x,t) =F(x,t), \qquad (x \in {\mathbb {A}}_f), \end{aligned}$$ with appropriate initial conditions $$u(x,0)=f(x), u_t(x,0)=g(x),$$ and external force F(x, t), is solved separating variables and using the Fourier expansion of functions in $$L^2({\mathbb {A}}_f)$$ , with respect to the wavelet basis.

We analyse possibility to extend a quantum operation (sub-unital normal completely positive linear map on the algebra $$\mathfrak{B}(\mathcal{H})$$ of bounded operators on a separable Hilbert space $$\mathcal{H}$$ ) to the space of all operators on $$\mathcal{H}$$ relatively bounded w.r.t. a given positive unbounded operator. We show that a quantum operation $$\Phi$$ can be uniquely extended to a bounded linear operator on the Banach space of all $$\sqrt{G}$$ -bounded operators on $$\mathcal{H}$$ provided that the operation $$\Phi$$ is $$G$$ -limited: the predual operation $$\Phi_{*}$$ maps the set of positive trace class operators $$\rho$$ with finite value of $$\textrm{Tr}\rho G$$ into itself. Assuming that $$G$$ has discrete spectrum of finite multiplicity we prove that for a wide class of quantum operations the existence of the above extension implies the $$G$$ -limited property. Applications to the theory of Bosonic Gaussian channels are considered.

Unbounded operators can transform arbitrarily small vectors into arbitrarily large vectors—a phenomenon known as instability. Stabilization methods strive to approximate a value of an unbounded operator by applying a family of bounded operators to rough approximate data that do not necessarily lie within the domain of unbounded operator. In this paper we shall be concerned with the stable method of computing values of unbounded operators having perturbations and the stability is established for this method.

In this paper, we give an example of a closed unbounded operator whose square's domain and adjoint's square domain are equal and trivial. Then, we come up with an essentially self-adjoint whose square has a trivial domain.

Let l be a Banach sequence space with a monotone norm in which the canonical system (e_{n}) is an unconditional basis. We show that if there exists a continuous linear unbounded operator between l-K\"{o}the spaces, then there exists a continuous unbounded quasi-diagonal operator between them. Using this result, we study in terms of corresponding K\"{o}the matrices when every continuous linear operator between l-K\"{o}the spaces is bounded. As an application, we observe that the existence of an unbounded operator between l-K\"{o}the spaces, under a splitting condition, causes the existence of a common basic subspace.

In a Hilbert space we consider the equation $$du(t)/dt=A(t)u(t)+F(t,u(t))$$ , where A(t) is an unbounded operator, having a bounded strong derivative, and F is a continuous mapping. We derive norm estimates for solutions of the considered equation. These estimates give us the $$L^2$$ -stability and absolute stability conditions. To the best of our knowledge an absolute stability test for nonautonomous evolution equations has been obtained for the first time. Our main tool is the norm estimate for the derivative of a solution of the time-dependent Lyapunov equation.

For abstract integrodifferential equations with unbounded operator coefficients in a Hilbert space, the correct solvability of initial value problems is studied and the spectral analysis of operator functions being symbols of these equations is performed. This makes it possible to represent strong solutions of the equations under consideration as series in exponentials corresponding to spectral points of the operator functions. The equations in question are abstract forms of linear partial integrodifferential equations arising in the theory of viscoelasticity and in a number of other important applications.

In this paper, we introduce the notion of abstract local operator algebras and operator modules, and provide a representation theorem for them which extends the BRS theorem for operator algebras. Furthermore, we give a new proof for the representation theorem of local operator systems. Also, we investigate the Haagerup tensor product of local operator spaces and Morita equivalence of local operator algebras.

This paper deals with a new description of the one sided operator matrix form, as a generalization of the case of the unbounded operator matrix with the non diagonal domain, to investigate some advances in the analysis of some essential spectra under weaker hypotheses then the one provided in the works of [17, 33]. An example of differential equations is tested to ensure the validity of the abstract results.

We consider the Caffarelli–Silvestre extension problem, i.e., a Bessel type ODE in a Banach space X with a closed and typically unbounded operator A as right-hand side and point out a couple of consequences arising from the assumption of the well-posedness of the problem. In the end a conjecture is stated concerning the implications of analyticity of the solution of the extension problem.

