A numerical method based on piecewise Bernstein polynomials for a class of optimal control problems constrained by third-kind FIDEs

This paper presents an efficient strategy for solving fractional optimal control problems governed by third-kind fractional integro-differential equations. The system features a singular coefficient vanishing at the initial time and a nonlinear Volterra integral term, representing a challenging class of fractional dynamical systems with memory effects. We develop a direct collocation method based on orthonormal piecewise Bernstein polynomials (BPs), combining approximation power with numerical stability. The methodology proceeds as follows: First, we construct orthonormal piecewise BPs over a partitioned interval and derive spectral error estimates. Second, we obtain the Riemann-Liouville fractional integral operational matrix for this basis. The state and control variables are expanded in these basis functions, reducing the infinite-dimensional problem to a finite-dimensional constrained optimisation. Fractional dynamics are enforced at shifted Chebyshev collocation points, and nonlinear constraints are incorporated via Lagrange multipliers. The effectiveness of the scheme is demonstrated through two benchmark problems with known exact solutions. Numerical results confirm spectral convergence, with errors reaching machine precision for higher polynomial degrees. The method handles singular coefficients and hereditary integrals consistently across different fractional orders, achieving high accuracy in both state and control approximations.