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.

Abstract. In this paper, we introduce the uncertain optimal control problem of determining a control that minimizes the expectation of an objective functional for a system with parameter uncertainty in both dynamics and objective. We present a computational framework for the numerical solution of this problem, wherein an independently drawn random sample is taken from the space of uncertain parameters, and the expectation in the objective functional is approximated by a sample average. The result is a sequence of approximating standard optimal control problems that can be solved using existing techniques. To analyze the performance of this computational framework, we develop necessary conditions for both the original and approximate problems and show that the approximation based on sample averages is consistent in the sense of Polak [Optimization: Algorithms and Consistent Approximations, Springer, New York, 1997]. This property guarantees that accumulation points of a sequence of global minimizers (stationary points) of the approximate problem are global minimizers (stationary points) of the original problem. We show that the uncertain optimal control problem can further be approximated in a consistent manner by a sequence of nonlinear programs under mild regularity assumptions. In numerical examples, we demonstrate that the framework enables the solution of optimal search and optimal ensemble control problems.Key words. optimal control, numerical methods, parameter uncertainty AMS subject classifications. Primary, 49M25; Secondary, 49K45 DOI. 10.1137/140983161 1. Introduction. In this paper we consider an extension of nonlinear optimal control for problems with Mayer-type objective functional to a setting with parameter uncertainty. We introduce the uncertain optimal control problem (UOCP), where the objective functional and system dynamics depend on stochastic parameters and the goal is to find a control that minimizes the expected value of the objective. The UOCP addresses a number of emerging applications in optimal control that require design of an open-loop control for an uncertain system such as those arising in optimal search or ensemble control. In the optimal search problem, the goal is to design a search plan to maximize the probability of detecting a moving target with unknown location or goals [30]. In the ensemble control problem, the goal is to determine a single open-loop control for a large number of structurally identical systems with parameter variation [27], which can be viewed as a single system with stochastic parameters [40]. In addition, existing control problems such as trajectory optimization may benefit from a problem formulation that incorporates inherent uncertainty in dynamical models, environment [17,46], and behavior of other agents [11,37,44]. In this paper, we develop a computational framework for the UOCP as well as necessary conditions for validation and verification of solutions.Specifically, the UOCP is the following problem: Find an initial state and control

Abstract-This paper focuses on the problem of optimizing the trajectories of multiple searchers attempting to detect a non-evading moving target whose motion is conditionally deterministic. This problem is a parameter-distributed optimal control problem, as it involves an integration over a space of stochastic parameters as well as an integration over the time domain. In this paper, we consider a wide range of discretization schemes to approximate the integral in the parameter space by a finite summation, which results in a standard controlconstrained optimal control problem that can be solved using existing techniques in optimal control theory. We prove that when the sequence of solutions to the discretized problem has an accumulation point, it is guaranteed to be an optimal solution of the original search problem. We also provide a necessary condition that accumulation points of this sequence must satisfy.