A Primal-Dual Method for Optimal Control and Trajectory Generation in High-Dimensional Systems

Presented is a method for efficient computation of the Hamilton-Jacobi (HJ) equation for time-optimal control problems using the generalized Hopf formula. Typically, numerical methods to solve the HJ equation rely on a discrete grid of the solution space and exhibit exponential scaling with dimension. The generalized Hopf formula avoids the use of grids and numerical gradients by formulating an unconstrained convex optimization problem. The solution at each point is completely independent, and allows a massively parallel implementation if solutions at multiple points are desired. This work presents a primal-dual method for efficient numeric solution and presents how the resulting optimal trajectory can be generated directly from the solution of the Hopf formula, without further optimization. Examples presented have execution times on the order of milliseconds and experiments show computation scales approximately polynomial in dimension with very small high-order coefficients.Comment: Updated references and funding sources. To appear in the proceedings of the 2018 IEEE Conference on Control Technology and Application

Optimal Trajectories of a UAV Base Station Using Hamilton-Jacobi Equations

International audienceWe consider the problem of optimizing the trajectory of an Unmanned Aerial Vehicle (UAV). Assuming a traffic intensity map of users to be served, the UAV must travel from a given initial location to a final position within a given duration and serves the traffic on its way. The problem consists in finding the optimal trajectory that minimizes a certain cost depending on the velocity and on the amount of served traffic. We formulate the problem using the framework of Lagrangian mechanics. We derive closed-form formulas for the optimal trajectory when the traffic intensity is quadratic (single-phase) using Hamilton-Jacobi equations. When the traffic intensity is bi-phase, i.e. made of two quadratics, we provide necessary conditions of optimality that allow us to propose a gradient-based algorithm and a new algorithm based on the linear control properties of the quadratic model. These two solutions are of very low complexity because they rely on fast convergence numerical schemes and closed form formulas. These two approaches return a trajectory satisfying the necessary conditions of optimality. At last, we propose a data processing procedure based on a modified K-means algorithm to derive a bi-phase model and an optimal trajectory simulation from real traffic data

Bregman iterative algorithms for ℓ1minimization with applications to compressed sensing

Abstract. We propose simple and extremely efficient methods for solving the Basis Pursuit problem min{�u�1: Au = f, u ∈ R n}, which is used in compressed sensing. Our methods are based on Bregman iterative regularization and they give a very accurate solution after solving only a very small number of instances of the unconstrained problem min u∈Rn μ�u�1 + 1 2 �Au − f k � 2 2, for given matrix A and vector f k. We show analytically that this iterative approach yields exact solutions in a finite number of steps, and present numerical results that demonstrate that as few as two to six iterations are sufficient in most cases. Our approach is especially useful for many compressed sensing applications where matrix-vector operations involving A and A ⊤ can be computed by fast transforms. Utilizing a fast fixed-point continuation solver that is solely based on such operations for solving the above unconstrained sub-problem, we were able to solve huge instances of compressed sensing problems quickly on a standard PC

Scribble1/AP2 Complex Coordinates NMDA Receptor Endocytic Recycling

The appropriate trafficking of glutamate receptors to synapses is crucial for basic synaptic function and synaptic plasticity. It is now accepted that NMDA receptors (NMDARs) internalize and are recycled at the plasma membrane but also exchange between synaptic and extrasynaptic pools; these NMDAR properties are also key to governing synaptic plasticity. Scribble1 is a large PDZ protein required for synaptogenesis and synaptic plasticity. Herein, we show that the level of Scribble1 is regulated in an activity-dependent manner and that Scribble1 controls the number of NMDARs at the plasma membrane. Notably, Scribble1 prevents GluN2A subunits from undergoing lysosomal trafficking and degradation by increasing their recycling to the plasma membrane following NMDAR activation. Finally, we show that a specific YxxR motif on Scribble1 controls these mechanisms through a direct interaction with AP2. Altogether, our findings define a molecular mechanism to control the levels of synaptic NMDARs via Scribble1 complex signaling