Indefinite risk-sensitive control

Robust risk‐sensitive control

“…and thus the asymptotic relation (39) holds with h 1 = 0. As A v is also negative, the proof that the asymptotic relation ( 40) also holds with h 2 = 0 proceeds very similarly to the above.…”

“…Proof. We only derive condition (39) as the derivation of condition (40) proceeds similarly. Under the probability measure P u it holds that x(t) ∼ N(𝜇 u (t), Σ u (t)).…”

“…Taking the limit as T → ∞ of logarithm of this expression divided by 𝛾 1 f 1 (T), gives (39). However, this is just the stability condition (34).…”

“…For risk-sensitive control with partial observations see, for example, References 14-16, for discrete-time systems see, for example, References 17,18, for connections with robust control see, for example, References 16,19-21, for the risk-sensitive maximum principle see, for example, References 22-24, for the risk-sensitive control of mean-filed systems see, for example, References 24-26 and 27, for the Hamilton-Jacobi-Bellman equation of risk-sensitive control see Reference 28, for the risk-sensitive differential games see, for example, References 29-35, and for more general exponential criteria that admit explicit closed-form solutions see References 36-39. The risk-sensitive control is particularly suitable for optimal investment problems, see, for example, References 36,[38][39][40][41][42][43] In this paper, we generalize the linear state-feedback stochastic mixed H 2 ∕H ∞ control problem by replacing the quadratic criteria (2) and (3) with the following two exponential-quadratic criteria:…”

“…The risk‐sensitive control problem for linear stochastic systems with additive noise was introduced by Jacobson 13 who found an explicit closed‐form solution in a linear state‐feedback in the case of full observations. For risk‐sensitive control with partial observations see, for example, References 14–16, for discrete‐time systems see, for example, References 17,18, for connections with robust control see, for example, References 16,19–21, for the risk‐sensitive maximum principle see, for example, References 22–24, for the risk‐sensitive control of mean‐filed systems see, for example, References 24–26 and 27, for the Hamilton‐Jacobi‐Bellman equation of risk‐sensitive control see Reference 28, for the risk‐sensitive differential games see, for example, References 29–35, and for more general exponential criteria that admit explicit closed‐form solutions see References 36–39. The risk‐sensitive control is particularly suitable for optimal investment problems, see, for example, References 36,38–43.…”