An Approximation Technique for Solving Linear-Quadratic Optimal Control Problems Using Chebyshev Polynomials*

We introduce an innovative numerical method for addressing Linear-Quadratic optimal control problems that come with control constraints. Initially, we reframe the LQ optimal control problem into an unconstrained problem by utilizing the constraint transcription technique. Next, we employ Chebyshev series to approximate both the state and control functions. To bolster the approximation precision, Chebyshev series are further leveraged to approximate the coefficient functions inherent to the linear dynamic system. According to the unique attributes of Chebyshev polynomials, the approximate problem can be seamlessly converted into a corresponding quadratic programming problem. As a result, any feasible solution to the approximate problem guarantees compliance with its dynamic system throughout the complete time horizon. To substantiate the effectiveness of our proposed methodology, we’ve tested it on a real-world scenario. The computational results underscore that our method outperforms the Chebyshev pseudo-spectral method in terms of approximation accuracy.