Understanding the long term behavior of solutions is a fundamental question in any time evolving system. This begins with questions of existence, and whether local existence can be extended globally. Of the solutions which exist globally, coherent structures (such as equilibria, traveling waves and periodic orbits) serve as emblematic examples of how solutions may behave. However, nonlinear differential equations are rarely solvable by hand. Instead of searching for arbitrary solutions, the dynamical systems viewpoint typically begins by identifying basic landmark invariant sets, and then studying connecting orbits between these. Often, it is just the topology of these landmarks which is robust; changes in topology signal important changes in the system.
With abstract theorems one can describe in great detail the dynamics on and around generic invariant sets. However, for a specific differential equation, verifying the hypotheses of such a theorem often requires hard quantitative analysis. In this talk I will discuss various ways one can develop a global understanding of how complex systems change over time, and bridge the gap between what can be proven mathematically and what can be computed numerically, with the aid of computer-assisted proofs.
Multi-Agent Path-Planning in a Moving Medium via Wasserstein Hamiltonian Flow
I’ll discuss a finite dimensional variational model for multi-agent path-planning in which a group of agents traverses from initial positions to a target distribution in a moving medium. The model is derived using the agent-based formulation of the Wasserstein Hamiltonian flow that transports between probability distributions while optimizing a running cost. The objective is the mismatch between the agents' final positions and the target distribution. The constraints are a system of Hamiltonian equations that provide the trajectories of the agents. The free variables on which the optimization is defined form a finite vector of the initial velocities for the agents. The model is solved numerically by the L-BFGS method in conjunction with a shooting strategy. Several simulation examples, including a time-dependent moving medium, are presented to illustrate the performance of the model.