What is it about?
In optimal control problems, often the precise problem data are uncertain. In order to deal with the uncertainty, probabilistic constraints make sense, where a probability for the allowed violation of inequality constraints is specified. In this paper we consider the control of a vibrating string with control action through Neumann boundary conditions. The aim is to find a control with minimal L^2-norm for which the probability that the energy of the string is below an a priori defined bound is larger than a given probability p. The uncertain initial conditions ar modelled as random Fourier series.
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Why is it important?
Uncertainty quantification is an important area where often, the uncertainty is included in some way as a penalty term in the objective function. This approach does not allow to specify precisely a priori the required probability that the constrain is satisfied. The use of probabilistic constraints allows to include the desired probability threshold as a problem parameter. For the optimization, the derivatives of the probability with respect to the control are used. Based upon a gradient formula, the evaluation uses a sperical radial decomposition approach which saves one dimension in the evalutaion of the corresponding integral.
Perspectives
This has been a fruit ful collaboration with my colleagues from the Weierstrass institute (WIAS) in Berlin that I hope to continue in the future. The cooperation took place in the framework of the TRR 154.
Martin Gugat
Friedrich-Alexander-Universitat Erlangen-Nurnberg
Read the Original
This page is a summary of: Optimal Neumann Boundary Control of a Vibrating String with Uncertain Initial Data and Probabilistic Terminal Constraints, SIAM Journal on Control and Optimization, January 2020, Society for Industrial & Applied Mathematics (SIAM),
DOI: 10.1137/19m1269944.
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