Infinite-Horizon LQR for Markov Jump Linear Systems: Application to Optimal Management of Correction Campaigns in Social Networks

We formulate and solve the infinite-horizon Linear Quadratic Regulator (LQR) problem for Markov Jump Linear Systems (MJLS) and apply it to the optimal suppression of a harmful rumour spreading across social-media platforms. The rumour dynamics are modelled by the Maki--Thompson (MT) equations, linearised around the fully-unexposed equilibrium $(X{=}1,\,Y{=}0)$, which is shared by all modes and yields a state $x\!=\!(X\!-\!1,\,Y)'$ that is continuous under platform switching. At this operating point the open-loop linearised spreading rate is $\mu_2^{(i)}\!=\!+\lambda_i\!>\!0$ in every mode, making the system mean-square (MS) \textit{unstable} (spectral abscissa $+3.1097$): without intervention the rumour spreads explosively under any switching sequence. The MJLS has three modes (X/Twitter, WhatsApp, Facebook/Meta) with Markovian migration. Mode-wise classical stabilisability of each pair $(A_i,B_i)$ is verified via the Hautus rank condition; the Sequential Riccati Iteration (SRI) confirms existence of the Coupled Algebraic Riccati Equation (CARE) solution in $39$ iterations ($\delta\!<\!10^{-12}$). The infinite-horizon LQR for MJLS (JLQR) moves the spectral abscissa from $+3.1097$ to $-1.4625$ (dominant time constant $\tau\!=\!0.684\,\mathrm{h}$), guaranteeing
$\mathbb{E}[\|x(t)\|^2]\!\to\!0$ under any switching sequence. For $x(0)\!=\!(-0.30,0.30)'$ ($X(0)\!=\!0.70$, $Y(0)\!=\!0.30$, $Z(0)\!=\!0$: 30\% actively spreading, no stiflers, X/Twitter active) the optimal cost is $J^*\!=\!0.7808$, validated by Monte Carlo
($M\!=\!3{,}000$, relative error $0.21\%$, $J^*$ inside the 95\% CI).