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on Qi Lü, Control theory of stochastic distributed parameter systems recent progress and open problems · 2022-ctrl-22

The statement

Let $T>0$ , let $(\Omega,\mathcal F,\{\mathcal F_t\}_{t\in[0,T]},\mathbb P)$ be a filtered probability space satisfying the usual conditions and carrying a one-dimensional Brownian motion $W$ , and let $H$ be a separable Hilbert space. Let $A:D(A)\subset H\to H$ generate a $C_0$ -semigroup, and write $\mathcal L(H)$ for the bounded linear operators on $H$ . Assume

F\in L^1_{\mathbb F}\bigl(0,T;L^2(\Omega;\mathcal L(H))\bigr),\qquad P_T\in L^2_{\mathcal F_T}(\Omega;\mathcal L(H)),

and

J,K\in L^4_{\mathbb F}\bigl(0,T;L^\infty(\Omega;\mathcal L(H))\bigr).

Consider the operator-valued backward stochastic evolution equation

\begin{cases} dP(t)=\bigl[-(A^*+J(t)^*)P(t)-P(t)(A+J(t))-K(t)^*P(t)K(t)\\ \hspace{5.8em}{}-(K(t)^*Q(t)+Q(t)K(t))+F(t)\bigr]dt+Q(t)\,dW(t),&t\in[0,T),\\ P(T)=P_T. \end{cases}

Define $\mathcal P[0,T]$ to consist of adapted operator-valued processes $P$ such that $\|P(\cdot)\|_{\mathcal L(H)}\in L^\infty_{\mathbb F}(0,T;L^2(\Omega))$ and, for every $t\in[0,T]$ and $\xi\in L^4_{\mathcal F_t}(\Omega;H)$ , the process $P(\cdot)\xi$ belongs to the space $D_{\mathbb F}([t,T];L^{4/3}(\Omega;H))$ of adapted, $L^{4/3}(\Omega;H)$ -càdlàg, essentially bounded processes, with a bound

\|P(\cdot)\xi\|_{D_{\mathbb F}([t,T];L^{4/3}(\Omega;H))} \le C\|\xi\|_{L^4_{\mathcal F_t}(\Omega;H)}

for a constant $C$ independent of $t$ and $\xi$ . Set

\widehat{\mathcal Q}[0,T] =\left\{Q:[0,T]\times\Omega\to\mathcal L(H):Q\text{ is adapted and }\|Q(\cdot)\|_{\mathcal L(H)}\in L^2_{\mathbb F}(0,T;L^2(\Omega))\right\}.

For $t\in[0,T]$ , $\xi_j\in L^4_{\mathcal F_t}(\Omega;H)$ , and $\psi_j,\widetilde\psi_j\in L^2_{\mathbb F}(t,T;L^4(\Omega;H))$ , $j=1,2$ , let $\varphi_j$ be the mild solution of

\begin{cases} d\varphi_j(s)=\bigl[(A+J(s))\varphi_j(s)+\psi_j(s)\bigr]ds +\bigl[K(s)\varphi_j(s)+\widetilde\psi_j(s)\bigr]dW(s),&s\in(t,T],\\ \varphi_j(t)=\xi_j. \end{cases}

A pair $(P,Q)\in\mathcal P[0,T]\times\widehat{\mathcal Q}[0,T]$ is called a transposition solution if, for every such choice of data,

\begin{aligned} &\mathbb E\langle P_T\varphi_1(T),\varphi_2(T)\rangle_H -\mathbb E\int_t^T\langle F(s)\varphi_1(s),\varphi_2(s)\rangle_H\,ds\\ &=\mathbb E\langle P(t)\xi_1,\xi_2\rangle_H +\mathbb E\int_t^T\langle P(s)\psi_1(s),\varphi_2(s)\rangle_H\,ds +\mathbb E\int_t^T\langle P(s)\varphi_1(s),\psi_2(s)\rangle_H\,ds\\ &\quad+\mathbb E\int_t^T\langle P(s)K(s)\varphi_1(s),\widetilde\psi_2(s)\rangle_H\,ds +\mathbb E\int_t^T\langle P(s)\widetilde\psi_1(s),K(s)\varphi_2(s)+\widetilde\psi_2(s)\rangle_H\,ds\\ &\quad+\mathbb E\int_t^T\langle Q(s)\widetilde\psi_1(s),\varphi_2(s)\rangle_H\,ds +\mathbb E\int_t^T\langle Q(s)\varphi_1(s),\widetilde\psi_2(s)\rangle_H\,ds. \end{aligned}

Prove that this operator-valued backward stochastic evolution equation admits a unique transposition solution $(P,Q)\in\mathcal P[0,T]\times\widehat{\mathcal Q}[0,T]$ .

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.