Computation of Reachable Sets Based on Hamilton-Jacobi-Bellman Equation with Running Cost Function

A novel method for computing reachable sets is proposed in this paper. In the proposed method, a Hamilton-Jacobi-Bellman equation with running cost function is numerically solved and the reachable sets of different time horizons are characterized by a family of non-zero level sets of the solution of the Hamilton-Jacobi-Bellman equation. In addition to the classical reachable set, by setting different running cost functions and terminal conditions of the Hamilton-Jacobi-Bellman equation, the proposed method allows to compute more generalized reachable sets, which are referred to as cost-limited reachable sets. In order to overcome the difficulty of solving the Hamilton-Jacobi-Bellman equation caused by the discontinuity of the solution, a method based on recursion and grid interpolation is employed. At the end of this paper, some examples are taken to illustrate the validity and generality of the proposed method.

Computation of Reachable Sets Based on Hamilton-Jacobi-Bellman Equation with Running Cost Function

A novel method for computing reachable sets is proposed in this paper. In the proposed method, a Hamilton-Jacobi-Bellman equation with running cost function is numerically solved and the reachable sets of different time horizons are characterized by a family of non-zero level sets of the solution of the Hamilton-Jacobi-Bellman equation. In addition to the classical reachable set, by setting different running cost functions and terminal conditions of the Hamilton-Jacobi-Bellman equation, the proposed method allows to compute more generalized reachable sets, which are referred to as cost-limited reachable sets. In order to overcome the difficulty of solving the Hamilton-Jacobi-Bellman equation caused by the discontinuity of the solution, a method based on recursion and grid interpolation is employed. At the end of this paper, some examples are taken to illustrate the validity and generality of the proposed method.

Interception of automated adversarial drone swarms in partially observed environments

As the prevalence of drones increases, understanding and preparing for possible adversarial uses of drones and drone swarms is of paramount importance. Correspondingly, developing defensive mechanisms in which swarms can be used to protect against adversarial Unmanned Aerial Vehicles (UAVs) is a problem that requires further attention. Prior work on intercepting UAVs relies mostly on utilizing additional sensors or uses the Hamilton-Jacobi-Bellman equation, for which strong conditions need to be met to guarantee the existence of a saddle-point solution. To that end, this work proposes a novel interception method that utilizes the swarm’s onboard PID controllers for setting the drones’ states during interception. The drone’s states are constrained only by their physical limitations, and only partial feedback of the adversarial drone’s positions is assumed. The new framework is evaluated in a virtual environment under different environmental and model settings, using random simulations of more than 165,000 swarm flights. For certain environmental settings, our results indicate that the interception performance of larger swarms under partial observation is comparable to that of a one-drone swarm under full observation of the adversarial drone.

Comparison Principle for Hamilton-Jacobi-Bellman Equations via a Bootstrapping Procedure

We study the well-posedness of Hamilton–Jacobi–Bellman equations on subsets of $${\mathbb {R}}^d$$ R d in a context without boundary conditions. The Hamiltonian is given as the supremum over two parts: an internal Hamiltonian depending on an external control variable and a cost functional penalizing the control. The key feature in this paper is that the control function can be unbounded and discontinuous. This way we can treat functionals that appear e.g. in the Donsker–Varadhan theory of large deviations for occupation-time measures. To allow for this flexibility, we assume that the internal Hamiltonian and cost functional have controlled growth, and that they satisfy an equi-continuity estimate uniformly over compact sets in the space of controls. In addition to establishing the comparison principle for the Hamilton–Jacobi–Bellman equation, we also prove existence, the viscosity solution being the value function with exponentially discounted running costs. As an application, we verify the conditions on the internal Hamiltonian and cost functional in two examples.

Existence and multiplicity for Hamilton-Jacobi-Bellman equation

<p style='text-indent:20px;'>This paper is concerned with the existence and multiplicity of constant sign solutions for the following fully nonlinear equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\{ \begin{array}{l} -\mathcal{M}_\mathcal{C}^{\pm}(D^2u) = \mu f(u) \ \ \ \ \text{in} \ \ \Omega,\\ u = 0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{on}\ \partial\Omega, \end{array} \right. \end{equation*} $\end{document} </tex-math> </disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M3">\begin{document}$ \Omega\subset\mathbb{R}^N $\end{document}</tex-math></inline-formula> is a bounded regular domain with <inline-formula><tex-math id="M4">\begin{document}$ N\geq3 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ \mathcal{M}_\mathcal{C}^{\pm} $\end{document}</tex-math></inline-formula> are general Hamilton-Jacobi-Bellman operators, <inline-formula><tex-math id="M6">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> is a real parameter. By using bifurcation theory, we determine the range of parameter <inline-formula><tex-math id="M7">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> of the above problem which has one or multiple constant sign solutions according to the behaviors of <inline-formula><tex-math id="M8">\begin{document}$ f $\end{document}</tex-math></inline-formula> at <inline-formula><tex-math id="M9">\begin{document}$ 0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M10">\begin{document}$ \infty $\end{document}</tex-math></inline-formula>, and whether <inline-formula><tex-math id="M11">\begin{document}$ f $\end{document}</tex-math></inline-formula> satisfies the signum condition <inline-formula><tex-math id="M12">\begin{document}$ f(s)s&gt;0 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M13">\begin{document}$ s\neq0 $\end{document}</tex-math></inline-formula>.</p>

Dynamic Multiagent Incentive Contracts: Existence, Uniqueness, and Implementation

Multiagent incentive contracts are advanced techniques for solving decentralized decision-making problems with asymmetric information. The principal designs contracts aiming to incentivize non-cooperating agents to act in his or her interest. Due to the asymmetric information, the principal must balance the efficiency loss and the security for keeping the agents. We prove both the existence conditions for optimality and the uniqueness conditions for computational tractability. The coupled principal-agent problems are converted to solving a Hamilton–Jacobi–Bellman equation with equilibrium constraints. Extending the incentive contract to a multiagent setting with history-dependent terminal conditions opens the door to new applications in corporate finance, institutional design, and operations research.