OICC Press Mathematical Sciences 2251-7456 18 1 (March 2024) 2022 08 20 Numerical solutions of two-dimensional PDE-constrained optimal control problems via bilinear pseudo-spectral method 10.1007/s40096-022-00488-8 EN Fereshteh Samadi Department of Mathematics, Payame Noor University (PNU), Tehran, IR Aghileh Heydari Department of Mathematics, Payame Noor University (PNU), Tehran, IR Sohrab Effati Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, IR Journal Article 2022 08 20 Abstract A bilinear pseudo-spectral method (BPSM) is proposed for solving two-dimensional parabolic optimal control problems (OCPs). Firstly, the OCP is converted to a partial differential equation system including the state equation of the main problem, the adjoint equation, and the gradient equation which should be solved. Secondly, the coupled system is discretized in the space domain by a BPSM based on Chebyshev polynomials and a coupled matrix differential equation is obtained. Lastly, by using the eigenvalue decomposition method, the equation is transformed to a set of independent coupled ordinary differential equation system in the new coordinate with opposite orientation and after another eigenvalue decomposition technique, these transformed equations are discretized in the time space by some higher-order formulas. The error estimate is investigated theoretically. Numerical results confirm the desirable accuracy and efficiency of the method.