Sustainable Optimal Controls for Nonlinear Partial Differential Equations with Applications

• Applicant: Professor Dr. Günter Leugering
• Subject Area: Mathematics
• Term: from 2019 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 405372818

• Final Report Year: 2021

Final Report Abstract

Sustainability has become a major concern in society and, hence, in science particularly in engineering. Infrastructures are prone to damage and failure in particular if the service life comes to an end or if an unforeseeable event takes place. We experience such events more frequently due to the global warming. The immediate question is how to possibly prevent damage to occur in the first place and otherwise as to whether the infrastructure is still under operational conditions after damage has taken place. Moreover, the operations themselves that are supported by the structure are the subject of possible concern as they may add to the evolution of damage intrinsically. In particular, a pipe network conveying gas or fluids in general, or a traffic network may suffer from damage once the transportation process is carried to its technical limits over a long time interval, leading to leakage or street deterioration, respectively. Control and process optimization, therefore, can no longer be considered independent of its impact on the underlying material, let alone the mutlilink structures constituted by that material. Controls reflecting these consideration were in the focus of the research funded by the DFG in this project, finally aiming at sustainable control regimes. Clearly, this aim is multidimensional in its facets and in fact very complex in the actual modeling, let alone optimal control schemes. The project, as being applied for the DFG-SFFRU funding, could only shed a first light on this challenging problem from a mathematical point of view. In fact, we see this as an emerging field of interdisciplinary research. To begin with, in order to fix ideas, the focus has been on simple damage models and simple governing partial differential equations, like linear wave equations, where degeneration takes place in the interior of the domain or systems of such equations, where degeneration occurs at the coupling interface. Minimum norm exact controllability can be seen a prototype for optimal control with state constraints. We pursued this format throughout the project. As a result, there were three major strings of research: Problems of exact boundary controllability for 1. 1-d wave equations with interior degeneration, 2. a star-like network in the plane realized via singular measures, 3. wave equations coupled by springs an masses. The first two lines of research have been completed. The third one is close to be completed.

Publications

• A Note on Weighted Sobolev Spaces Related to Weakly and Strongly Degenerate Differential Operators, J. of Optimization, Differential Equations and Their Applications (JODEA), Vol.27, Issue 2, 2019, 1-22
  P.I. Kogut, O.P. Kupenko, G. Leugering, Y. Wang
  
• Exact boundary controllability for a coupled system of quasilinear wave equations with dynamical boundary conditions, Nonlinear Analysis: Real World Applications. 49 (2019), 71-79
  G. Leugering and T. Li, Y. Wang
  
• Exact Boundary Controllability for the Spatial Vibration of String with Dynamical Boundary Conditions, Chin. Ann. Math., Series B, 41(2020), 325-334
  G. Leugering and T. Li, Y. Wang
  
• On an Initial Boundary-Value Problem for 1D Hyperbolic Equation with Interior Degeneracy: Series Solutions with the Continuously Differentiable Fluxes, J. of Optimization, Differential Equations and Their Applications (JODEA), Vol.28, Issue 1, 2020, 1-42
  V.L. Borsch, P.I. Kogut, G. Leugering
  
• The Exact Bounded Solution to an Initial Boundary Value Problem for 1D Hyperbolic Equation with Interior Degeneracy. I. Separation of Variables, J. of Optimization, Differential Equations and Their Applications (JODEA), Vol.28, Issue 2, 2020, 2-20
  V.L. Borsch, P.I. Kogut
  
• Nodal Profile Control for Networks of Geometrically Exact Beams, Journal de Mathematiques Pures et Appliquees, (2021), ISSN 0021-7824
  G. Leugering, C. Rodriguez, Y. Wang
  
• On Boundary Exact Controllability of One- Dimensional Wave Equations with Weak and Strong Interior Degeneration, Mathematical Methods in the Applied Sciences., 27 September 2021
  P. I. Kogut, O.P. Kupenko, G. Leugering
  
• Solutions to a Simplified Initial Boundary Value Problem for 1D Hyperbolic Equation with Interior Degeneracy, Journal of Optimization, Differential Equations and Their Applications (JODEA), 29 (1) (2021), 1-31
  V. L. Borsch, P.I. Kogut