STUDY OF FRACTIONAL ORDER DELAY CAUCHY NON-AUTONOMOUS EVOLUTION PROBLEMS VIA DEGREE THEORY

This work is devoted to derive some existence and uniqueness (EU) conditions for the solution to a class of nonlinear delay non-autonomous integro-differential Cauchy evolution problems (CEPs) under Caputo derivative of fractional order. The required results are derived via topological degree method (TDM). TDM is a powerful tool which relaxes strong compact conditions by some weaker ones. Hence, we establish the EU under the situation that the nonlinear function satisfies some appropriate local growth condition as well as of non-compactness measure condition. Furthermore, some results are established for Hyers–Ulam (HU) and generalized HU (GHU) stability. Our results generalize some previous results. At the end, by a pertinent example, the results are verified.

EMERGE Modular Robot: A Tool for Fast Deployment of Evolved Robots

This work presents a platform for evolution of morphology in full cycle reconfigurable hardware: The EMERGE (Easy Modular Embodied Robot Generator) modular robot platform. Three parts necessary to implement a full cycle process, i.e., assembling the modules in morphologies, testing the morphologies, disassembling modules and repeating, are described as a previous step to testing a fully autonomous system: the mechanical design of the EMERGE module, extensive tests of the modules by first assembling them manually, and automatic assembly and disassembly tests. EMERGE modules are designed to be easy and fast to build, one module is built in half an hour and is constructed from off-the-shelf and 3D printed parts. Thanks to magnetic connectors, modules are quickly attached and detached to assemble and reconfigure robot morphologies. To test the performance of real EMERGE modules, 30 different morphologies are evolved in simulation, transferred to reality, and tested 10 times. Manual assembly of these morphologies is aided by a visual guiding tool that uses AprilTag markers to check the real modules positions in the morphology against their simulated counterparts and provides a color feedback. Assembly time takes under 5 min for robots with fewer than 10 modules and increases linearly with the number of modules in the morphology. Tests show that real EMERGE morphologies can reproduce the performance of their simulated counterparts, considering the reality gap. Results also show that magnetic connectors allow modules to disconnect in case of being subjected to high external torques that could damage them otherwise. Module tracking combined with their easy assembly and disassembly feature enable EMERGE modules to be also reconfigured using an external robotic manipulator. Experiments demonstrate that it is possible to attach and detach modules from a morphology, as well as release the module from the manipulator using a passive magnetic gripper. This shows that running a completely autonomous, evolution of morphology in full cycle reconfigurable hardware of different topologies for robots is possible and on the verge of being realized. We discuss EMERGE features and the trade-off between reusability and morphological variability among different approaches to physically implement evolved robots.

Optimal energy decay rates for abstract second order evolution equations with nonautonomous damping

We consider an abstract second order non-autonomous evolution equation in a Hilbert space $H:$ $u''+Au+\gamma(t) u'+f(u)=0,$ where $A$ is a self-adjoint and nonnegative operator on $H$, $f$ is a conservative $H$-valued function with polynomial growth (not necessarily to be monotone), and $\gamma(t)u'$ is a time-dependent damping term. How exactly the decay of the energy is affected by the damping coefficient $\gamma(t)$ and the exponent associated with the nonlinear term $f$? There seems to be little development on the study of such problems, with regard to {\it non-autonomous} equations, even for strongly positive operator $A$. By an idea of asymptotic rate-sharpening (among others), we obtain the optimal decay rate of the energy of the non-autonomous evolution equation in terms of $\gamma(t)$ and $f$. As a byproduct, we show the optimality of the energy decay rates obtained previously in the literature when $f$ is a monotone operator.