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

Stochastic, Adaptive, and Learning Control · ICM 2022: Qi Lü, Control theory of stochastic distributed parameter systems recent progress and open problems

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Problem statement

Let T>0T>0, let (Ω,F,{Ft}t∈[0,T],P)(\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 WW, and let HH be a separable Hilbert space. Let A:D(A)⊂H→HA:D(A)\subset H\to H generate a C0C_0-semigroup, and write L(H)\mathcal L(H) for the bounded linear operators on HH. Assume

F∈LF1(0,T;L2(Ω;L(H))),PT∈LFT2(Ω;L(H)), 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∈LF4(0,T;L∞(Ω;L(H))). 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

{dP(t)=[−(A∗+J(t)∗)P(t)−P(t)(A+J(t))−K(t)∗P(t)K(t)−(K(t)∗Q(t)+Q(t)K(t))+F(t)]dt+Q(t) dW(t),t∈[0,T),P(T)=PT. \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 P[0,T]\mathcal P[0,T] to consist of adapted operator-valued processes PP such that ∥P(⋅)∥L(H)∈LF∞(0,T;L2(Ω))\|P(\cdot)\|_{\mathcal L(H)}\in L^\infty_{\mathbb F}(0,T;L^2(\Omega)) and, for every t∈[0,T]t\in[0,T] and ξ∈LFt4(Ω;H)\xi\in L^4_{\mathcal F_t}(\Omega;H), the process P(⋅)ξP(\cdot)\xi belongs to the space DF([t,T];L4/3(Ω;H))D_{\mathbb F}([t,T];L^{4/3}(\Omega;H)) of adapted, L4/3(Ω;H)L^{4/3}(\Omega;H)-càdlàg, essentially bounded processes, with a bound

∥P(⋅)ξ∥DF([t,T];L4/3(Ω;H))≤C∥ξ∥LFt4(Ω;H) \|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 CC independent of tt and ξ\xi. Set

Q^[0,T]={Q:[0,T]×Ω→L(H):Q is adapted and ∥Q(⋅)∥L(H)∈LF2(0,T;L2(Ω))}. \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∈[0,T]t\in[0,T], ξj∈LFt4(Ω;H)\xi_j\in L^4_{\mathcal F_t}(\Omega;H), and ψj,ψ~j∈LF2(t,T;L4(Ω;H))\psi_j,\widetilde\psi_j\in L^2_{\mathbb F}(t,T;L^4(\Omega;H)), j=1,2j=1,2, let φj\varphi_j be the mild solution of

{dφj(s)=[(A+J(s))φj(s)+ψj(s)]ds+[K(s)φj(s)+ψ~j(s)]dW(s),s∈(t,T],φj(t)=ξj. \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)∈P[0,T]×Q^[0,T](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,

E⟨PTφ1(T),φ2(T)⟩H−E∫tT⟨F(s)φ1(s),φ2(s)⟩H ds=E⟨P(t)ξ1,ξ2⟩H+E∫tT⟨P(s)ψ1(s),φ2(s)⟩H ds+E∫tT⟨P(s)φ1(s),ψ2(s)⟩H ds+E∫tT⟨P(s)K(s)φ1(s),ψ~2(s)⟩H ds+E∫tT⟨P(s)ψ~1(s),K(s)φ2(s)+ψ~2(s)⟩H ds+E∫tT⟨Q(s)ψ~1(s),φ2(s)⟩H ds+E∫tT⟨Q(s)φ1(s),ψ~2(s)⟩H ds. \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)∈P[0,T]×Q^[0,T](P,Q)\in\mathcal P[0,T]\times\widehat{\mathcal Q}[0,T].

