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MTH-222.4 Control Theory for PDEs

Syllabus

Controllability and Stabilizability of ODEs - Various concepts of controllability, Reachable states. Kalman criterion, Fattorini-Hautus test. Various concepts of observability. Duality between controllability and observability. Observability inequalities and cost of the control. Controllability of non-autonomous linear systems. Local and Global Controllability for nonlinear ODE systems. Feedback stabilization of linear and non-linear ODE systems, construction of the feedback control operator.

Semigroup theory - Time-dependent Sobolev spaces, Unbounded operators and their adjoint. Strongly continuous semigroups and generators. Theorems of Lumer-Philips, Stone, and Hillie-Yoshida. Nonhomogeneous linear evolution equations, Mild Solutions.

Controllability and stabilizability of PDEs - Abstract linear control system, admissibility, and wellposedness. Controllability and observability concepts. Duality between controllability and observability. Fourier Series Methods in control of PDEs, Method of moments, Ingham inequalities and its applications to control theory. Carleman estimates and controllability of parabolic equations. Propagation of singularities and controllability of hyperbolic equations. Feedback stabilization of linear PDEs. Riccati based feedback stabilization. Controllability and stabilizability of non-linear PDEs.