A tutorial on solutions , nonsmooth analysis , and stability

A tutorial on solutions, nonsmooth analysis, and stability Discontinuous dynamical systems arise in a large number of applications. In optimal control problems, open-loop bang-bang controllers switch discontinuously between extreme values of the inputs to generate minimum-time trajectories from the initial to the final states [1]. Thermostats implement on-off controllers to regulate room temperature [2]. When the temperature is above the desired value, the controller switches the cooling system on. Once the temperature reaches a preset value, the controller switches the cooling system off. The controller is therefore a discontinuous function of the room temperature. In nonsmooth mechanics, the motion of rigid bodies is subject to velocity jumps and force discontinuities as a result of friction and impact [3, 4]. In the robotic manipulation of objects by means of mechanical contact [5], discontinuities occur naturally from interaction with the environment. Discontinuities are also intentionally designed to achieve regulation and stabilization. Sliding mode control [6, 7] uses discontinuous feedback controllers for stabilization. The design procedure for sliding mode control begins by identifying a surface in the state space with the property that the dynamics of the system restricted to this surface are easily stabilizable. Feedback controllers are then synthesized on each side of the surface to steer the solutions of the system toward the surface. The resulting closed-loop system is discontinuous on the surface. In robotics [8], it is of interest to induce emergent behavior in a swarm of robots by prescribing interaction rules among individual agents. Simple laws such as " move away from the nearest other robot or environmental boundary " give rise to discontinuous dynamical systems. For example, consider a robot placed in a corner of a room. On opposite sides of the bisector line between the two walls forming the corner, the " move away " law translates into different velocity vectors for the robot. The dynamical system is thus discontinuous along the bisector line. In optimization [9, 10], continuous-time algorithms that perform generalized gradient descent, which is a discontinuous function of the state, are used when the objective function is not smooth. In adaptive control [11], switching algorithms are employed to select the most appropriate controller from a given finite family in order to enhance robustness, ensure boundedness of the estimates, and prevent the system from stepping into undesired regions of the state space. Many control systems cannot be stabilized by continuous state-dependent feedback. As a consequence …