Abstract
We consider the problem of recovery of unknown right-hand sides of the first-order linear systems of ordinary differential equations with periodic coefficients from indirect observations of their solutions on a finite system of points and intervals. Under the assumption that right-hand sides and deterministic errors in observations are subjected to some quadratic restrictions, we obtain a posteriori estimates of right-hand sides which are compatible with observations data.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Notes
1 Here and in what follows we assume that if a function is piecewise continuous and has at a certain point
a discontinuity of the first kind, we denote by
the jump of function
at this point.
2 Let us verify, for example, (Equation19(19)
(19) ). We have
that proves (Equation19
(19)
(19) ).
3 It is easy to see that the unique solvability of problems (Equation23(22)
(22) ) and (Equation24
(23)
(23) ) follows from the unique solvability of problems (Equation1
(1)
(1) ) and (Equation2
(2)
(2) ) since normalized fundamental matrix
of the system
adjoint to (Equation3
(3)
(3) ) satisfies the condition (see [Citation8,Citation9])
(1)
(1) It is known that
.
4 In fact, by virtue of generalized Cauchy–Schwarz inequality [Citation12, p.186],
and this inequality is transformed to an equality on the element