Pessimistic-value optimal control of uncertain fractional-order systems: a new orthonormal Bernoulli wavelet method

For uncertain fractional order control systems, expected value models mainly describe average performance and may lead to considerable decision risk in some risk sensitive settings. The pessimistic value criterion provides an alternative based on a prescribed confidence level and is suitable for risk averse decisions requiring conservative performance assessment. This paper formulates a fractional optimal control problem governed by an uncertain fractional differential equation under the pessimistic value criterion and transforms it into an equivalent deterministic problem through the α-path. To solve the resulting deterministic problem, a class of orthonormal Bernoulli wavelets is constructed, and the corresponding fractional integration operational matrix is derived. By combining wavelet approximation with the Galerkin method, the equivalent fractional optimal control problem is reduced to a finite dimensional system of algebraic equations, which is solved by a Newton type iterative algorithm. The error bound and convergence properties of the proposed method are also analysed. Two numerical examples, including comparisons with Fibonacci wavelet and finite difference methods, show state approximation errors of order between 10−8 and 10−10 in the high order cases, while a CO2 emission control application illustrates the practical applicability of the proposed method.