Differential-operator Equations Research Articles

Optimal control of undamped Sobolev-type retarded systems

The optimal control problem for a system whose evolution is described by a Sobolev-type second-order retarded operator-differential equation is studied. The main assumption is that a restriction is imposed on the derivatives of the resolvent of the quadratic operator pencil on a ray in the right half-plane. Several applications to systems described by non-Kovalevskaya-type partial differential equations are considered.

Unraveling mirror properties in time-delayed quantum feedback scenarios

We derive in the Heisenberg picture a widely used phenomenological coupling element to treat feedback effects in quantum optical platforms. Our derivation is based on a microscopic Hamiltonian, which describes the mirror-emitter dynamics based on a dielectric, a mediating fully quantized electromagnetic field and a single two-level system in front of the dielectric. The dielectric is modelled as a system of identical two-state atoms. The Heisenberg equation yields a system of describing differential operator equations, which we solve in the Weisskopf–Wigner limit. Due to a finite round-trip time between emitter and dielectric, we yield delay differential operator equations. Our derivation motivates and justifies the typical phenomenologicalassumed coupling element and allows, furthermore, a generalization to a variety of mirrors, such as dissipative mirrors or mirrors with gain dynamics.

On the Hamiltonian property of linear dynamical systems in Hilbert space

Conditions for the operator differential equation \(\dot x = Ax\) possessing a quadratic first integral (1/2)(Bx, x) to be Hamiltonian are obtained. In the finite-dimensional case, it suffices to require that ker B ⊂ ker A*. For a bounded linear mapping x → Ωx possessing a first integral, sufficient conditions for the preservation of the (possibly degenerate) Poisson bracket are obtained.

Теорія крайових задач для операторно-диференціальних рівнянь

Теорія крайових задач для операторно-диференціальних рівнянь

Variation of Parameters for Causal Operator Differential Equations

The operator T from a domain D into the space of measurable functions is called a nonanticipating (causal) operator if the past information is independent from the future outputs. We will study the solution x(t) of a nonlinear operator differential equation where its changes depends on the causal operator T, and semigroup of operator A(t), and all initial parameters (t0, x0) . The initial information is described x(t)=φ(t) for almost all t ≤t0 and φ(t0) =φ0. We will study the nonlinear variation of parameters (NVP) for this type of nonanticipating operator differential equations and develop Alekseev type of NVP.

On Solvability of Third-Order Operator Differential Equation with Parabolic Principal Part in Weighted Space

Sufficient conditions are found for the correct and unique solvability of a class of third-order parabolic operator differential equations, whose principal parts have multiple characteristics, in a Sobolev-type space with exponential weight. The estimates for the norms of intermediate derivative operators are obtained and the relationship between these estimates and solvability conditions is established. Besides, the connection is found between the order of exponential weight and the lower bound for the spectrum of abstract operator appearing in the principal part of the equation.

On the Solvability of the Boundary-Value Problem for the First Order Operator-Differential Equations with Multiple Characteristics

In this paper, a class of operator-differential equation of the first order with multiple characteristics is considered, for which the initial boundary value problem on the semi-axis is well-posed and uniquely solvable in the Sobolev space.

Optimal Control of Nonsmooth, Semilinear Parabolic Equations

This paper is concerned with an optimal control problem governed by a semilinear, nonsmooth operator differential equation. The nonlinearity is locally Lipschitz-continuous and directionally differentiable but not Gâteaux-differentiable. By employing the limited differentiability properties of the control-to-state map, first-order necessary optimality conditions in qualified form are established, which are equivalent to the purely primal condition saying that the directional derivative of the reduced objective in feasible directions is nonnegative. The paper ends with the application of the general results to a semilinear heat equation.

Solving Two-Point Boundary Value Problems for a Wave Equation via the Principle of Stationary Action and Optimal Control

A new approach to solving two-point boundary value problems for a wave equation is developed. This new approach exploits the principle of stationary action to reformulate and solve such problems in the framework of optimal control. In particular, an infinite dimensional optimal control problem is posed so that the wave equation dynamics and temporal boundary data of interest are captured via the characteristics of the associated Hamiltonian and choice of terminal payoff, respectively. In order to solve this optimal control problem for any such terminal payoff, and hence solve any two-point boundary value problem corresponding to the boundary data encapsulated by that terminal payoff, a fundamental solution to the optimal control problem is constructed. Specifically, the optimal control problem corresponding to any given terminal payoff can be solved via a max-plus convolution of this fundamental solution with the specified terminal payoff. Crucially, the fundamental solution is shown to be a quadratic functional that is defined with respect to the unique solution of a set of operator differential equations, and computable using spectral methods. An example is presented in which this fundamental solution is computed and applied to solve a two-point boundary value problem for the wave equation of interest.

On generalized solutions of a boundary value problem for operator-differential equations on a finite interval

On generalized solutions of a boundary value problem for operator-differential equations on a finite interval

Спектральные характеристики нелокальной задачи для двух линейных систем дифференциальных уравнений в частных производных

Изучается граничная задача для линейной системы дифференциальных уравнений, записанная в виде дифференциально-операторного уравнения $$ aD_t u(t)+bBu(t)=f(t) $$ с нелокальными граничными условиями по $t$. Такую краевую задачу для линейной системы дифференциальных уравнений (в том числе и в частных производных) мы условимся называть нелокальной. Цель работы состоит в изучении спектральных характеристик дифференциальных операторов, порожденных нелокальной задачей для двух линейных систем дифференциальных уравнений в частных производных, рассматриваемых в ограниченной области конечномерного евклидова пространства.

