Our team has made several contributions in the description of the limit behaviour, as the mesh sizes tend to zero, of numerical schemes for wave and Schrödinger equations from a control theoretical perspective. These results show that, in particular, filtering the high frequency numerical spurious solutions is necessary (and a good remedy) to assure the convergence of numerical schemes from a control perspective. These results also provide insight into the link between conservative finite and infinite-dimensional dynamical systems and their asymptotic behaviour.
In this post, we use IpOpt and AMPL to simulate optimal controls on a nonlinear ODE system with unbounded interactions. The restriction and initial guess on the state variables are critical for this problem to operate minimization algorithm. From the data calculated from AMPL, we interpret and visualize it using Matlab.
We design the LQR controller and solve it with linearized model.
Summary of example objective The goal of this tutorial is to use LQR theory applied to a model of collective behavior. The model choosen shares a formal structure with the semidiscretization of the semilinear 1d heat equation.
Guidance by repulsion model describing the behaviour of two agents, a driver and an evader
Various numerical approximation methods are discussed with the aim of recoving the large time asymptotic properties of the hypoelliptic Kolmogorov model