Numerical method and application of optimal control problem

Abstract

In this thesis, we mainly study a numerical method for solving optimal control problems and apply it to Cucker-Smale dynamics and data assimilation.

Pontryagin's maximum principle results in two-point boundary problem. One numerical method that is easy to employ for such problems is the so-called "forward-backward sweep" method. However this method is not always convergent especially when applied to non-linear systems. We extend the ``regularised forward-backward sweep iteration method'' from the continuous setting to the discrete setting for solving optimal control problems. The continuous problem is discretized by using a variational integrator which yields a symplectic method. The regularised forward-backward iteration method depends on a regularization parameter ρ. We provide the proof that when ρ is large enough, the forward-backward sweep method is convergent all the time. According to the proof, the parameter ρ depends on the length of the time window and the Lipschitz constant. Numerical experiments illustrate convergence, which may still be slow, especially for large ρ.

The Cucker-Smale model is a model of flocking, in which a group of agents attempt to synchronize into uniform motion. When the model could not converge naturally, some extra control may be added to the Cucker-Smale model to make the dynamic. It is interesting to consider `sparse control'. The optimal control cost functional combines distance to velocity consensus and the magnitude of the control in a class of so-called l_p-l_q-norms. In this chapter, we focus on discussing the l_2-l_ 2-norm, l_1-l_1-norm, l_2-l_1-norm on the control. The results in these three different cases show that the optimal controls become 0 or asymptotically tend to 0, after a finite time period. We find the optimal control is unbounded in l_1-norm, therefore we implemented constrained controls. Meanwhile, to avoid slow convergence due to discontinuity of the control, we add a soft-constraint δ and study the effect of the smoothness of the control on the convergence of the regularized forward-backward sweep iteration. Under the condition of l_1-l_1 norm, the experiment shows that the control is a bang-bang controller, either zero or sharply constrained, before the control totally goes to 0. The optimal control under the l_2-l_1 norm is also sparse.

In Chapter 5, we proposed a new data assimilation algorithm, to utilizes the probability distribution of an ensemble of controlled particles to quantify uncertainty in stochastic systems. Especially, the controlled dynamical system for the particles is deterministic. The method is defined as an optimal control problem. The cost function is composed of the norm of the control function and the Wasserstein distance on the observation space. Two different sampling processes were studied. With the first situation, we take many noisy samples of the observable along a distinct sample path. Alternatively, we take single observations of an ensemble of sample paths. Experiments with a (bi-modal) double well potential indicate good results, with no evidence of ensemble collapse. We also compare to the EnKF for deterministic ensemble simulation of the Lorenz-63 model to illustrate the advantage of the Wasserstein distance for dynamics on a strange attractor.