Modification of the Dynamic Regularization Method for Linear Parabolic Equations
- PDF / 402,810 Bytes
- 11 Pages / 612 x 792 pts (letter) Page_size
- 31 Downloads / 211 Views
L THEORY
Modification of the Dynamic Regularization Method for Linear Parabolic Equations V. I. Maksimov1∗ 1
Institute of Mathematics and Mechanics, Ural Branch, Russian Academy of Sciences, Yekaterinburg, 620108 Russia e-mail: ∗ [email protected] Received April 23, 2020; revised April 23, 2020; accepted May 14, 2020
Abstract—We consider the problem of reconstructing distributed inputs (disturbances) in linear parabolic equations. An algorithm for solving this problem is given. An upper bound for the convergence rate is established for the case in which the input is a function of bounded variation. The algorithm combines the optimal preset and positional control methods and permits reconstruction based on inaccurate measurements of solutions of the equations at discrete time instants. DOI: 10.1134/S0012266120110063
INTRODUCTION In the present paper, we study the problem of reconstructing distributed inputs (disturbances) in linear parabolic equations from the results of inaccurate measurements of the values of solutions of these equations at some time instants. It is assumed that a parabolic equation with an unknown right-hand side, treated as a disturbance, is considered on a given finite time interval. It is required to construct an algorithm for reconstructing one of the possible disturbances that generate the measured solution. Since the solution is measured at discrete (rather frequent) time instants with an error, the exact reconstruction of the corresponding disturbance is, generally speaking, impossible due to the measurement error. Therefore, we propose to construct some approximation to it. We require that the approximation be arbitrarily close to the reconstructed input provided that the measurement errors and the distance between the solution measurement times are sufficiently small. This problem lies in the class of inverse problems of control system dynamics and, in a more general context, belongs in the framework of the theory of ill-posed problems. The procedure developed in the present paper is essentially that the reconstruction algorithm is represented in the form of a control algorithm for some auxiliary dynamical system called a model. The control in the model is adapted to the results of current observations in such a way that its implementation in time “approximates” the unknown input. 1. STATEMENT OF THE PROBLEM Let V and H be real Hilbert spaces. The space V is embedded in the space H densely and continuously, V ⊂ H = H ∗ ⊂ V ∗ . By | · |V and | · |H we denote the norms on V and H, respectively, and ( · , · ) and h · , · i are the inner product on H and the duality between V and V ∗ . Consider the parabolic equation x(t) ˙ + Ax(t) = B f˜(t),
t ∈ T = [0, ϑ],
(1)
with the initial condition x(0) = x0 ∈ H. Here ϑ ∈ (0, +∞) is fixed, A : V → V ∗ is a given linear continuous and symmetric operator satisfying the coercivity condition hAy, yi + ω|y|2H ≥ c|y|2V 1452
for each y ∈ V
(2)
MODIFICATION OF THE DYNAMIC REGULARIZATION METHOD
1453
for some constants c > 0 and ω ∈ R, f˜(·) is th
Data Loading...
