This paper presents numerical methods for discretizing continuous-time linear-quadratic optimal control problems (LQ-OCPs). The key contributions are:
Formulation of differential equation systems for LQ discretization: The authors derive systems of differential equations that can be used to compute the discrete-time equivalents of continuous-time LQ-OCPs.
Numerical methods for solving the differential equation systems: Three numerical methods are introduced - ordinary differential equation (ODE) methods, matrix exponential methods, and a novel step-doubling method. These methods are used to solve the differential equation systems and obtain the discrete-time LQ-OCP.
Distribution of stochastic costs: For stochastic LQ-OCPs, the authors show that the stochastic cost function follows a generalized chi-squared distribution, and provide expressions for its expectation and variance.
The paper first introduces deterministic and stochastic LQ-OCPs, and formulates the corresponding differential equation systems for discretization. It then presents the three numerical methods in detail, highlighting their properties and computational efficiency. Finally, the methods are tested and compared through numerical experiments, demonstrating that the discrete-time LQ-OCP derived using the proposed techniques is equivalent to the original continuous-time problem.
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