Homogeneous differential-operator equations in locally convex spaces

We describe a general method that allows us to find solutions to homogeneous differential-operator equations with variable coefficients by means of continuous vector-valued functions. The “homogeneity” is not interpreted as the triviality of the right-hand side of an equation. It is understood in the sense that the left-hand side of an equation is a homogeneous function with respect to operators appearing in that equation. Solutions are represented as functional vector-valued series which are uniformly convergent and generated by solutions to a kth order ordinary differential equation, by the roots of the characteristic polynomial and by elements of a locally convex space. We find sufficient conditions for the continuous dependence of the solution on a generating set. We also solve the Cauchy problem for the considered equations and specify conditions for the existence and the uniqueness of the solution. Moreover, under certain hypotheses we find the general solution to the considered equations. It is a function which yields any particular solution. The investigation is realized by means of characteristics of operators such as the order and the type of an operator, as well as operator characteristics of vectors, namely, the operator order and the operator type of a vector relative to an operator. We also use a convergence of operator series with respect to an equicontinuous bornology.

Invariance of the order and type of a sequence of operators

In the paper, the invariance property of characteristics (the order and type) of an operator and of a sequence of operators with respect to a topological isomorphism is proved. These characteristics give precise upper and lower bounds for the expressions ‖An(x)‖p and enable one to state and solve problems of operator theory in locally convex spaces in a general setting. Examples of such problems are given by the completeness problem for the set of values of a vector function in a locally convex space, the structure problem for a subspace invariant with respect to an operator A, the problem of applicability of an operator series to a locally convex space, the theory of holomorphic operator-valued functions, the theory of operator and differential-operator equations in nonnormed spaces, and so on. However, the immediate evaluation of characteristics of operators (and of sequences of operators) directly by definition is practically unrealizable in spaces with more complicated structure than that of countably normed spaces, due to the absence of an explicit form of seminorms or to their complicated structure. The approach that we use enables us to find characteristics of operators and sequences of operators using the passage to the dual space, by-passing the definition, and makes it possible to obtain bounds for the expressions ‖An(x)‖p even if an explicit form of seminorms is unknown.

Solvability conditions of a boundary value problem with operator coefficients and related estimates of the norms of intermediate derivative operators

Sufficient conditions for the proper and unique solvability in the Sobolev space of vector functions of the boundary value problem for a certain class of second-order elliptic operator differential equations on a semiaxis are obtained. The boundary condition at zero involves an abstract linear operator. The solvability conditions are established by using properties of operator coefficients. The norms of intermediate derivative operators, which are closely related to the solvability conditions, are estimated.

Solvability of a nonlocal boundary-value problem for the operator-differential equation with weak nonlinearity in a refined scale of Sobolev spaces

A nonlocal boundary-value problem for the differential equation with weak nonlinearity and with differential operator B = (B 1, …, B p ), where \( {B}_j\equiv {z}_j\frac{\partial }{\partial {z}_j}\;\mathrm{and}\;j=1,\dots, p \) is considered. By using the Nash–Moser iterative scheme, the solvability conditions for the present problem in the Hilbert H¨ormander spaces of functions of many complex variables forming a refined Sobolev scale of spaces is established.

Nonlocal Multipoint Problem for Differential-Operator Equations of Order 2n

We study a nonlocal multipoint problem for differential-operator equations of order 2n.The spectral properties of the operator of this problem are analyzed and the conditions for the existence and uniqueness of its solution are established. The solution of the problem is found in the form of expansion in series in the system of eigenfunctions. It is also proved that the system of eigenfunctions of the analyzed problem forms a Riesz basis.

On a differential game in an abstract parabolic system

We consider the game problem of approach for a system whose dynamics is described by a differential operator equation in a Hilbert space. The equation is written in an implicit form with generally noninvertible operator multiplying the derivative. It is assumed that the characteristic operator pencil corresponding to the linear part of the equation satisfies a constraint of parabolic type in a right half-plane. Using the method of resolving functionals, we obtain sufficient conditions for the approach of a dynamical vector of the system to a cylindrical terminal set. Applications to systems described by partial differential equations are considered.

Differential equations in spaces of abstract stochastic distributions

Stochastic Ito equations with additive and multiplicative noise in separable Hilbert spaces are studied by reducing them to differential-operator equations in spaces of generalized Hilbert space-valued random variables. Results on the existence and uniqueness of solutions in these spaces are obtained by using the S-transform technique and methods of the theory of semigroups of linear operators.

Well-posed boundary-value problems, right hyperbolicity, and exponential dichotomy

A relationship between the existence of well-posed boundary-value problems and exponential dichotomy for functional equations and linear differential-operator equations on a half-line is considered. It is shown that well-posed boundary-value problems can exist for equations without the exponential dichotomy property.

Abstract elliptic equations with integral boundary conditons

This paper focuses on nonlocal integral boundary value problems for elliptic differential-operator equations. Here given conditions guarantee that maximal regularity and Fredholmness in Lp spaces. These results are applied to the Cauchy problem for abstract parabolic equations, its infinite systems and boundary value problems for anisotropic partial differential equations in mixed Lp norm