Method of Extremal Shift in Problems of Reconstruction of an Input for Parabolic Variational Inequalities

Abstract

The problem of dynamical reconstruction of a control in a parabolic variational inequality [1][2][3][4] through inaccurate measurement of phase state is considered. This problem belongs to the class of dynamical inverse problems which consist in reconstruction of unknown input (a control) of dynamical systems from (inaccurate) measurement of outputs.

The method of extremal shift in control problems for evolution variational inequalities under disturbances

“…Therefore, by the rule of forming the control u h (•) (see (21), ( 22)), we conclude from ( 27) that, for a.a.…”

“…Moreover, the proofs of these theorems are simplified. Note that the algorithms described above (as well as all the algorithms from [20,19,21,16,18,23,24]) were designed to solve control problems on a bounded time interval [t 0 , ϑ]. If the length ϑ−t 0 of this interval increases, computational and measurement errors are accumulated.…”

“…These algorithms adaptively take into account inaccurate measurements of phase trajectories and they are regularizing in the following sense: the more accurate is an incoming information signal, the better is output. Other problems of feedback control of parabolic variational inequalities were considered from the position of the approach above in [20,21,16]. Namely, a differential game of guidance of a solution of a parabolic variational inequality to a given target set at a fixed time was considered in [20].…”

“…Namely, a differential game of guidance of a solution of a parabolic variational inequality to a given target set at a fixed time was considered in [20]. A problem of designing an algorithm for dynamical reconstruction of a right-hand part was studied in [21], a problem of tracking a prescribed solution under instantaneous constraints (in the form of sets) imposed on both disturbances and control, in [16].…”

“…In the case when a control u as well as a disturbance v in inequality (1) are subject to instantaneous constraints (u ∈ P , v ∈ P , where P ⊂ U is a given bounded and closed set), the problem in question is solved by means of the method of extremal shift [14]. Namely, if ϑ < +∞ and the control u = u h (•) in the right-hand part of inequality ( 1) is calculated by the formula 4) then, as it follows from [21,16], for any ε > 0 one can find numbers h 1 > 0 and δ 1 > 0 such that the inequality…”