Exact Analytic Solutions for Optimal Control Problems Under Multiplicative Noise

Abstract Control-dependent (multiplicative) noise makes it difficult to achieve optimal control because large control signals amplify noise. This paper considers a minimal (one-dimensional) system that includes multiplicative noise and solves the optimal control problem for arbitrary cost functions. In a limit when the control-cost approaches zero, this formulation becomes analytically solvable. The analysis reveals several important properties that are absent in traditional LQ anlysis. In particular, multiplicative noise makes optimal solutions depend on the global features of cost functions.

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