This paper describes and proves the consistency of a flexible numerical method for producing solutions to state and control constrained control problems with parameter dependencies. This method allows for the use of a variety of underlying discretisation schemes, which can be catered to differing numerical challenges of specific problems, such as rapid convergence or large parameter spaces. The paper first provides a broad formulation for optimal control problems with parameter dependencies which includes multiple types of state, control, and end time constraints to enable a wide scope of application. For this formulation, the consistency of these methods for state and control constrained problems is then proved. Finally, a numerical example of an optimal search problem with constraints is demonstrated.

This paper introduces a new concept of partial observability for nonlinear systems. This new approach enables a quantitative analysis on the observability of an individual state variable and unknown parameters of a nonlinear dynamics even when the system is not observable in the traditional sense. The paper develops theoretical properties of partial observability and its computational algorithms. These results are applied and validated on a nonstandard estimation problem of detecting the internal cooperating strategy of a particular adversarial swarm model. Partial observability analysis on the parameters that define the cooperating strategy reveals interesting findings. For example, some parameters are observable even when the swarm is at a steady state that is not observable in traditional sense. It is also shown that observability of the internal cooperating strategy depends on both the swarm trajectory and the time window of the measurement. Motivated by these findings, a variational method of estimation based on the dynamic optimization of a cost function is proposed. Simulation results show that the proposed estimation method outperforms Kalman filters. The results in this paper provide useful tools for applications involving adversarial swarms, including defense against swarm attacks and herding of biological swarms.

This paper focuses on a non-standard constrained nonlinear optimal control problem in which the objective functional involves an integration over a space of stochastic parameters as well as an integration over the time domain. The research is inspired by the problem of optimizing the trajectories of multiple searchers attempting to detect non-evading moving targets. In this paper, we propose a framework based on the approximation of the integral in the parameter space for the considered uncertain optimal control problem. The framework is proved to produce a zeroth-order consistent approximation in the sense that accumulation points of a sequence of optimal solutions to the approximate problem are optimal solutions of the original problem. In addition, we demonstrate the convergence of the corresponding adjoint variables. The accumulation points of a sequence of optimal state-adjoint pairs for the approximate problem satisfy a necessary condition of Pontryagin Minimum Principle type, which facilitates assessment of the optimality of numerical solutions.

This paper addresses the problem of optimal defense of a high-value unit (HVU) against a large-scale swarm attack. We discuss multiple models for intra-swarm cooperation strategies and provide a framework for combining these cooperative models with HVU tracking and adversarial interaction forces. We show that the problem of defending against a swarm attack can be cast in the framework of uncertain parameter optimal control. We discuss numerical solution methods, then derive a consistency result for the dual problem of this framework, providing a tool for verifying computational results. We also show that the dual conditions can be computed numerically, providing further computational utility. Finally, we apply these numerical results to derive optimal defender strategies against a 100-agent swarm attack.