Boundary Arc and Boundary Point Stabilization of Reaction-Diffusion Equations on the Unit Disk

The uncontrolled diffusion equation on the unit disk under homogeneous Neumann boundary conditions is only neutrally stable as zero is an eigenvalue. If we add linear reaction terms the resulting equation can be unstable. To stabilize such a reaction-diffusion equation to a uniform state which we conveniently take to be zero, we pose and solve a Linear Quadratic Regulator (LQR). We consider two types of control actuation, boundary arc control and boundary point control. With boundary arc control we assume that there are <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$m$</tex-math></inline-formula> arcs on the boundary of the disk and we can control the time varying flux to be spatially constant across each arc so the control is <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$m$</tex-math></inline-formula> dimensional. Boundary point control is the idealized limit of boundary arc control as the length of each arc shrinks to zero but the gain factor multiplying the control inversely goes to infinity. For each type of actuation we derive a Riccati PDE whose solution is the Fredholm kernel of the optimal cost of the LQR. We also compute the Fredholm kernel of the corresponding optimal feedback and the first few closed loop eigenvalues. We show that optimal feedback moves the neutrally stable eigenvalue into the open left half plane but has little effect on the other eigenvalues as they are already sufficiently stable. We also show how a nonlinear reaction-diffusion equation can be stabilized by nonLinear nonQuadatic Regulation (nLnQR) which is an extension of Al'brekht's method to nonlinear distributed parameter systems. We close with an example of how a distributed predator prey system can be stabilized to its equilibrium point. We show that two unilateral controls (controls that can only take nonpositive values) can be almost as stabilizing as one bilateral control (control that can take on both positive and negative values).