The following result can be proved using standard arguments as in [26, 28].
Lemmas 1 and 2 can be proved using standard arguments as the Lemmas 1 and 2 in [13], respectively.
The well-posedness of the discrete and continuous problems can be proved using standard argument of Lax-Milgram in appropriate functional spaces; refer, for instance, to [34] for a special case of fractional Maxwell model.
Corollary 6 All the state components x i ( t ) of system (1) with piecewise constant feedback function (9) in case 2′, i ∈ N ˜ 3, will flow to the interval [ 3, + ∞ ) when t → + ∞. Corollaries 4-6 can be proved using standard arguments as Theorems 4-6.
Corollary 3 All the state components x i ( t ) of system (1) with piecewise constant feedback function (9) in case 1′, i ∈ N 3, will flow to the interval [ 3, + ∞ ) when t → + ∞. Corollaries 1-3 can be proved using standard arguments as Theorems 1-3. Remark 2 Theorems 1-3 and Corollaries 1-3 are obtained based on Mexican-hat-type feedback function (8) and piecewise constant feedback function (9).
Although the overall system is infinite dimensional, convergence of the observer is proven using a standard Lyapunov approach along with classic mathematical tools such as Cauchy series, Parseval equality, and compact embeddings of Hilbert spaces.
Surface-enhanced Raman scattering property is proved using 4-mercaptobenzoic acid (4-MBA) as standard analyte and the detection concentration level is down to 10−10 M.
Since local existence results can be proved by using standard arguments, say, Galerkin method, we only deal with the a priori estimates.
The next lemma is proved by using standard Schwartz's arguments in [38].
