Optimal variable-order fractional PID controllers for dynamical systems

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“…The parameters A,B are constant matrices, f : Rn → Rn is a continuously differentiable function such that its gradient is Lipschitz continuous over the domain, and y0 ∈ Rn is a specified constant vector. In the problem formulation (5)- (7), the objective is to find the control function u * (t) and the corresponding state trajectory y * (t) satisfying Eqs. (6)- (7), which minimize the quadratic cost functional (5).…”

“…The fractional calculus (FC), as a branch of mathematical analysis, investigates the extension of deriva-tives and integrals to non-integer orders [1][2][3]. Nowadays, the new aspects of the FC are growing fast and its applications are found in different areas like chaos synchronization [4], diffusion equation [5], biology [6], control [7] and economic [8]. Additionally, the utility of the FC in the optimal control problems (OCPs) has attracted the attention of many researchers.…”

On the Ulam‐Hyers stabilities of the solutions of Ψ‐Hilfer fractional differential equation with abstract Volterra operator

“…() With new definitions of fractional derivatives and integrals,() there has arisen some applications in a few territories of learning. () There has gained prominence interest in the investigation of existence and Ulam‐Hyers stabilities of differential equations and fractional integrals, be they of the impulsive type, functional or of Volterra. () Recently, Sousa et al() investigated the existence, uniqueness, and stabilities of Ulam‐Hyers of fractional differential equations.…”

“…[20][21][22][23] With new definitions of fractional derivatives and integrals, 24,25 there has arisen some applications in a few territories of learning. [26][27][28][29][30][31][32] There has gained prominence interest in the investigation of existence and Ulam-Hyers stabilities of differential equations and fractional integrals, be they of the impulsive type, functional or of Volterra. 18,[33][34][35][36] Recently, Sousa et al 37-43 investigated the existence, uniqueness, and stabilities of Ulam-Hyers of fractional differential equations.In this sense, by means of the Ψ-Hilfer fractional derivative, we introduced a class of nonlinear fractional differential equation with the abstract Volterra operator, in order to investigate a new class of Ulam-Hyers stabilities, contributing with new results and consequently for the growth of the area.Consider the fractional differential equation with abstract Volterra operator in a Banach space given by…”

Fractional Order Version of the Hamilton–Jacobi–Bellman Equation

“…Applications of FC have been explored to various fields of science and engineering, including control engineering [10], chaotic systems [11,12], reservoir engineering [13], diffusive processes [14], and so on [15]. For the design of variableorder fractional proportional-integral-derivative controllers for linear dynamical systems, see [16]. For applications of fractional calculus on the nutrition of pregnant women and the health of newborns and nursing mothers, see [17]; fractional models of HIV-infection and their potential to extract new hidden features of biological complex systems are investigated in [18].…”