The linear response of a dynamical system refers to changes to properties ofthe system when small external perturbations are applied. We consider thelittle-studied question of selecting an optimal perturbation so as to (i)maximise the linear response of the equilibrium distribution of the system,(ii) maximise the linear response of the expectation of a specified observable,and (iii) maximise the linear response of the rate of convergence of the systemto the equilibrium distribution. We also consider the inhomogeneous ortime-dependent situation where the governing dynamics is not stationary and onewishes to select a sequence of small perturbations so as to maximise theoverall linear response at some terminal time. We develop the theory forfinite-state Markov chains, provide explicit solutions for some illustrativeexamples, and numerically apply our theory to stochastically perturbeddynamical systems, where the Markov chain is replaced by a matrixrepresentation of an approximate annealed transfer operator for the randomdynamical system.
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