Impulsive dynamical systems is a well-established area of dynamical systems theory, and it is used in this work to analyze several basic properties of reset control systems: existence and uniqueness of solutions, and continuous dependence on the initial condition (well-posedness). The work scope is about reset control systems with a linear and time-invariant base system, and a zero-crossing resetting law. A necessary and sufficient condition for existence and uniqueness of solutions, based on the well-posedness of reset instants, is developed. As a result, it is shown that reset control systems (with strictly proper plants) do no have Zeno solutions. It is also shown that full reset and partial reset (with a special structure) always produce well-posed reset instants. Moreover, a definition of continuous dependence on the initial condition is developed, and also a sufficient condition for reset control systems to satisfy that property. Finally, this property is used to analyze sensitivity of reset control systems to sensor noise. This work also includes a number of illustrative examples motivating the key concepts and main results. Index TermsReset control systems, impulsive dynamical systems, hybrid systems. I. INTRODUCTIONReset control systems trace back to the seminal work of Clegg [21], that introduced a nonlinear integrator that sets its output to zero whenever its input is zero. Almost two decades later, the works by Horowitz and coworkers ( [30], [31]) propose design methods to incorporate a Clegg integrator (CI), and also a first order reset element (FORE), into a control loop. In the late 90s, the term reset controller is finally coined in the works by Hollot, Chait and coworkers ( [14]), to describe a 'linear and time invariant system with mechanisms and laws to reset their states to zero', being the main motivation its use for overcoming fundamental limitations of linear and time invariant (LTI) control systems.Impulsive and hybrid systems are active areas of dynamical systems theory that have been developed in the last three decades ( [4] [49]). Since reset controller dynamics is a combination of time and event based dynamics, it is not surprising that in the last decade different impulsive/hybrid dynamical system formulations were used for modeling and analysis of reset control systems. The survey [44] emphasizes the diversity of hybrid systems formulations: hybrid automata, switched systems, piecewise models, complementary systems, hybrid inclusions, · · · . There are two main frameworks that has been successfully used for modeling reset control systems: the framework of impulsive dynamical systems (IDS) [28], used in [9] and references therein; and the framework of hybrid inclusions (HI) developed in [26], used in [1], [41], [42]. Finally, another formulation of reset systems as hybrid automata has been investigated in [43].From a control practice point of view, an important issue in the different impulsive/hybrid systems formulations, directly related with their solution concept, is wel...

Reset control systems are a special type of state-dependent impulsive dynamic systems, in which the time evolution depends both on continuous dynamics between resets and the discrete dynamics corresponding to the resetting times. This work is devoted to investigate well-posedness of reset control systems, taking as starting point the classical definition of Clegg and Horowitz. Well-posedness is related to the existence and uniqueness of solutions, and in particular to the resetting times to be well defined and distinct. A sufficient condition is developed for a reset system to have well-posed resetting times, which is also a sufficient condition for avoiding Zeno solutions and, thus, for a reset control system to be well-posed.

In this note we develop the theory of the quantum Pontryagin principle for continuous measurements and feedback. The analysis is carried out under the assumption of compatible events in the output channel. The plant is a quantum system, which generally is in a mixed state, coupled to a continuous measurement channel. The Pontryagin Maximum Principle is derived in both the Schrödinger picture and Heisenberg picture, in particular in statistical moment coordinates. To avoid solving stochastic equations we derive a LQG scheme which is more suitable for control purposes.

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