Seminar Details
2012-04-17 (14.00) : LQ-optimal control of infinite dimensional linear systems with boundary control and observation
At Euler building
Organized by Mathematical Engineering
Speaker :
Jérémy DEHAYE (FUNDP Namur)
Abstract :
Distributed parameter systems based on partial differential equations can be used to describe the behaviour of many physical setups, e.g. convection-diffusion-reaction systems with a linear or linearized reaction rate and inputs and sensors located on the spatial boundary. These systems can generally be written as abstract infinite-dimensional linear differential systems with boundary control and observation, and the associated operators are thus unbounded.
Under these conditions, difficulties can arise when trying to solve some specific control problems, including the linear-quadratic optimal control problem and the design of the associated stabilizing state feedback. In particular, the method of spectral factorization by symmetric extraction provides an alternative to the resolution of the operator Riccati equation for the LQ-optimal control problem but its convergence relies on the fact that the control and observation operators are bounded.
In order to solve this problem, we consider, under appropriate conditions on the initial equations, a change of variables and an extended abstract linear differential system with bounded control and observation operators. An adapted cost functional can be associated with this extended system and can be interpreted in the framework of the initial one. Solving the LQ-optimal control problem for this cost functional provides a stabilizing feedback at the cost of an approximation in the measurement of the output and the feedback not being the optimal one in general.
As an application, we try to solve the LQ-optimal control problem for a convection-diffusion-reaction system with both distributed and boundary control, and boundary observation. The involved differential operator belongs to the class of Sturm-Liouville operators that are densely defined in the Lebesgue space of square-integrable functions. These systems are used in the modelling of e.g. biochemical processes, such as nonisothermal tubular reactors with axial dispersion, biochemical reactors, UV disinfection and many others.
The main tools of functional analysis used in this framework are the
theories of C_0-semigroups and Riesz-spectral systems, playing a
central role when Sturm-Liouville systems are involved.