On the Time-domain Decomposition of Parabolic Optimal Control Problems

In the above, ŷ = ŷ(t) and ŷT are the target trajectory and target state, and the functions u and y = y(t,u) are called the control and the state, respectively. (For the purpose of analysis, we will use an appropriate change of variables to subsume any mass matrices that appear into the matrices A, B, C and D.) We will focus on the case where there are no control or state constraints, and where the governing equation is parabolic, i.e., when A is positive semi-definite, but not necessarily symmetric. A formulation similar to the above has been used for a variety of problems where the goal is to drive a mechanical system to a desired state while minimizing the cost: it has been used for the control of fluid flow modelled by the Navier-Stokes equations [4, 23], boundary control problems for the wave equation [14] and quantum control (see [18] and references therein).