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Question 1

Let T>0T>0, let (Ω,F,{Ft}t∈[0,T],P)(\Omega,\mathcal F,\{\mathcal F_t\}_{t\in[0,T]},\mathbb P) carry a one-dimensional Brownian motion WW, and assume that {Ft}\{\mathcal F_t\} is the usual augmentation of the natural filtration generated by WW. Let HH be a separable infinite-dimensional Hilbert space, let A:D(A)⊂H→HA:D(A)\subset H\to H generate a C0C_0-semigroup, and let L(H)\mathcal L(H) denote the bounded linear operators on HH. Given

F∈LF1(0,T;L2(Ω;L(H))),PT∈LFT2(Ω;L(H)), F\in L^1_{\mathbb F}(0,T;L^2(\Omega;\mathcal L(H))),\qquad P_T\in L^2_{\mathcal F_T}(\Omega;\mathcal L(H)),

and

J,K∈LF4(0,T;L∞(Ω;L(H))), J,K\in L^4_{\mathbb F}(0,T;L^\infty(\Omega;\mathcal L(H))),

consider

{dP(t)=[−(A∗+J(t)∗)P(t)−P(t)(A+J(t))−K(t)∗P(t)K(t)−(K(t)∗Q(t)+Q(t)K(t))+F(t)]dt+Q(t) dW(t),t∈[0,T),P(T)=PT. \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 P[0,T]\mathcal P[0,T] as the set of adapted operator-valued processes PP such that ∥P(⋅)∥L(H)∈LF∞(0,T;L2(Ω))\|P(\cdot)\|_{\mathcal L(H)}\in L^\infty_{\mathbb F}(0,T;L^2(\Omega)) and, for every t∈[0,T]t\in[0,T] and ξ∈LFt4(Ω;H)\xi\in L^4_{\mathcal F_t}(\Omega;H), the process P(⋅)ξP(\cdot)\xi is adapted, L4/3(Ω;H)L^{4/3}(\Omega;H)-càdlàg, essentially bounded on [t,T][t,T], and satisfies

∥P(⋅)ξ∥L∞(t,T;L4/3(Ω;H))≤C∥ξ∥LFt4(Ω;H) \|P(\cdot)\xi\|_{L^\infty(t,T;L^{4/3}(\Omega;H))}\le C\|\xi\|_{L^4_{\mathcal F_t}(\Omega;H)}

with CC independent of tt and ξ\xi. Define

Q^[0,T]={Q:[0,T]×Ω→L(H):Q is adapted and ∥Q(⋅)∥L(H)∈LF2(0,T;L2(Ω))}. \widehat{\mathcal Q}[0,T]=\{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))\}.

For t∈[0,T]t\in[0,T], ξj∈LFt4(Ω;H)\xi_j\in L^4_{\mathcal F_t}(\Omega;H), and ψj,ψ~j∈LF2(t,T;L4(Ω;H))\psi_j,\widetilde\psi_j\in L^2_{\mathbb F}(t,T;L^4(\Omega;H)), let φj\varphi_j, j=1,2j=1,2, solve

{dφj(s)=[(A+J(s))φj(s)+ψj(s)]ds+[K(s)φj(s)+ψ~j(s)]dW(s),s∈(t,T],φj(t)=ξj. \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}

Determine whether there exists a pair (P,Q)∈P[0,T]×Q^[0,T](P,Q)\in\mathcal P[0,T]\times\widehat{\mathcal Q}[0,T] such that, for every such choice of test data,

E⟨PTφ1(T),φ2(T)⟩H−E∫tT⟨F(s)φ1(s),φ2(s)⟩H ds=E⟨P(t)ξ1,ξ2⟩H+E∫tT⟨P(s)ψ1(s),φ2(s)⟩H ds+E∫tT⟨P(s)φ1(s),ψ2(s)⟩H ds+E∫tT⟨P(s)K(s)φ1(s),ψ~2(s)⟩H ds+E∫tT⟨P(s)ψ~1(s),K(s)φ2(s)+ψ~2(s)⟩H ds+E∫tT⟨Q(s)ψ~1(s),φ2(s)⟩H ds+E∫tT⟨Q(s)φ1(s),ψ~2(s)⟩H ds. \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}

Source reference: Qi Lü, “Control Theory of Stochastic Distributed Parameter Systems: Recent Progress and Open Problems,” in Proceedings of the International Congress of Mathematicians 2022, Vol. 4, EMS Press, 2023, pp. 299–323, Section 3.2 and Open Problem (8), https://ems.press/content/book-chapter-files/33340.

