Overview
A domain-extension framework has been developed for output-feedback boundary stabilization of reaction-diffusion equations. This method is applicable to arbitrary bounded Lipschitz domains, including non-convex and multiply connected geometries. The core concept involves embedding the plant, which is initially posed on an irregular domain, into a target domain for which a stabilizing design is already established. This process results in the extension of the plant into a virtual domain, and the overall system dynamics are configured to match the known closed-loop behavior on the target domain.
Research Context
The stabilization of partial differential equations (PDEs), particularly reaction-diffusion systems, is a critical area in control theory. Previous methods, such as backstepping, transform plant dynamics into a stable target system. This work builds upon such principles by introducing an embedding approach where the physical plant's irregular domain is extended into a more manageable, known target domain. The plant's boundary can have both actuated and uncontrolled portions. The proposed framework allows for uncontrolled boundary portions provided they are shared with the target boundary and possess the same boundary-condition type.
Approach
The domain-extension framework operates by inserting the plant into a target domain, examples of which include balls or rectangles, where a stabilizing control design is already defined. Every portion of the boundary through which this extension occurs must facilitate actuation and provide a complementary collocated measurement. The space between the original plant domain and the chosen target domain is populated by a virtual copy of the plant dynamics. This virtual component is then coupled to the physical plant via interface conditions. The combined state of the physical plant and the virtual domain is designed to evolve identically to the known closed-loop system dynamics of the target domain.
Well-posedness and exponential stability of the physical state are ensured through a restriction process. The design data required for this method are inherited from the target design. For constant-coefficient plants situated on balls and rectangles, these design data are in closed-form. The online operation of the virtual PDE involves computational demands comparable to those of a full-order PDE observer, which is a standard element in output-feedback designs.
To support this framework, a new explicit Neumann-actuated backstepping law has been developed for n-balls, thereby expanding the available repertoire of target designs. Output feedback is achieved by lifting the target-domain observer. This observer is driven by collocated interface measurements that are relayed through the virtual domain.
Findings
- The domain-extension framework enables output-feedback boundary stabilization for reaction-diffusion equations on arbitrary bounded Lipschitz domains.
- This method is applicable to non-convex and multiply connected geometries.
- The approach embeds the plant into a target domain (e.g., ball, rectangle) where a stabilizing design is pre-known.
- Well-posedness and exponential stability of the physical state are achieved by restriction from the extended system.
- Offline design data are derived from the target design and are closed-form for constant-coefficient plants on balls and rectangles.
- Online simulation of the virtual PDE has computational characteristics similar to a full-order PDE observer.
- A new explicit Neumann-actuated backstepping law for n-balls enhances available target designs.
- Output feedback is obtained by lifting a target-domain observer, driven by collocated interface measurements transmitted via the virtual domain.
Potential Applications
The designs were tested through numerical experiments on various complex geometries, including star-shaped, horseshoe, and multiply connected domains. These experiments involved a partitioned plant/controller implementation and a shared-wall cavity.