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Model-predictive control (MPC) is a powerful framework for controlling dynamic systems under constraints, but it remains challenging to deploy on resource-constrained platforms, especially for problems involving conic constraints. To address this, we extend recent work developing fast, structure-exploiting, cached ADMM solvers for embedded applications, to provide support for second-order cones, as well as C++ code generation from Python, MATLAB, and Julia for easy deployment. Microcontroller benchmarks show that our solver provides up to a two-order-of-magnitude speedup, ranging from 10.6x to 142.7x, over state-of-the-art embedded solvers on QP and SOCP problems, and enables us to fit order-of-magnitude larger problems in memory. We validate our solver’s deployed performance through simulation and hardware experiments, including conically-constrained trajectory tracking on a 27g Crazyflie quadrotor. To get started with Conic-TinyMPC, visit our documentation, examples, and the open-source codebase at tinympc.org. 

The paper is concerned with a scalar balance law, where the source term depends on a control function a(t). Given a bonded, measurable control, it is proved that, for generic initial data \bar u, the solution has finitely many shocks, interacting at most two at a time. Moreover, at the terminal time T no shock interaction occurs and no new shock is formed. In addition, a family of optimal control problems is considered, including a running cost and a terminal cost. An example is constructed where the optimal solution contains two shocks merging exactly at the terminal time T. Such behavior persists under any suitably small perturbation of the flux, source and cost functions, and of the initial data. This shows that generic solutions of optimization problems have different qualitative properties, compared with generic solutions to Cauchy problems. 

The proliferation of latency-critical and compute-intensive edge applications is driving increases in computing demand and carbon emissions at the edge. To better understand carbon emissions at the edge, we analyze granular carbon intensity traces at intermediate "mesoscales," such as within a single US state or among neighboring countries in Europe, and observe significant variations in carbon intensity at these spatial scales. Importantly, our analysis shows that carbon intensity variations, which are known to occur at large continental scales (e.g., cloud regions), also occur at much finer spatial scales, making it feasible to exploit geographic workload shifting in the edge computing context. Motivated by these findings, we propose CarbonEdge, a carbon-aware framework for edge computing that optimizes the placement of edge workloads across mesoscale edge data centers to reduce carbon emissions while meeting latency SLOs. We implement CarbonEdge and evaluate it on a real edge computing testbed and through large-scale simulations for multiple edge workloads and settings. Our experimental results on a real testbed demonstrate that CarbonEdge can reduce emissions by up to 78.7% for a regional edge deployment in central Europe. Moreover, our CDN-scale experiments show potential savings of 49.5% and 67.8% in the US and Europe, respectively, while limiting the one-way latency increase to less than 5.5 ms. 

We consider a class of deterministic mean field games, where the state associated with each player evolves according to an ODE which is linear w.r.t. the control. Existence, uniqueness, and stability of solutions are studied from the point of viewof generic theory. Within a suitable topological space of dynamics and cost functionals, we prove that, for “nearly all” mean field games (in the Baire category sense) the best reply map is single-valued for a.e. player. As a consequence, the mean field game admits a strong (not randomized) solution. Examples are given of open sets of games admitting a single solution, and other open sets admitting multiple solutions. Further examples show the existence of an open set of MFG having a unique solution which is asymptotically stable w.r.t. the best reply map, and another open set of MFG having a unique solution which is unstable. We conclude with an example of a MFG with terminal constraints which does not have any solution, not even in the mild sense with randomized strategies.