Risk-adjusted feedback control under uncertainty
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A control in feedback form is derived for linear quadratic, time-invariant optimal control problems subject to parabolic partial differential equations with coefficients depending on a countably infinite number of parameters. It is shown that the parametric feedback operator, which can be obtained by solving an associated operator-valued Riccati equation, depends analytically on the parameters, provided that the system operator depends analytically on the parameters. These novel parametric regularity results allow the application of quasi-Monte Carlo methods to efficiently compute an a priori chosen feedback law based on the expected value. Moreover, under moderate assumptions on the input parameters, quasi-Monte Carlo methods achieve superior convergence rates compared to ordinary Monte Carlo methods, independently of the dimension of the input parameters. Extensions toward risk-averse feedback design based on the entropic risk measure are outlined.