Novel closed-loop controllers for fractional linear quadratic time-varying systems (Q6559155)

The author considers the fractional linear-quadratic regulator problem on the given interval \([0,t_f]\) with the equation of motion involving the left fractional Caputo derivative of order \(\alpha\in(0,1]\), denoted by \N\[\N{}_0^C\!D_t^\alpha f(t):=\frac{1}{\Gamma(1-\alpha)}\int_0^t (t-\rho)^{-\alpha}\dot{f}(\rho)\,\mathrm{d}\rho, \N\]\Nwhere \(\dot{f}(\rho)\) is the usual derivative of the function \(f(\rho)\) with respect to \(\rho\). The author makes statements about the existence and construction of the closed-loop optimal controllers for the fractional order \(\alpha\) satisfying \(0.9<\alpha\leq1\). Such statements are not based on mathematical rigor, however. \N\NThe method for the derivation of the first variation (page~371) is just heuristical in optimal control theory. In general, it does not correspond to a~satisfactory mathematical proof. The statement of Lemma~3.1 contains an~approximation sign \(\simeq\), which is not defined, the notion of ``constant terms'' has no meaning in the context of this lemma (moreover, the values \(\Delta_{0f}^0\) and \(\Delta_{0f}^1\) and \(\Delta_{0f}^2\) defined on page~373 depend also on the function \(g\)), and the assumption that \(\alpha\in(0.9,1]\) is not used in the proof. The proof of Lemma~3.1 itself contains mathematical errors, e.g., when dealing with the integration by parts over the interval \([0,t]\) involving the improper integral, where the substitution for \(\rho=t\) in the second equality sign on page 372 is invalid since the term \((t-\rho)^{-\alpha}\) is raised to a~negative power. The terms \(\Delta_{0f}^1-\Delta_{0f}^2\) appearing in the derivation of the first variation \(\delta J_a(\mathbf{u}^*(t))\) on page~374 middle disappear in the subsequent analysis of \(\delta J_a(\mathbf{u}^*(t))=0\).\N\NIn conclusion, what is stated as the main contribution of the paper, is not supported by rigorous and correct mathematical arguments.