Vol 25, No 131 (2020)
Articles
Fundamental operator-function of an integro-differential operator with derivatives of functionals in Banach spaces
Abstract
In this paper, we consider a generalized integro-differential operator with derivatives of functionals, which has in its construction an invertible operator in the linear part free of derivatives. The research uses previously obtained results in the field of fundamental operator functions in Banach spaces, as well as the properties of generalized functions, operators, and functionals. For an integro-differential operator with derivatives of functionals in Banach spaces, a fundamental operator-function in the terminology of Jordan sets is obtained and the conditions for the existence of this fundamental operator-function are revealed.
Russian Universities Reports. Mathematics. 2020;25(131):249-262
249-262
On the regularization of classical optimality conditions in a convex optimal control problem with state constraints
Abstract
We consider the regularization of classical optimality conditions in a convex optimal control problem for a linear system of ordinary differential equations with pointwise state constraints such as equality and inequality, understood as constraints in the Hilbert space of square-integrable functions. The set of admissible task controls is traditionally embedded in the space of square-integrable functions. However, the target functional of the optimization problem is not, generally speaking, strongly convex. Obtaining regularized classical optimality conditions is based on a technique involving the use of two regularization parameters. One of them is used for the regularization of the dual problem, while the other is contained in a strongly convex regularizing addition to the target functional of the original problem. The main purpose of the obtained regularized Lagrange principle and the Pontryagin maximum principle is the stable generation of minimizing approximate solutions in the sense of J. Varga for the purpose of practical solving the considered optimal control problem with pointwise state constraints.
Russian Universities Reports. Mathematics. 2020;25(131):263-273
263-273
To estimating linear functionals values over solutions of systems with aftereffect
Abstract
For a wide class of linear functional differential systems with Volterra operators, a constructive technique is proposed to obtain estimates of linear functionals values over solutions in conditions of uncertainty of external perturbations. It can be applied to solutions of boundary value problems with arbitrary number of boundary conditions as well as to description of attainability sets in control problems with respect to given on-target functionals. External perturbations are constrained by a given linear inequalities system on the main time segment. The technique is based on the results of general theory of functional differential equations about the solvability of boundary value problems with general linear boundary conditions and the representation of solutions. The problem under consideration is reduced to the generalized moment problem. Therewith the results on the properties of the Cauchy matrix to systems with aftereffect are of essential importance. The general form of functionals allows one to cover many cases being topical in applications such as multipoint, integral ones, as well as hybrids of those.
Russian Universities Reports. Mathematics. 2020;25(131):274-283
274-283
On adjoint operators for fractional differentiation operators
Abstract
On a linear manifold of the space of square summable functions on a finite segment vanishing at its ends, we consider the operator of left-sided Caputo fractional differentiation. We prove that the adjoint for it is the operator of right-sided Caputo fractional differentiation. Similar results are established for the Riemann-Liouville fractional differentiation operators. We also demonstrate that the operator, which is represented as the sum of the left-sided and the right-sided fractional differentiation operators is self adjoint. The known properties of the Caputo and Riemann-Liouville fractional derivatives are used to substantiate the results.
Russian Universities Reports. Mathematics. 2020;25(131):284-289
284-289
On a representation of the solvability set in the retention problem
Abstract
The paper provides another iterative method for constructing a resolving set in the game problem of retaining the movements of an abstract dynamic system in given phase constraints. In the iterative procedure, instead of the program absorption operator, it is proposed to use a family of absorption operators for individual program disturbances. Such an approach is based on theorems on the existence and representation of common fix-points of a family of mappings.
Russian Universities Reports. Mathematics. 2020;25(131):290-298
290-298
On the stability of a system of two linear hybrid functional differential systems with aftereffect
Abstract
We consider a system of two hybrid vector equations containing linear difference (defined on a discrete set) and functional differential (defined on a half-axis) parts. To study it, a model system of two vector equations is chosen, one of which is linear difference with aftereffect (LDEA), and the other is a linear functional differential with aftereffect (LFDEA). Two equivalent representations of this system are shown: the first representation in the form of LFDEA, the second - in the form of LDEA. This allows us to study the stability issues of the system under consideration using the well-known results on the stability of LFDEA and LDEA. Using the results of the article [Gusarenko S. A. On the stability of a system of two linear differential equations with delayed argument // Boundary value problems. Interuniversity collection of scientific papers. Perm: PPI, 1989. P. 3-9], two examples are shown when a joint system of four equations will be stable with respect to the right side. In the first example, we use the LFDEA for which sufficient conditions for the sign-definiteness of the elements of the 2 × 2 Cauchy matrix function are known (in terms of the LFDEA coefficients). In the second example, LFDEA is given such that LFDEA is a system of linear ordinary differential equations (LODE) of the second order. In both cases, estimates of the components of the Cauchy matrix function are known. An exponential estimate with a negative exponent is given for the components of the Cauchy matrix function of LDEA.
Russian Universities Reports. Mathematics. 2020;25(131):299-306
299-306
Nondifferential Kuhn-Tucker theorems in constrained extremum problems via subdifferentials of nonsmooth analysis
Abstract
The paper is devoted to obtaining Kuhn-Tucker theorems in nondifferential form in constrained extremum problems in a Hilbert space. The constraints of the problems are specified by operators whose images are also embedded in a Hilbert space. These constraints contain parameters that are additively included in them. The basis for obtaining nondifferential Kuhn-Tucker theorems is the so-called perturbation method. The article consists of two main sections. The first of them is devoted to obtaining the nondifferential Lagrange principle in the case when the constrained extremum problem is convex. In this case, the Kuhn-Tucker theorem is its “regular part”. Various statements are also presented here that relate the Lagrange multipliers to the subdifferentiability properties of the convex value function of the problem. The main purpose of the first section is to trace how the classical construction of the Lagrange function in its regular and nonregular forms is “generated” by subdifferentials and asymptotic subdifferentials of the value function. This circumstance and the results of the first section make it possible to transfer the natural bridge from the convex parametric constrained extremum problems to similar nonlinear parametric problems of the second section in which the value function, generally speaking, is not convex. The central role here is played not by subdifferentials in the sense of convex analysis, but by subdifferentials of nonsmooth (nonlinear) analysis. As a result, in this case, the so-called modified (not classical) Lagrange function acts as the main construction. Its construction depends entirely on how subdifferentiability is understood in the sense of nonsmooth (nonlinear) analysis.
Russian Universities Reports. Mathematics. 2020;25(131):307-330
307-330
Scaling invariance of the strict KP hierarchy
Abstract
In this paper we show first of all that for solutions of the strict KP hierarchy it is sufficient to work in a standard setting. Further we discuss a minimal realization of the hierarchy and present the scaling invariance of the Lax equations of the hierarchy.
Russian Universities Reports. Mathematics. 2020;25(131):331-340
331-340