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Summary

Nonlinear aspects of wave propagation are investigated. Special attention is given to magnetic slabs and tubes, deriving the Benjamin-Ono equation for the slow mode in a slab and the Leibovich-Roberts equation for the slow mode in a tube. Soliton solutions are obtained and illustrated under various solar conditions. The role of Whitham’s equation is explored. Dissipative effects are also added, and shown to lead to the Benjamin-Ono-Burgers equation. Approximate solutions are given and illustrated for solar conditions. The roles of viscous and thermal damping of weakly nonlinear slow waves (sound waves) are also explored, and the effect of gravity is examined. Both standing waves and propagating waves are looked at. Finally, the nonlinear kink mode is presented.

A Lagrange–Newton–SQP method is analyzed for the optimal control of the Burgers equation. Distributed controls are given, which are restricted by pointwise lower and upper bounds. The convergence of the method is proved in appropriate Banach spaces. This proof is based on a weak second-order sufficient optimality condition and the theory of Newton methods for generalized equations in Banach spaces. For the numerical realization a primal-dual active set strategy is applied. Numerical examples are included.

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13 - Nonlinear Aspects

Summary

The SQP method for control constrained optimal control of the Burgers equation

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2 results

13 - Nonlinear Aspects

Summary

The SQP method for control constrained optimal control of the Burgers equation