TY - JOUR

T1 - Solvability, completeness and computational analysis of a perturbed control problem with delays

AU - Byszewski, Ludwik

AU - Blackmore, Denis

AU - Balinsky, Alexander A.

AU - Prykarpatski, Anatolij K.

AU - Luśtyk, Miroslaw

N1 - Publisher Copyright:
© 2020, Horizon Research Publishing. All rights reserved.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020/3

Y1 - 2020/3

N2 - As a first step, we provide a precise mathematical framework for the class of control problems with delays (which we refer to as the control problem) under investigation in a Banach space setting, followed by careful definitions of the key properties to be analyzed such as solvability and complete controllability. Then, we recast the control problem in a reduced form that is especially amenable to the innovative analytical approach that we employ. We then study in depth the solvability and completeness of the (reduced) nonlinearly perturbed linear control problem with delay parameters. The main tool in our approach is the use of a Borsuk-Ulam type fixed point theorem to analyze the topological structure of a suitably reduced control problem solution, with a focus on estimating the dimension of the corresponding solution set, and proving its completeness. Next, we investigate its analytical solvability under some special, mildly restrictive, conditions imposed on the linear control and nonlinear functional perturbation. Then, we describe a novel computational projection-based discretization scheme of our own devising for obtaining accurate approximate solutions of the control problem along with useful error estimates. The scheme effectively reduces the infinite-dimensional problem to a sequence of solvable finite-dimensional matrix valued tasks. Finally, we include an application of the scheme to a special degenerate case of the problem wherein the Banach-Steinhaus theorem is brought to bear in the estimation process.

AB - As a first step, we provide a precise mathematical framework for the class of control problems with delays (which we refer to as the control problem) under investigation in a Banach space setting, followed by careful definitions of the key properties to be analyzed such as solvability and complete controllability. Then, we recast the control problem in a reduced form that is especially amenable to the innovative analytical approach that we employ. We then study in depth the solvability and completeness of the (reduced) nonlinearly perturbed linear control problem with delay parameters. The main tool in our approach is the use of a Borsuk-Ulam type fixed point theorem to analyze the topological structure of a suitably reduced control problem solution, with a focus on estimating the dimension of the corresponding solution set, and proving its completeness. Next, we investigate its analytical solvability under some special, mildly restrictive, conditions imposed on the linear control and nonlinear functional perturbation. Then, we describe a novel computational projection-based discretization scheme of our own devising for obtaining accurate approximate solutions of the control problem along with useful error estimates. The scheme effectively reduces the infinite-dimensional problem to a sequence of solvable finite-dimensional matrix valued tasks. Finally, we include an application of the scheme to a special degenerate case of the problem wherein the Banach-Steinhaus theorem is brought to bear in the estimation process.

KW - Computational scheme

KW - Convergence

KW - Delay

KW - Perturbed linear control problem

KW - Solvability

KW - Stability

UR - http://www.scopus.com/inward/record.url?scp=85083586901&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85083586901&partnerID=8YFLogxK

U2 - 10.13189/ms.2020.080215

DO - 10.13189/ms.2020.080215

M3 - Article

AN - SCOPUS:85083586901

VL - 8

SP - 187

EP - 200

JO - Mathematics and Statistics

JF - Mathematics and Statistics

SN - 2332-2071

IS - 2

ER -