Doctor of Philosophy (PhD)
Impulsive systems arise when dynamics produce discontinuous trajectories. Discontinuties occur when movements of states happen over a small interval that resembles a point-mass measure. We adopt the formalism in which the controlled dynamic inclusion is the sum of a slow and a fast time velocities belonging to two distinct vector fields. Fast time velocities are controlled by a vector valued Borel measure. The trajectory of impulsive systems is a function of bounded variation. To give a definition of solutions, a notion of graph completion of the control measure is needed. In the nonimpulsive case, a solution can be defined as a limit of a sequence of approximate arcs which converge to an absolutely continuous arc. Even in simple cases it shows that this is not a good way to define solutions of the impulsive systems. The key point is that the approximate controls converge to two different graph completions. Introduction contains examples in which we discuss the need for the impulsive systems, their relation to the hybrid systems. A paradox related to the convergence of approximate arcs is illustrated. Chapter 1 contains preliminary results in nonimpulsive systems and mathematical analysis in general. Chapter 2 precisely defines impulsive systems, discusses two different solution concepts and proves properties of graph completions. Chapter 3 is entirely dedicated to adaptation of the Euler approximating schemes to the impulsive system. Two different schemes are offered. For one of them a measure which drives the system needs to be specified. We used it to show that the approximate trajectories graph-converge to a solution. The other sampling technique constructs a measure along with the solution. We use it in Chapter 4. Chapter 4 deals with issues when a trajectory remains within a closed set. This property is called invariance. Notions of weak and strong invariance for the impulsive systems are introduced and proximal characterizations are proved. In the case of weak invariance, two proofs are offered: one based on a sampling technique from Chapter 3 and other based on selections theory. The final chapter of this thesis discusses directions in future research.
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Zabic, Stanislav, "Impulsive systems" (2005). LSU Doctoral Dissertations. 2157.
Peter R. Wolenski