  DNA amplification is the process of replication of a specified DNA sequence
\emph{in vitro} through time-dependent manipulation of its external
environment. A theoretical framework for determination of the optimal dynamic
operating conditions of DNA amplification reactions, for any specified
amplification objective, is presented based on first-principles biophysical
modeling and control theory. Amplification of DNA is formulated as a problem in
control theory with optimal solutions that can differ considerably from
strategies typically used in practice. Using the Polymerase Chain Reaction
(PCR) as an example, sequence-dependent biophysical models for DNA
amplification are cast as control systems, wherein the dynamics of the reaction
are controlled by a manipulated input variable. Using these control systems, we
demonstrate that there exists an optimal temperature cycling strategy for
geometric amplification of any DNA sequence and formulate optimal control
problems that can be used to derive the optimal temperature profile. Strategies
for the optimal synthesis of the DNA amplification control trajectory are
proposed. Analogous methods can be used to formulate control problems for more
advanced amplification objectives corresponding to the design of new types of
DNA amplification reactions.
