Linear quadratic optimal control and zero-sum game for uncertain time-delay systems based on pessimistic value

The pessimistic value is an important criterion distinct from the expected value, suitable for addressing uncertain optimization. This paper studies the linear quadratic optimal control problem and zero-sum game for uncertain time-delay systems based on the pessimistic value. First, we introduce an uncertain time-delay system and formulate the associated optimal control problem. By using algebraic transformation methods, we convert the time-delay problem into an equivalent uncertain optimal control problem without time delay. We then propose a recursive equation that transforms the uncertain optimal control problem into a backward difference equation and provide the analytical solution for the uncertain linear quadratic optimal control problem. Within this framework, we extend the main results to a linear quadratic zero-sum game and propose the corresponding solution algorithm. Finally, we demonstrate the algorithm's solving process through numerical simulations, further verifying the applicability of our findings.