Progress & source assessment

Source-reported assessment from the ICM catalogue snapshot. This is not an independent review by Frontier Science.

The problem is unresolved as of August 1, 2026. The missing part is existence of the operator-valued process QQ with the stated LF2(0,T;L2(Ω;L(H)))L^2_{\mathbb F}(0,T;L^2(\Omega;\mathcal L(H))) regularity for general L(H)\mathcal L(H)-valued terminal datum PTP_T and forcing FF. Uniqueness in the transposition class is known, but the available general existence theorems use weaker relaxed or VV-transposition notions, while genuine transposition solutions are known only under additional regularity hypotheses.

  • Qi Lü and Xu Zhang, General Pontryagin-Type Stochastic Maximum Principle and Backward Stochastic Evolution Equations in Infinite Dimensions, SpringerBriefs in Mathematics, Springer, 2014, DOI: 10.1007/978-3-319-06632-5; arXiv:1204.3275. Theorem 4.1 proves uniqueness of a transposition solution for general L(H)\mathcal L(H)-valued data. Theorem 4.2 proves existence and uniqueness only when PTP_T and FF take values in the Hilbert space of Hilbert–Schmidt operators. For general bounded-operator-valued data, Chapter 6 proves existence and uniqueness only in the weaker relaxed transposition sense and explicitly states that existence of a genuine transposition solution is not proved.

  • Qi Lü and Xu Zhang, “Transposition Method for Backward Stochastic Evolution Equations Revisited, and Its Application,” Mathematical Control and Related Fields 5(3) (2015), 529–555, DOI: 10.3934/mcrf.2015.5.529; arXiv:1405.4454. Theorem 3.1 again gives at most one genuine transposition solution for general operator-valued data, and Theorem 3.2 gives well-posedness for Hilbert–Schmidt-valued terminal data and forcing. The paper explicitly says that well-posedness in its genuine transposition sense remains open.

  • Qi Lü and Xu Zhang, “Operator-Valued Backward Stochastic Lyapunov Equations in Infinite Dimensions, and Its Application,” Mathematical Control and Related Fields 8(1) (2018), 337–381, DOI: 10.3934/mcrf.2018014. This paper establishes well-posedness in the VV-transposition and relaxed transposition senses. These notions provide useful substitutes for optimal-control applications but do not produce the pair (P,Q)(P,Q) in the exact spaces and duality formulation required by the problem.

  • Qi Lü, “Control Theory of Stochastic Distributed Parameter Systems: Recent Progress and Open Problems,” in Proceedings of the International Congress of Mathematicians 2022, Vol. 4, EMS Press, 2023, pp. 299–323, direct URL: https://ems.press/content/book-chapter-files/33340. Open problem (8) asks for unique well-posedness of this equation in the genuine transposition sense and notes that only a very special case was known.

  • Guangdong Jing, “Properties for Transposition Solutions to Operator-Valued BSEEs, and Applications to Robust Second Order Necessary Conditions for Controlled SEEs,” Journal of Mathematical Analysis and Applications 549(1) (2025), Article 129445, DOI: 10.1016/j.jmaa.2025.129445; arXiv:2403.15710. This recent follow-up continues to use relaxed transposition and VV-transposition solutions for the second-order adjoint equation. It develops their regularity and optimal-control applications but does not establish existence of the genuine L(H)\mathcal L(H)-valued transposition solution required here.

  • Liangying Chen and Wilhelm Stannat, “Transposition Approach to Optimal Control of McKean–Vlasov SPDEs,” preprint, 2026, arXiv:2603.06245. Section 4 treats the operator-valued second-order adjoint equation by the relaxed transposition framework and invokes its established relaxed well-posedness. Thus even this June 2026 application does not supply the genuine operator-valued process QQ required in the present problem.

Thus the strongest established results are uniqueness for the exact solution concept, existence and uniqueness for Hilbert–Schmidt-valued data, and general well-posedness only for weaker substitute notions. The general existence assertion in the statement remains open.

References
  1. [1]ICM Conjectures · 2022-ctrl-22 link
  2. [2]ICM 2022: Qi Lü, Control theory of stochastic distributed parameter systems recent progress and open problems link

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