Deterministic Dicke state preparation with continuous measurement and control

We characterize the long-time projective behavior of the stochastic master equation describing a continuous, collective spin measurement of an atomic ensemble both analytically and numerically. By adding state based feedback, we show that it is possible to prepare highly entangled Dicke states deterministically.Comment: Additional information is available at http://minty.caltech.edu/Ensemble

Optimal strategies for driving a mobile agent in a guidance by repulsion model

We present a guidance by repulsion model based on a driver-evader interaction where the driver, assumed to be faster than the evader, follows the evader but cannot be arbitrarily close to it, and the evader tries to move away from the driver beyond a short distance. The key ingredient allowing the driver to guide the evader is that the driver is able to display a circumvention maneuver around the evader, in such a way that the trajectory of the evader is modified in the direction of the repulsion that the driver exerts on the evader. The evader can thus be driven towards any given target or along a sufficiently smooth path by controlling a single discrete parameter acting on driver's behavior. The control parameter serves both to activate/deactivate the circumvention mode and to select the clockwise/counterclockwise direction of the circumvention maneuver. Assuming that the circumvention mode is more expensive than the pursuit mode, and that the activation of the circumvention mode has a high cost, we formulate an optimal control problem for the optimal strategy to drive the evader to a given target. By means of numerical shooting methods, we find the optimal open-loop control which reduces the number of activations of the circumvention mode to one and which minimizes the time spent in the active~mode. Our numerical simulations show that the system is highly sensitive to small variations of the control function, and that the cost function has a nonlinear regime which contributes to the complexity of the behavior of the system, so that a general open-loop control would not be of practical interest. We then propose a feedback control law that corrects from deviations while preventing from an excesive use of the circumvention mode, finding numerically that the feedback law significantly reduces the cost obtained with the open-loop control

Can local single-pass methods solve any stationary Hamilton-Jacobi-Bellman equation?

The use of local single-pass methods (like, e.g., the Fast Marching method) has become popular in the solution of some Hamilton-Jacobi equations. The prototype of these equations is the eikonal equation, for which the methods can be applied saving CPU time and possibly memory allocation. Then, some natural questions arise: can local single-pass methods solve any Hamilton-Jacobi equation? If not, where the limit should be set? This paper tries to answer these questions. In order to give a complete picture, we present an overview of some fast methods available in literature and we briefly analyze their main features. We also introduce some numerical tools and provide several numerical tests which are intended to exhibit the limitations of the methods. We show that the construction of a local single-pass method for general Hamilton-Jacobi equations is very hard, if not impossible. Nevertheless, some special classes of problems can be actually solved, making local single-pass methods very useful from the practical point of view.Comment: 19 page

Optimal pilot decisions and flight trajectories in air combat

The thesis concerns the analysis and synthesis of pilot decision-making and the design of optimal flight trajectories. In the synthesis framework, the methodology of influence diagrams is applied for modeling and simulating the maneuvering decision process of the pilot in one-on-one air combat. The influence diagram representations describing the maneuvering decision in a one sided optimization setting and in a game setting are constructed. The synthesis of team decision-making in a multiplayer air combat is tackled by formulating a decision theoretical information prioritization approach based on a value function and interval analysis. It gives the team optimal sequence of tactical data that is transmitted between cooperating air units for improving the situation awareness of the friendly pilots in the best possible way. In the optimal trajectory planning framework, an approach towards the interactive automated solution of deterministic aircraft trajectory optimization problems is presented. It offers design principles for a trajectory optimization software that can be operated automatically by a nonexpert user. In addition, the representation of preferences and uncertainties in trajectory optimization is considered by developing a multistage influence diagram that describes a series of the maneuvering decisions in a one-on-one air combat setting. This influence diagram representation as well as the synthesis elaborations provide seminal ways to treat uncertainties in air combat modeling. The work on influence diagrams can also be seen as the extension of the methodology to dynamically evolving decision situations involving possibly multiple actors with conflicting objectives. From the practical point of view, all the synthesis models can be utilized in decision-making systems of air combat simulators. The information prioritization approach can also be implemented in an onboard data link system.reviewe

Convexity Applications in Single and Multi-Agent Control

The focus of this dissertation is in the application of convexity for control problems; specifically, single-agent problems with linear or nonlinear dynamics and multi-agent problems with linear dynamics. A mixture of convex and non-convex constraints for optimal control problems is also considered. The main contributions of this dissertation include: 1) a convexification of single-agent problems with linear dynamics and annular control constraint, 2) a technique for controlling bounded nonlinear single-agent systems, and 3) a technique for solving multi-agent pursuit-evasion games with linear dynamics and convex control and state constraints. The first result shows that for annularly constrained linear systems, controllability is a sufficient condition for a free or fixed time problem to be solvable as a sequence of convex optimization problems. The second result shows that if a nonlinear system is bounded and “ordered”, it is possible to use a convex combination of bounding linear systems to design a control for the nonlinear system. The third result takes advantage of a convex reachable set computation for each agent in solving games using a geometrical approach. Altogether, the theoretical and computational results demonstrate the significance of convex analysis in solving non-convex control problems

Cyclic Pursuit: Symmetry, Reduction and Nonlinear Dynamics

In this dissertation, we explore the use of pursuit interactions as a building block for collective behavior, primarily in the context of constant bearing (CB) cyclic pursuit. Pursuit phenomena are observed throughout the natural environment and also play an important role in technological contexts, such as missile-aircraft encounters and interactions between unmanned vehicles. While pursuit is typically regarded as adversarial, we demonstrate that pursuit interactions within a cyclic pursuit framework give rise to seemingly coordinated group maneuvers. We model a system of agents (e.g. birds, vehicles) as particles tracing out curves in the plane, and illustrate reduction to the shape space of relative positions and velocities. Introducing the CB pursuit strategy and associated pursuit law, we consider the case for which agent i pursues agent i+1 (modulo n) with the CB pursuit law. After deriving closed-loop cyclic pursuit dynamics, we demonstrate asymptotic convergence to an invariant submanifold (corresponding to each agent attaining the CB pursuit strategy), and proceed by analysis of the reduced dynamics restricted to the submanifold. For the general setting, we derive existence conditions for relative equilibria (circling and rectilinear) as well as for system trajectories which preserve the shape of the collective (up to similarity), which we refer to as pure shape equilibria. For two illustrative low-dimensional cases, we provide a more comprehensive analysis, deriving explicit trajectory solutions for the two-particle "mutual pursuit" case, and detailing the stability properties of three-particle relative equilibria and pure shape equilibria. For the three-particle case, we show that a particular choice of CB pursuit parameters gives rise to remarkable almost-periodic trajectories in the physical space. We also extend our study to consider CB pursuit in three dimensions, deriving a feedback law for executing the CB pursuit strategy, and providing a detailed analysis of the two-particle mutual pursuit case. We complete the work by considering evasive strategies to counter the motion camouflage (MC) pursuit law. After demonstrating that a stochastically steering evader is unable to thwart the MC pursuit strategy, we propose a (deterministic) feedback law for the evader and demonstrate the existence of circling equilibria for the closed-loop pursuer-evader dynamics