Location
MC 6460
Candidate
Thomas Newton | Applied Mathematics, University of Waterloo
Title
Optimal Actuator Design for Nonlinear PDE Control
Abstract
Control theory studies how to influence dynamical systems so that they behave in a desired way. In optimal control, this is achieved by designing inputs that minimize a cost function. When systems exhibit both spatial and temporal dynamics, they are naturally modelled by partial differential equations (PDEs), where performance depends not only on how control is applied in time, but also where it is applied in space. This leads to the problem of optimal actuator design: selecting the location, shape, or distribution of actuators to achieve the best possible controller performance.
This talk introduces this problem through a simple example based on the heat equation for a fixed initial condition, which illustrates the key ideas from PDE-constrained optimization and control. In particular, we show how the problem can be formulated as an abstract optimization over infinite dimensional function spaces, then how adjoint equations are used to compute derivatives and characterize optimal solutions.
We then extend this framework to a more realistic setting, where the actuator design must perform well over a range of possible initial conditions. This leads to a worst-case design problem with a nested minimax structure. We outline the resulting optimality conditions, derived use of tools such as Danskin’s theorem to handle differentiation through the maximization over initial conditions.
In the linear setting, these conditions simplify significantly, allowing the problem to be expressed in terms of Riccati equations with useful convex relaxations. However, for nonlinear dynamics the problem remains difficult, both analytically and computationally. The talk concludes with a brief discussion of current challenges and future work for tackling the nonlinear problem.