a b s t r a c tThis 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.

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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 discretization 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.