Final Report Summary - NUMERIWAVES (New analytical and numerical methods in wave propagation)
NUMERIWAVES has constituted a significant step forward in the development of analytical and numerical methods for wave propagation, with real life and industrial applications in mind. Special emphasis has been done on problems related to control and design. Some of the main achievements are as follows.
• Motivated by the problem of boundary control/observation of waves, we have developed the microlocal approach to the numerical approximation of wave equations, based on earlier works on microlocal defect and Wigner measures. When dealing with control problems for continuous wave equations, according to intuition, efficient sensors and controllers need to absorb all rays of Geometric Optics, to capture the energy of all possible solutions and frequencies. Likewise, to preserve the control theoretical properties of the continuous models at the numerical level, uniformly with respect to the mesh-size parameters, numerical approximation schemes need to guarantee that all numerical solutions reach the sensor/actuator location. However, the behavior of numerical solutions can be rather complex, due to their interaction with the numerical mesh, generating spurious high-frequency numerical solutions. We have designed non-uniform meshes preserving the control theoretical properties of continuous waves (joint work with S. Ervedoza and A. Marica).
• In a series of joint articles with Y. Privat and E. Trélat we have addressed the problem of the optimal location of sensors and actuators for wave and heat processes. In particular we have analyzed the existence of classical locations for sensors and actuators in opposition to the possible emergence of relaxation phenomena, according to which sensors/actuators need to be distributed everywhere in the domain where the processes under consideration evolve. Implementing a randomization procedure the problem has been reduced to optimal observation and control of the eigenfunctions of the Laplacian generating the dynamics of the system. While for the wave equation the lack of dissipation leads often to relaxation phenomena, because of the intrinsic damping optimal locations are determined by a finite number of eigenfunctions for heat-line dynamics.
• In collaboration with L. Ignat and A. Pozo we also analyzed the large time asymptotic behavior of the discrete solutions of numerical approximation schemes for scalar hyperbolic conservation laws. We proved that, at the numerical level, the large time dynamics depends on the amount of numerical viscosity introduced by the scheme: while Engquist-Osher and Godunov yield the continuous N-wave asymptotic profiles, the Lax-Friedrichs scheme produces viscous self-similar ones. In subsequent work, in collaboration with N. Allahverdi and A. Pozo, we have developed the control theoretical consequences of these facts, analyzing the different performance of the various numerical schemes when dealing with optimal control problems in long time horizons, both for the Burgers equations, and the generalized model arising in sonic-boom minimization for supersonic aircrafts.
• In a series of joint articles with M. Gugat, A. Porretta and E. Trélat, we have analyzed the property of turnpike, according to which, in long time horizons, optimal controls and controlled trajectories nearly coincide with the steady-state ones. Sharp sufficient conditions for turnpike to hold have been derived both in the finite-dimensional and in the infinite-dimensional setting.
• In collaboration with A. Porretta we addressed the problem of the decay rate of numerical approximation schemes for the Kolmogorov model, a paradigm of hypoelliptic Partial Differential Equation (PDE), as time tends infinity. We have shown that, provided the various differential operators entering in the PDE are properly approximated, the decay rate of PDE solutions is preserved at the numerical level. This has been done by constructing augmented Lyapunov functionals that take advantage of the interaction of the various differential operators at the numerical level, exhibited through the corresponding Lie brackets.
The fundamental research developed within the project has then been applied in sonic-boom minimization for supersonic aircrafts and the management of hydraulic resources.
• Motivated by the problem of boundary control/observation of waves, we have developed the microlocal approach to the numerical approximation of wave equations, based on earlier works on microlocal defect and Wigner measures. When dealing with control problems for continuous wave equations, according to intuition, efficient sensors and controllers need to absorb all rays of Geometric Optics, to capture the energy of all possible solutions and frequencies. Likewise, to preserve the control theoretical properties of the continuous models at the numerical level, uniformly with respect to the mesh-size parameters, numerical approximation schemes need to guarantee that all numerical solutions reach the sensor/actuator location. However, the behavior of numerical solutions can be rather complex, due to their interaction with the numerical mesh, generating spurious high-frequency numerical solutions. We have designed non-uniform meshes preserving the control theoretical properties of continuous waves (joint work with S. Ervedoza and A. Marica).
• In a series of joint articles with Y. Privat and E. Trélat we have addressed the problem of the optimal location of sensors and actuators for wave and heat processes. In particular we have analyzed the existence of classical locations for sensors and actuators in opposition to the possible emergence of relaxation phenomena, according to which sensors/actuators need to be distributed everywhere in the domain where the processes under consideration evolve. Implementing a randomization procedure the problem has been reduced to optimal observation and control of the eigenfunctions of the Laplacian generating the dynamics of the system. While for the wave equation the lack of dissipation leads often to relaxation phenomena, because of the intrinsic damping optimal locations are determined by a finite number of eigenfunctions for heat-line dynamics.
• In collaboration with L. Ignat and A. Pozo we also analyzed the large time asymptotic behavior of the discrete solutions of numerical approximation schemes for scalar hyperbolic conservation laws. We proved that, at the numerical level, the large time dynamics depends on the amount of numerical viscosity introduced by the scheme: while Engquist-Osher and Godunov yield the continuous N-wave asymptotic profiles, the Lax-Friedrichs scheme produces viscous self-similar ones. In subsequent work, in collaboration with N. Allahverdi and A. Pozo, we have developed the control theoretical consequences of these facts, analyzing the different performance of the various numerical schemes when dealing with optimal control problems in long time horizons, both for the Burgers equations, and the generalized model arising in sonic-boom minimization for supersonic aircrafts.
• In a series of joint articles with M. Gugat, A. Porretta and E. Trélat, we have analyzed the property of turnpike, according to which, in long time horizons, optimal controls and controlled trajectories nearly coincide with the steady-state ones. Sharp sufficient conditions for turnpike to hold have been derived both in the finite-dimensional and in the infinite-dimensional setting.
• In collaboration with A. Porretta we addressed the problem of the decay rate of numerical approximation schemes for the Kolmogorov model, a paradigm of hypoelliptic Partial Differential Equation (PDE), as time tends infinity. We have shown that, provided the various differential operators entering in the PDE are properly approximated, the decay rate of PDE solutions is preserved at the numerical level. This has been done by constructing augmented Lyapunov functionals that take advantage of the interaction of the various differential operators at the numerical level, exhibited through the corresponding Lie brackets.
The fundamental research developed within the project has then been applied in sonic-boom minimization for supersonic aircrafts and the management of hydraulic resources.