Controller synthesis offers a correct-by-construction methodology to ensure the correctness and reliability of safety-critical cyber-physical systems (CPS). Controllers are classified based on the types of controls they employ, which include reset controllers, feedback controllers, and switching logic controllers. Reset controllers steer the behavior of a CPS to achieve system objectives by restricting its initial set and redefining its reset map associated with discrete jumps. Although the synthesis of feedback controllers and switching logic controllers has received considerable attention, research on reset controller synthesis is still in its early stages, despite its theoretical and practical significance. This paper outlines our recent efforts to address this gap. Our approach reduces the problem to computing differential invariants and reach-avoid sets. For polynomial CPS, the resulting problems can be solved by further reduction to convex optimizations. Moreover, considering the inevitable presence of time delays in CPS design, we further consider synthesizing reset controllers for CPS that incorporate delays.

The method of equivariant moving frames is employed to construct and completely classify the differential invariants for the action of the projective group on functions defined on the two-dimensional projective plane. While there are four independent differential invariants of order $\leq 3$, it is proved that the algebra of differential invariants is generated by just two of them through invariant differentiation. The projective differential invariants are, in particular, of importance in image processing applications.

Image feature points are detected as pixels which locally maximise a detector function, two commonly used examples of which are the (Euclidean) image gradient and the Harris–Stephens corner detector. A major limitation of these feature detectors is that they are only Euclidean-invariant. In this work, we demonstrate the application of a 2D equi-affine-invariant image feature point detector based on differential invariants as derived through the equivariant method of moving frames. The fundamental equi-affine differential invariants for 3D image volumes are also computed.

In this article, we study the homomorphisms between scalar generalized Verma modules. We conjecture that any homomorphism between scalar generalized Verma modules is a composition of elementary homomorphisms. The purpose of this article is to confirm the conjecture for some parabolic subalgebras under the assumption that the infinitesimal characters are regular.