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<p>This dissertation presents a novel distributed optimal control (DOC) problem formulation
that is applicable to multiscale dynamical systems comprised of numerous interacting
systems, or agents, that together give rise to coherent macroscopic behaviors, or
coarse dynamics, that can be modeled by partial differential equations (PDEs) on larger
spatial and time scales. The DOC methodology seeks to obtain optimal agent state
and control trajectories by representing the system's performance as an integral cost
function of the macroscopic state, which is optimized subject to the agents' dynamics.
The macroscopic state is identified as a time-varying probability density function
to which the states of the individual agents can be mapped via a restriction operator.
Optimality conditions for the DOC problem are derived analytically, and the optimal
trajectories of the macroscopic state and control are computed using direct and indirect
optimization algorithms. Feedback microscopic control laws are then derived from
the optimal macroscopic description using a potential function approach.</p><p>The
DOC approach is demonstrated numerically through benchmark multi-agent trajectory
optimization problems, where large systems of agents were given the objectives of
traveling to goal state distributions, avoiding obstacles, maintaining formations,
and minimizing energy consumption through control. Comparisons are provided between
the direct and indirect optimization techniques, as well as existing methods from
the literature, and a computational complexity analysis is presented. The methodology
is also applied to a track coverage optimization problem for the control of distributed
networks of mobile omnidirectional sensors, where the sensors move to maximize the
probability of track detection of a known distribution of mobile targets traversing
a region of interest (ROI). Through extensive simulations, DOC is shown to outperform
several existing sensor deployment and control strategies. Furthermore, the computation
required by the DOC algorithm is proven to be far reduced compared to that of classical,
direct optimal control algorithms.</p>
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