Numerical solution of optimal control problems governed by integro‐differential equations

Abstract

In this paper, an efficient hybrid approximation scheme for solving optimal control problems governed by integro-differential equations is proposed. The current approach is based on a generalization of the hybrid of block-pulse functions and Legendre's polynomials. An upper bound for the generalized hybrid functions with respect to the maximum norm is acquired and its convergence is demonstrated. The optimal control problem under study is transcribed to a mathematical programming one. Two illustrative examples are considered to verify the capability and reliability of the proposed procedure.

State‐dependent indirect hp‐pseudospectral method for rigid spacecraft attitude optimal control problem

“…However, the standard optimal nonlinear control technique needs to solve the complex Hamilton–Jacobi–Bellman (HJB) equation, which is quite computationally intense [7]. The analytic optimal controllers seldom exist for nonlinear problems, whereas numerical approaches are commonly utilized in nonlinear optimal control [8].…”

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

“…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 [5–29]…”

“…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.…”

“…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.…”

Solution of optimal control problems governed by volterra integral and fractional integro-differential equations

“…Moreover, some numerical schemes are proposed for solving OCPs by integer and non-integer integro-differential equations such as the methods based on fractional-order Legendre functions (Rabiei et al (2018)), discrete Hahn polynomials (Mohammadi et al (2022)), Orthonormal piecewise Bernoulli functions (Heydari et al (2022)), hybrid of block-pulse functions and Legendre polynomials (Marzban (2020)), spectral method and grey wolf optimizer (Khanduzi et al (2020)), Chebyshev approximations (El-Kady and Moussa (2013)), etc.…”