A Riccati Approach to Mixed $H_2/H_\infty$ Closed-Loop Games for Infinite-Dimensional Stochastic Systems

Abstract

This paper studies a finite-horizon mixed $H_2/H_\infty$ feedback Nash game for stochastic evolution equations on a separable Hilbert space. The drift generator is unbounded, the remaining coefficients are bounded, and the one-dimensional Brownian diffusion depends on the state, control, and disturbance. The $H_2$ channel is an LQ state--control energy. For the disturbance channel, the stochastic LQ uniform-convexity characterization yields equivalence between strict induced $L^2$ attenuation and unique strongly regular mild Riccati solvability. Simultaneous bounded-generator approximation of the Lyapunov equation and the state justifies quadratic identities for strongly continuous mild operator solutions. These identities verify both Nash inequalities and full-output strict attenuation from a strongly regular coupled Riccati pair. An invertible feedback block and a contraction argument establish locally unique coupled solutions on a sufficiently short terminal interval for every positive attenuation level. For a specified stochastic heat-equation model at $\gamma=0.09$, a uniform invariant rectangle further proves full-horizon existence, bounded infinite-dimensional feedbacks, and operator-norm spectral convergence. Numerical computations reproduce the projected gains and compare selected best responses. Global coupled solvability for general coefficients remains an explicit hypothesis.