Risk-sensitive optimal control for MJLS has no closed-form Riccati solution and is not equivalent to H-infinity control

Statement

For continuous-time Markov jump linear systems (MJLS) with linear regime-dependent dynamics, quadratic regime-dependent costs, and Gaussian process noise, the risk-sensitive (LEQG) optimal control problem does not admit a closed-form solution as a coupled set of modified Riccati equations, and risk-sensitive control is not equivalent to H∞ minimax control. This is in sharp contrast to the jump-free LQG/LEQG/H∞ setting, where the three problems are linked by the Jacobson 1973 exponential transformation and a known H∞ duality (Glover-Doyle, Whittle, Başar-Bernhard).

A direct consequence: there is no large-deviation limit and no risk-neutral limit of the would-be risk-sensitive solution, because no closed-form solution exists to take limits of.

Evidence summary

Moon and Başar (2016) provide the structural derivation:

1. HJB non-closure. Apply dynamic programming to the risk-sensitive criterion J = (2/γ) log E[exp((γ/2) ∫ ℓ ds)] conditional on (x_t, θ_t). Substitute the quadratic-in-state ansatz V(x, θ) = x' P(θ) x + q(θ). After applying the exponential transformation that works in jump-free LEQG, the resulting equation contains a residual term Σ_{θ' ≠ θ} q_{θθ'} [exp((γ/2)(x' P(θ') x − x' P(θ) x)) − 1] which is not a quadratic in x for any choice of P(·). The ansatz fails to close the recursion, so no modified Riccati equation indexed by regime can solve the problem.

2. H∞ structural mismatch. Independently derive the MJLS H∞ minimax controller via a switching dissipation inequality. The resulting coupled Riccati system has a sign structure and coupling pattern that does not match any candidate risk-sensitive Riccati system. No parameter substitution recovers Jacobson’s equivalence.

3. Limit corollary. Because no closed-form risk-sensitive solution exists, neither the small-γ (risk-neutral LQG) limit nor the large-γ (large-deviation) limit of the standard LEQG analysis can be carried out for MJLS.

The evidence is strong but single-source: it is a structural derivation, not an empirical claim, but it has not yet been independently re-derived in this wiki by other papers. Status is set to weakly_supported (not supported) to reflect single-source evidence; confidence is 0.75 because the structural argument is internally clean and the negative result is consistent with the broader pattern that exponential-transformation tricks rarely survive regime switching.

Conditions and scope

• Setting: continuous-time MJLS with finite-state Markov chain, regime-dependent linear (A, B, G) matrices, regime-dependent quadratic cost weights (M, R), Gaussian Brownian-motion driving noise.
• Risk criterion: exponential-of-integral cost with positive risk parameter γ > 0.
• Out of scope: discrete-time MJLS (structurally similar but not directly proven), non-quadratic costs, nonlinear dynamics, partial observation.
• What is not ruled out: numerical HJB solvers, approximate Riccati schemes with explicit suboptimality bounds, special parameter regimes (slow regime switching, weak coupling) where approximate closure may hold, alternative robust formulations (distributional robustness, ambiguity-set methods).

Counter-evidence

None recorded in this wiki yet. Open questions: does any non-quadratic value-function ansatz close the MJLS-LEQG HJB? Are there special regime structures (e.g. block-diagonal generators) that admit closure?

Linked ideas

None yet.

Open questions

• Tractable approximate risk-sensitive MJLS controllers with provable suboptimality bounds.
• Discrete-time MJLS analog of the impossibility result.
• Slow-switching / weak-coupling asymptotic regimes where Jacobson equivalence approximately holds.
• Implications for robust filtering / robust state estimation in MJLS observation models — does the same obstruction apply to risk-sensitive MJLS Kalman filtering?