In this chapter, some recent stability and robust stability results on linear time-delay systems are outlined. The goal of this guided tour is to give (without entering the details) a wide overview of the state of the art of the techniques encountered in time-delay system stability problems. In particular, two specific stability problems with respect to delay (delay.independent and respectively delay-dependent) axe analyzed and some references where the reader can find more details and proofs are pointed out. The references list is not intended to give a complete literature survey, but rather to be a source for a more complete bibliography. In order to simplify the presentation several examples have been considered.

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This paper concerns an optimal control problem defined on a class of switched-mode hybrid dynamical systems. The system's mode is changed (switched) whenever the state variable crosses a certain surface in the state space, henceforth called a switching surface. These switching surfaces are parameterized by finite-dimensional vectors called the switching parameters. The optimal control problem is to minimize a cost functional, defined on the state trajectory, as a function of the switching parameters. The paper derives the gradient of the cost functional in a costate-based formula that reflects the special structure of hybrid systems. It then uses the formula in a gradient-descent algorithm for solving an obstacle-avoidance problem in robotics.

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SUMMARYThis paper considers the problem of guaranteed cost control for uncertain neutral delay systems with a quadratic cost function. The system under consideration is subject to norm-bounded time-varying parametric uncertainty appearing in all the matrices of the state-space model. The problem we address is the design of a state feedback controller such that the closed-loop system is not only stable but also guarantees an adequate level of performance for all admissible uncertainties. A sufficient condition for the existence of guaranteed cost controllers is given in terms of a linear matrix inequality (LMI). When this condition is feasible, the desired state feedback controller gain matrices can be obtained via convex optimization. An illustrative example is provided to demonstrate the effectiveness of the proposed approach.