We study a non-standard infinite horizon, infinite dimensional linear–quadratic control problem arising in the physics of non-stationary states (see e.g. Bertini et al. (2004, 2005)): finding the minimum energy to drive a given stationary state $\bar{x}=0$ (at time $t=-\infty$) into an arbitrary non-stationary state $x$ (at time $t=0$). This is the opposite to what is commonly studied in the literature on null controllability (where one drives a generic state $x$ into the equilibrium state $\bar{x}=0$). Consequently, the Algebraic Riccati Equation (ARE) associated with this problem is non-standard since the sign of the linear part is opposite to the usual one and since its solution is intrinsically unbounded. Hence the standard theory of AREs does not apply. The analogous finite horizon problem has been studied in the companion paper (Acquistapace and Gozzi, 2017). Here, similarly to such paper, we prove that the linear selfadjoint operator associated with the value function is a solution of the above mentioned ARE. Moreover, differently to Acquistapace and Gozzi (2017), we prove that such solution is the maximal one. The first main result (Theorem 5.8) is proved by approximating the problem with suitable auxiliary finite horizon problems (which are different from the one studied in Acquistapace and Gozzi (2017)). Finally in the special case where the involved operators commute we characterize all solutions of the ARE (Theorem 6.5) and we apply this to the Landau–Ginzburg model.

我们研究了非平稳状态物理学中出现的非标准无限视界、无限维线性二次控制问题（参见例如 Bertini 等人 (2004, 2005)）：找到驱动给定静止状态的最小能量 $\bar{X}=0$ （当时 $吨=-\infty$) 进入任意的非平稳状态 $X$ （当时 $吨=0$）。这与关于零可控性的文献中通常研究的相反（其中一个驱动通用状态$X$ 进入平衡状态 $\bar{X}=0$）。因此，与此问题相关的代数 Riccati 方程 (ARE) 是非标准的，因为线性部分的符号与通常的符号相反，并且其解本质上是无界的。因此，ARE 的标准理论不适用。在配套论文（Acquistapace 和 Gozzi，2017 年）中研究了类似的有限视界问题。在这里，与此类论文类似，我们证明与值函数相关的线性自伴随算子是上述 ARE 的解。此外，与 Acquistapace 和 Gozzi (2017) 不同的是，我们证明了这样的解是最大的。第一个主要结果（定理 5.8）是通过用合适的辅助有限视界问题（不同于 Acquistapace 和 Gozzi（2017）研究的问题）来逼近问题来证明的。