97 research outputs found
Bilateral boundary control of an input delayed 2-D reaction-diffusion equation
In this paper, a delay compensation design method based on PDE backstepping
is developed for a two-dimensional reaction-diffusion partial differential
equation (PDE) with bilateral input delays. The PDE is defined in a rectangular
domain, and the bilateral control is imposed on a pair of opposite sides of the
rectangle. To represent the delayed bilateral inputs, we introduce two 2-D
transport PDEs that form a cascade system with the original PDE. A novel set of
backstepping transformations is proposed for delay compensator design,
including one Volterra integral transformation and two affine Volterra integral
transformations. Unlike the kernel equation for 1-D PDE systems with delayed
boundary input, the resulting kernel equations for the 2-D system have singular
initial conditions governed by the Dirac Delta function. Consequently, the
kernel solutions are written as a double trigonometric series with
singularities. To address the challenge of stability analysis posed by the
singularities, we prove a set of inequalities by using the Cauchy-Schwarz
inequality, the 2-D Fourier series, and the Parseval's theorem. A numerical
simulation illustrates the effectiveness of the proposed delay-compensation
method.Comment: 11 pages, 3 figures(including 8 sub-figures
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Challenges in Optimization with Complex PDE-Systems (hybrid meeting)
The workshop concentrated on various aspects of optimization problems with systems of nonlinear partial differential equations (PDEs) or variational inequalities (VIs) as constraints. In particular, discussions around several keynote presentations in the areas of optimal control of nonlinear or non-smooth systems, optimization problems with functional and discrete or switching variables leading to mixed integer nonlinear PDE constrained optimization, shape and topology optimization, feedback control and stabilization, multi-criteria problems and multiple optimization problems with equilibrium constraints as well as versions of these problems under uncertainty or stochastic influences, and the respectively associated numerical analysis as well as design and analysis of solution algorithms were promoted. Moreover, aspects of optimal control of data-driven PDE constraints (e.g. related to machine learning) were addressed
Gain Scheduling with a Neural Operator for a Transport PDE with Nonlinear Recirculation
To stabilize PDE models, control laws require space-dependent functional
gains mapped by nonlinear operators from the PDE functional coefficients. When
a PDE is nonlinear and its "pseudo-coefficient" functions are state-dependent,
a gain-scheduling (GS) nonlinear design is the simplest approach to the design
of nonlinear feedback. The GS version of PDE backstepping employs gains
obtained by solving a PDE at each value of the state. Performing such PDE
computations in real time may be prohibitive. The recently introduced neural
operators (NO) can be trained to produce the gain functions, rapidly in real
time, for each state value, without requiring a PDE solution. In this paper we
introduce NOs for GS-PDE backstepping. GS controllers act on the premise that
the state change is slow and, as a result, guarantee only local stability, even
for ODEs. We establish local stabilization of hyperbolic PDEs with nonlinear
recirculation using both a "full-kernel" approach and the "gain-only" approach
to gain operator approximation. Numerical simulations illustrate stabilization
and demonstrate speedup by three orders of magnitude over traditional PDE
gain-scheduling. Code (Github) for the numerical implementation is published to
enable exploration.Comment: 16 pages, 5 figure
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Numerical Techniques for Optimization Problems with PDE Constraints
The development, analysis and implementation of efficient and robust numerical techniques for optimization problems associated with partial differential equations (PDEs) is of utmost importance for the optimal control of processes and the optimal design of structures and systems in modern technology. The successful realization of such techniques invokes a wide variety of challenging mathematical tasks and thus requires the application of adequate methodologies from various mathematical disciplines. During recent years, significant progress has been made in PDE constrained optimization both concerning optimization in function space according to the paradigm ’Optimize first, then discretize’ and with regard to the fast and reliable solution of the large-scale problems that typically arise from discretizations of the optimality conditions. The contributions at this Oberwolfach workshop impressively reflected the progress made in the field. In particular, new insights have been gained in the analysis of optimal control problems for PDEs that have led to vastly improved numerical solution methods. Likewise, breakthroughs have been made in the optimal design of structures and systems, for instance, by the socalled ’all-at-once’ approach featuring simultaneous optimization and solution of the underlying PDEs. Finally, new methodologies have been developed for the design of innovative materials and the identification of parameters in multi-scale physical and physiological processes
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