Optimal control with probabilistic constraints

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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Photo by Stefan Widua on Unsplash

Photo by Stefan Widua on Unsplash

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.

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