发表机构
Huzhou Normal University; Shandong University of Science and Technology（湖州师范学院; 山东科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对无限维随机系统提出 Riccati 方法，解决混合 $H_2/H_\infty$ 闭环 Nash 博弈，通过强正则耦合 Riccati 对实现严格衰减，并证明局部及全时域解的存在性。
AI 中文摘要
本文研究可分离 Hilbert 空间上随机发展方程的有限时域混合 $H_2/H_\infty$ 反馈 Nash 博弈。漂移生成元无界，其余系数有界，一维 Brown 扩散依赖于状态、控制和扰动。$H_2$ 通道是 LQ 状态-控制能量。对于扰动通道，随机 LQ 一致凸性刻画给出了严格诱导 $L^2$ 衰减与唯一强正则 mild Riccati 可解性之间的等价性。Lyapunov 方程和状态的同时有界生成元逼近证明了强连续 mild 算子解的二次恒等式。这些恒等式从强正则耦合 Riccati 对验证了 Nash 不等式和全输出严格衰减。可逆反馈块和压缩论证在足够短的终端区间上为每个正衰减水平建立了局部唯一耦合解。对于指定的随机热方程模型，在 $\gamma=0.09$ 时，一致不变矩形进一步证明了全时域存在性、有界无限维反馈和算子范数谱收敛。数值计算重现了投影增益并比较了选定的最优响应。一般系数的全局耦合可解性仍是一个明确的假设。
英文摘要
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 $γ=0.09$, 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.
