  In this paper, we address the distributed prescribed-time convex optimization
(DPTCO) problem for a class of nonlinear multi-agent systems (MASs) under
undirected connected graph. A cascade design framework is proposed such that
the DPTCO implementation is divided into two parts: distributed optimal
trajectory generator design and local reference trajectory tracking controller
design. The DPTCO problem is then transformed into the prescribed-time
stabilization problem of a cascaded system. Changing Lyapunov function method
and time-varying state transformation method together with the sufficient
conditions are proposed to prove the prescribed-time stabilization of the
cascaded system as well as the uniform boundedness of internal signals in the
closed-loop systems. The proposed framework is then utilized to solve robust
DPTCO problem for a class of chain-integrator MASs with external disturbances
by constructing a novel variables and exploiting the property of time-varying
gains. The proposed framework is further utilized to solve the adaptive DPTCO
problem for a class of strict-feedback MASs with parameter uncertainty, in
which backstepping method with prescribed-time dynamic filter is adopted. The
descending power state transformation is introduced to compensate the growth of
increasing rate induced by the derivative of time-varying gains in recursive
steps and the high-order derivative of local reference trajectory is not
required. Finally, theoretical results are verified by two numerical examples.
