The Douglas–Rachford (DR) algorithm is a powerful tool used to find zeros of monotone inclusions. This algorithm, as well as other splitting and projection methods, is commonly used to solve convex optimisation problems though few have applied this method to solve optimal control problems, and fewer still have applied it to solve continuous-time optimal control problems. We consider the application of the DR algorithm to solve linear-quadratic (LQ) optimal control problems beginning with a minimum-energy control-constrained LQ control problem before solving a control and state-constrained LQ control problem. These optimal control problems are challenging because they are infinite-dimensional optimisation problems that cannot be solved analytically due to the constraints on the variables.
The advantage provided by our approach is that we can derive the projection operators needed for the DR algorithm (or any other projection-type algorithm) for the original LQ control problem rather than for a discrete representation of the problem. By reformulating the optimal control problem as minimising the sum of two functionals, this becomes a problem of finding a zero of the sum of the subdifferentials of these two functionals. Once the problem has been expressed in this form, we can derive the projection operators for these two functionals and then apply the DR algorithm. This approach leads to improved numerical performance when compared with a direct discretisation method like the AMPL–Ipopt suite.
This thesis is a thorough investigation of the application of the DR algorithm to LQ control problems. Along with deriving the projection operators for LQ control problems, we also explore the relationship between the dual of a weighted minimum-energy control problem and the DR algorithm. In doing so, we obtain an expression for the fixed point of the DR operator which can be used to construct an optimality check to verify a numerical solution. We also investigate the behaviour of the DR and PR algorithms in the inconsistent case, that is, when there are no zeros to the monotone inclusion problem. In this case, we prove the strong convergence of the shadow sequences associated with the DR and PR algorithms under mild assumptions.
We carry out many numerical experiments throughout this thesis to demonstrate the performance of the DR algorithm for an array of practical examples including an harmonic oscillator (damped and undamped), machine tool manipulator and mass–spring system. Though the focus of the thesis is the application of the DR algorithm, we include experiments with other algorithms including Dykstra’s algorithm, the Aragón Artacho–Campoy (AAC) algorithm and the Peaceman–Rachford (PR) algorithm for comparison. For the DR, PR and AAC algorithms, we experiment with values for parameters introduced to further improve the performance of the algorithms. There is no theory available to optimise the parameters so the large testbed of examples presented in this thesis provides some insight into the best choices for these parameters.
Some of the research has appeared in [Reference Burachik, Caldwell and Kaya1–Reference Burachik, Caldwell, Kaya, Moursi and Saurette5].
