Abstract
We study a system of infinitely many Riccati equations that arise from a cumulant control problem, which is a generalization of regulator problems, risk‐sensitive controls, minimal cost variance controls, and k‐cumulant controls. We obtain estimates for the existence intervals of solutions of the system. In particular, new existence conditions are derived for solutions on the horizon of the cumulant control problem.
Highlights
IntroductionConsider a linear control system and a quadratic cost function: dx Ax Bu dt Gdw, t ∈ t0, tf ; x t0 x0, tf
We study a system of infinitely many Riccati equations that arise from a cumulant control problem, which is a generalization of regulator problems, risk-sensitive controls, minimal cost variance controls, and k-cumulant controls
Theorem 3.4 below shows that the cumulant control problem is well posed for any sequence μ with a small ρ μ
Summary
Consider a linear control system and a quadratic cost function: dx Ax Bu dt Gdw, t ∈ t0, tf ; x t0 x0, tf. The cumulant control problem, considered in 1 , is to find a control u that minimizes the combined cumulant κ defined in 1.3 This problem leads to the following system of infinitely many equations of Riccati type: H1 A BK T H1 H1 A BK Q KT RK 0, H1 tf Qf , i−1. In 9 the norm of a solution of a coupled matrix Riccati equation was shown to satisfy a differential inequality similar to 1.11. Estimates for maximal existence interval of a classical Riccati equation had been obtained in 10 in terms of upper and lower solutions. For the coupled Riccati equation associated with the minimal cost variance control, some implicit sufficient conditions had been given in 11 for the existence of a solution. We use the trace tr H to bound the solution of system 1.4 and 1.5 , which generally leads to a better estimate for the existence interval
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