A novel increment approach for optimal control problem of fractional‐order (1, 2] nonlinear systems

Abstract

This paper deals with fractional optimal control governed by semilinear equations using the increment approach. We have considered controlled object as with the initial conditions: where α ∈ (1, 2], s(τ) is state variable in , B(τ) ∈ Lp, ϱ(0, T), and . Let the m‐dimensional control vector function be C(τ) and defined as . Assume the function η(τ, C(τ)) satisfies Caratheodory condition and defined on . We have obtained our results with the help of the adjoint equation and Pontryagin's maximum condition. For better understanding, we have included one example.

New discussion concerning to optimal control for semilinear population dynamics system in Hilbert spaces

“…In recent years, the existence and uniqueness of mild solution, optimal control, timeoptimal control approximate control, and exact control for fractional-order, integer-order, integro -differential system, neutral system, etc. have been studied by many researcher's articles [1][2][3][7][8][9][11][12][13][14][15][16][17][18][19][20][21][22][23][24][26][27][28][29][30][32][33][34][35]. In [6], the authors obtained the existence and optimal control results using Krasnoselskii's fixed point theorem and minimizing sequence concept for the second-order stochastic differential equations having mixed fractional Brownian motion.…”

A note on the existence and optimal control for mixed Volterra–Fredholm‐type integrodifferential dispersion system of third order

“…Many research works [19–27,30–44] have explored the existence and uniqueness, optimal control, and time‐optimal control for fractional‐order, integer‐order, integrodifferential system, neutral system, and so on in recent years. The authors of an earlier study [36] used Krasnoselskii's fixed‐point theorem and the minimizing sequence notion to achieve existence and optimal control conclusions for a second‐order stochastic differential equations with mixed‐fractional Brownian motion.…”

“…Patel et al. [44] derived results for optimal control using the increment approach. They established the result by constructing the adjoint equation and Pontryagin's maximum principle.…”

“…As a result, an integrodifferential system is created by adding an integration term to the differential system to include the memory possessions in these frameworks. Integrodifferential systems have been widely employed in viscoelastic mechanics, fluid dynamics, thermoelastic contact, control theory, heat conduction, industrial mathematics, financial mathematics, biological models, and other domains, one can refer to earlier studies Many research works [19][20][21][22][23][24][25][26][27][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44] have explored the existence and uniqueness, optimal control, and timeoptimal control for fractional-order, integer-order, integrodifferential system, neutral system, and so on in recent years. The authors of an earlier study [36] used Krasnoselskii's fixed-point theorem and the minimizing sequence notion to achieve existence and optimal control conclusions for a second-order stochastic differential equations with mixed-fractional Brownian motion.…”

“…With the help of Henry-Gronwall inequality, Mohan et al [27] established the continuous dependence of integrodifferential system. Patel et al [44] derived results for optimal control using the increment approach. They established the result by constructing the adjoint equation and Pontryagin's maximum principle.…”