Cost distribution shaping: the relation between Bode integral, entropy, risk-sensitivity, and cost cumulant control

The cost function in stochastic optimal control is viewed as a random variable. Then the classical linear-quadratic-Gaussian control, entropy control, risk-sensitive control, and cost cumulant control can be viewed as the cost distribution shaping methods. We would survey the existing relations between entropy, Bode integral, and risk-sensitive cost function. Furthermore, we would relate the cost cumulants with information theoretic entropy, and Bode integral. The interpretation of cost cumulant control is given in terms of the control entropy minimization. The paper also relates information theoretic entropy with exponential-of-integral cost function using a Lagrange multiplier and calculus of variations. Finally, the logarithmic-exponential-of-integral cost function is related to the information theoretic entropy using large deviation theory.