This paper investigates an optimal control problem governed by variable-order Caputo fractional dynamics. The main difficulty arises from the fact that, for a type-Ⅰ variable-order Caputo derivative, the corresponding adjoint operator is not obtained by a direct replacement of the constant fractional order in the standard right-sided Riemann–Liouville derivative. Instead, it must be derived from the dual kernel induced by the variable-order Caputo operator. Based on this observation, we formulate a right-sided variable-order adjoint operator and establish the associated integration-by-parts formula. Under suitable regularity assumptions, first-order necessary optimality conditions are derived for a class of quadratic optimal control problems with nonlinear variable-order fractional dynamics. The resulting optimality system consists of a forward variable-order Caputo state equation, a backward adjoint equation involving the dual-kernel operator, a stationarity condition, and a natural transversality condition. A predictor–corrector forward–backward sweep method is then developed for the numerical approximation of the coupled system. The adjoint equation is treated by a backward Volterra-type approximation motivated by the fractional memory structure. Numerical examples, including linear-quadratic, nonlinear scalar, and coupled nonlinear systems, illustrate the feasibility and mesh-refinement behavior of the proposed approach.
Citation: Huaqing Ma, Wenrui Wang, Changyuan Chen. Variable-order Caputo fractional optimal control problems: optimality conditions and a forward–backward numerical method[J]. Electronic Research Archive, 2026, 34(11): 7833-7863. doi: 10.3934/era.2026336
This paper investigates an optimal control problem governed by variable-order Caputo fractional dynamics. The main difficulty arises from the fact that, for a type-Ⅰ variable-order Caputo derivative, the corresponding adjoint operator is not obtained by a direct replacement of the constant fractional order in the standard right-sided Riemann–Liouville derivative. Instead, it must be derived from the dual kernel induced by the variable-order Caputo operator. Based on this observation, we formulate a right-sided variable-order adjoint operator and establish the associated integration-by-parts formula. Under suitable regularity assumptions, first-order necessary optimality conditions are derived for a class of quadratic optimal control problems with nonlinear variable-order fractional dynamics. The resulting optimality system consists of a forward variable-order Caputo state equation, a backward adjoint equation involving the dual-kernel operator, a stationarity condition, and a natural transversality condition. A predictor–corrector forward–backward sweep method is then developed for the numerical approximation of the coupled system. The adjoint equation is treated by a backward Volterra-type approximation motivated by the fractional memory structure. Numerical examples, including linear-quadratic, nonlinear scalar, and coupled nonlinear systems, illustrate the feasibility and mesh-refinement behavior of the proposed approach.
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Huaqing Ma, Wenrui Wang, Changyuan Chen. Variable-order Caputo fractional optimal control problems: optimality conditions and a forward–backward numerical method[J]. Electronic Research Archive, 2026, 34(11): 7833-7863. doi: 10.3934/era.2026336
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