This paper discusses the in-domain feedback stabilization of reaction-diffusion PDEs with Robin boundary conditions in the presence of an uncertain time- and spatially-varying delay in the distributed actuation. The proposed control design strategy consists of a constant-delay predictor feedback designed based on the known nominal value of the control input delay and is synthesized on a finite-dimensional truncated model capturing the unstable modes of the original infinite-dimensional system. By using a small-gain argument, we show that the resulting closed-loop system is exponentially stable provided that the variations of the delay around its nominal value are small enough. The proposed proof actually applies to any distributed-parameter system associated with an unbounded operator that 1) generates a $C_0$-semigroup on a weighted space of square integrable functions over a compact interval; and 2) is self-adjoint with compact resolvent., Comment: Accepted for publication in Automatica as a brief paper

We investigate the well-posedness of initial-value problems for abstract integrodifferential equations with unbounded operator coefficients in a Hilbert space and provide the spectral analysis of operator-functions describing symbols of such equations. These equations are an abstract form of linear partial integrodifferential equations arising in the viscoelasticity theory and other important applications. We establish the localization and the spectrum structure of operator-functions describing symbols of these equations.

For abstract integro-differential equations with unbounded operator coefficients in a Hilbert space, we study the well-posed solvability of initial problems and carry out spectral analysis of the operator functions that are symbols of these equations. This allows us to represent the strong solutions of these equations as series in exponentials corresponding to points of the spectrum of operator functions. The equations under study are the abstract form of linear integro-partial differential equations arising in viscoelasticity and several other important applications.

In this paper, we develop and study algorithms for approximately solving linear algebraic systems: ${{\mathcal{A}}}_h^\alpha u_h = f_h$, $ 0< \alpha

The product of a matrix function and a vector is used to solve evolution equations numerically. Hashimoto and Nodera (ANZIAM J 58:C149–C161, 2016) proposed the Shift-invert Rational Krylov method for computing these products. However, since matrices produced by evolution equations behave like unbounded operators in infinite-dimensional spaces, an analysis with the unbounded operator is essential. In this paper, the Shift-invert Rational Krylov method is extended to be applied to unbounded operators.

It is known that the nonlinear nonhomogeneous backward Cauchy problem ut(t) + Au(t) = f (t, u(t)), 0 ⩽ t < τ with u(τ) = φ, where A is a densely defined positive self-adjoint unbounded operator on a Hilbert space, is ill-posed in the sense that small perturbations in the final value can lead to large deviations in the solution. We show, under suitable conditions on φ and f, that a solution of the above problem satisfies an integral equation involving the spectral representation of A, which is also ill-posed. Spectral truncation is used to obtain regularized approximations for the solution of the integral equation, and error analysis is carried out with exact and noisy final value φ. Also stability estimates are derived under appropriate parameter choice strategies. This work extends and generalizes many of the results available in the literature, including the work by Tuan (2010) for linear homogeneous final value problem and the work by Jana and Nair (2016b) for linear nonhomogeneous final value problem.

Integro-differential equations with unbounded operator coefficients in a Hilbert space are considered. Such equations arise in viscoelasticity theory, thermal physics, and homogenization problems in multiphase media. Initial–boundary value problems for the indicated equations are proved to be well posed, and their spectral analysis is performed.

We study the well-posedness of initial-value problems for abstract integrodifferential equations with unbounded operator coefficients in Hilbert spaces and provide a spectral analysis of operator functions that are symbols of the specified equations. These equations represent an abstract form of linear partial integrodifferential equations arising in viscoelasticity theory and other important applications. For the said integrodifferential equations, we obtain well-posedness results in weighted Sobolev spaces of vector functions defined on the positive semiaxis and valued in a Hilbert space. For the symbols of the said equations, we find the localization and the structure of the spectrum.

We study a problem with periodic boundary conditions for a $2n$-order differential equation whose coefficients are non-self-adjoint operators. It is established that the operator of the problem has two invariant subspaces generated by the involution operator and two subsystems of the system of eigenfunctions which are Riesz bases in each of the subspaces. For a differential-operator equation of even order, we study a problem with non-self-adjoint boundary conditions which are perturbations of periodic conditions. We study cases when the perturbed conditions are Birkhoff regular but not strongly Birkhoff regular or nonregular. We found the eigenvalues and elements of the system $V$ of root functions of the operator which is complete and contains an infinite number of associated functions. Some sufficient conditions for which this system $V$ is a Riesz basis are obtained. Some conditions for the existence and uniqueness of the solution of the problem with homogeneous boundary conditions are obtained.

We study the problems of existence and representations of the solutions bounded on the entire axis for both linear and nonlinear differential equations with unbounded operator coefficients in the Frechet and Banach spaces under the condition of exponential dichotomy on the semiaxes of the corresponding homogeneous equation.