Optimal Control of DNA Amplification

In “Dynamics and Control of DNA Sequence Amplification”, we have formulated the PCR optimal control problem and discussed the strategies for solving it. In this chapter, we implement those strategies and solve a simple PCR optimal control problem. The foundational concepts of Optimal Control were developed as early as the 1750s in classical mechanics as the principle of least action, a variational approach that minimizes the action (the time integral of a Lagrange cost) of a moving particle to derive the equations of motion in Hamilton’s form. A more general and formal version of the optimal control problem, especially from a control engineering perspective, was developed by Pontryagin, who also derived the conditions for the optimality which now comprise Pontryagin Maximum Principle (PMP) ([1, 2]). Optimal control is used in diverse applications ranging from satellite control to chemical plant control, from quantum control to biological process control, etc. As a result of this, there are several classes of optimal control problems such as constrained optimal control, time optimal control, etc. The complexity of optimal control problems pose challenging issues in developing efficient numerical schemes. There are tremendous research efforts directed towards developing various numerical schemes to solve various optimal control problems. Though the primary objective of our work is to solve a PCR optimal control problem, in this chapter we also briefly review several numerical approaches that are available to solve a large scale dynamic optimization problem so that it will be useful in future development of an efficient software platform for solving PCR optimal control problems. Development of an efficient numerical scheme for the optimal control of PCR is beyond the scope of this work. The primary goal of this work is the development of a foundation for the derivation of optimal control laws for amplification of DNA sequences through application of the PMP to sequence-dependent models of DNA amplification. When applied to a given DNA sequence, these laws prescribe the optimally controlled dynamics of amplification of that sequence.