Control of hyperbolic and parabolic equations on networks and singular limits
We study the controllability properties of transport equations and of parabolic equations with vanishing diffusivity posed on a tree-shaped network. Using a control localized on the exterior nodes, we obtain a null-controllability result for both systems. The hyperbolic proof relies on the method of...
| Authors: | ; ; ; ; |
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| Format: | Article |
| Publication date: | 2025 |
| Country: | España |
| Institution: | Universidad del País Vasco |
| Repository: | Addi. Archivo Digital para la Docencia y la Investigación |
| OAI Identifier: | oai:addi.ehu.eus:10810/73759 |
| Online Access: | http://hdl.handle.net/10810/73759 |
| Access Level: | Open access |
| Keyword: | Controllability Cost of controllability Advection-diffusion equations Vanishing viscosity Networks |
| Summary: | We study the controllability properties of transport equations and of parabolic equations with vanishing diffusivity posed on a tree-shaped network. Using a control localized on the exterior nodes, we obtain a null-controllability result for both systems. The hyperbolic proof relies on the method of characteristics; while the parabolic one on duality arguments and Carleman inequalities. In particular, we estimate the cost of the null-controllability of advection-diffusion equations with diffusivity ε > 0 and study its asymptotic behavior when ε → 0+. More specifically, we show that the cost of null-controllability decays exponentially for a time sufficiently large and explodes for short times. The core of the proof consists in proving an observability estimate keeping track of the viscosity parameter by relying on a suitable Carleman inequality. |
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